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\begin{document}










\title[Diagonal cycles and Euler systems II]{Diagonal cycles and Euler systems II: 
\\ The Birch and Swinnerton-Dyer conjecture \\ for Hasse-Weil-Artin $L$-functions}

\author{Henri Darmon and Victor Rotger}

\begin{abstract}
This article establishes
new  cases
 of the Birch and Swinnerton-Dyer conjecture  in analytic rank $0$,
  for   elliptic curves over $\Q$ viewed
 over the fields cut out by certain self-dual
    Artin representations of dimension at most $4$. When the associated $L$-function vanishes (to even order $\ge 2$) at its central point, two canonical 
classes in the corresponding  Selmer group are constructed  and shown to be linearly 
independent assuming the non-vanishing of a Garrett-Hida $p$-adic $L$-function at a point lying outside its range of classical interpolation. The key tool
for both results is the study of certain
  $p$-adic families of global  Galois cohomology classes arising
from 
 Gross-Kudla-Schoen  diagonal  cycles   in a tower of
triple products of  
modular curves.  
  \end{abstract}

\address{H. D.: Department of Mathematics, McGill University, Montr\'eal, Canada}
\email{darmon@math.mcgill.ca}
\address{V. R.: Departament de Matem\`{a}tica Aplicada II, Universitat Polit\`{e}cnica de Catalunya, Barcelona, Spain}
\email{victor.rotger@upc.edu}


%\subjclass{11G18, 14G35}


\maketitle
\tableofcontents


\section*{Introduction}

 Let $E$ be an elliptic curve   over $\Q$ and let
\begin{equation*}
\varrho:  G_\Q= \Gal(\bar\Q/\Q) \lra  \Aut_L(V_\varrho) \simeq \GL_n(L)
\end{equation*}
be an Artin representation with coefficients in a finite extension $L\subset \bar\Q \subset \C$ of $\Q$, factoring though the Galois group of a finite extension $H/\Q$.
 The Hasse-Weil-Artin $L$-series $L(E,\varrho,s)$  of $E$ twisted by $\varrho$,  defined on the 
right half-plane $\mathrm{Re}(s)>3/2$  
by an absolutely convergent Euler product of degree $2n$,  is expected to admit an analytic continuation to the whole complex plane. 
%and to satisfy a functional equation relating its values at  $s$ and $2-s$. 

 Let $E(H)^\varrho := \Hom_{G_\Q}(V_\varrho,E(H)\otimes L)$ denote the $\varrho$-{\em isotypical component} of the Mordell-Weil group  $E(H)$, and define the  analytic and algebraic rank of  the twist of $E$ by $\varrho$ as
$$
r_{\mathrm{an}}(E,\varrho) = \ord_{s=1} L(E,\varrho,s), \qquad   \quad r(E,\varrho)= \dim_{L} E( H)^{\varrho}.
$$
A Galois-equivariant refinement of the  Birch and Swinnerton-Dyer 
conjecture predicts that
\begin{equation}
 \label{eqn:bsd-gen}
 r_{\mathrm{an}}(E,\varrho)  \stackrel{?}{=} r(E,\varrho).
 \end{equation}



%The implication $L(E,\varrho,1) \ne 0 \Rightarrow E(H)^\varrho = 0$ has been proved 
%\begin{enumerate}
%\item[(i)]  when $n=1$, i.e., when $\varrho$ is a Dirichlet character, by the work of Kato \cite{kato};
%\item[(ii)] in many cases where $\varrho$ is  induced from a {\em ring class character} of 
%an imaginary quadratic field, by a series of works building 
%on the breakthroughs of Gross-Zagier \cite{GZ} and Kolyvagin \cite{Ko};
% \item[(iii)] in certain settings where $\varrho$ is induced from a character of order $\le 2$ of a 
 %totally real field, by extending the Gross-Zagier-Kolyvagin strategy to
%such fields,  in which
 %the key ingredients---namely, CM points on Shimura curves---continue to be available.
 %(See \cite{Zh} and also
 %\cite{Ne3} for a statement of some of the most definitive results in these last two directions.)
%\end{enumerate}


The main object of this article is to study conjecture
 \eqref{eqn:bsd-gen} when  $\varrho=\varrho_1\otimes\varrho_2$, where 
$$\varrho_1,\varrho_2: G_\Q \lra \GL_2(L)$$ 
are odd,  irreducible  two-dimensional Artin representations 
 satisfying  $\det(\varrho_1) =\det(\varrho_2)^{-1}$. This last condition implies
 that $\varrho$ has real 
 traces, i.e.,  is isomorphic to its  dual, or contragredient, representation. 
 
Thanks to the major advances  achieved in the last two decades,
the
 elliptic curve $E$ and the Artin representations $\varrho_1$ and $\varrho_2$
 are  known to be 
 associated to cuspidal newforms 
 $$ f\in S_2(N_f), \qquad  g\in S_1(N_g,\chi), \qquad  h\in S_1(N_h,\chi^{-1})$$
of weight $2$, $1$, $1$, respectively. The nebentype character $\chi=\det(\varrho_1)$ of $g$ is an odd  Dirichlet character whose 
 conductor  divides both   $N_g$ and $N_h$.

The $L$-function  $L(E,\varrho,s)$ can 
  be identified 
 with the    triple-product convolution
  $L(f,g,h,s)$, whose analytic continuation and functional equation 
  follow from
  the work of Garrett, Piatetski-Shapiro and Rallis. In particular, the analytic rank $r_{\rm an}(E,\varrho)$ is 
  defined unconditionally, suggesting that   \eqref{eqn:bsd-gen} 
  might repay closer scrutiny in this setting. 

It is assumed throughout this work
  that the level $N_f$ of $f$ is relatively prime  
 to $N_gN_h$.  This implies  (cf.~\cite[thm.~1.4]{prasad})
 that all the local signs  in the functional equation 
 of $L(f,g,h,s)= L(E,\varrho,s)$ are $+1$, 
 hence that the same is true for the global sign, implying that  $r_{\mathrm{an}}(E,\varrho)$ is {\em  even}.
 Subject to  these hypotheses, the main results of this article are Theorems A and B below. (Cf.\,Theorem \ref{theoremA} and Theorem \ref{theoremB} for somewhat more general statements.)
 
\medskip\noindent
 {\bf Theorem A}.   {\em
If $L(E,\varrho,1)\ne 0$ then $E(H)^{\varrho} = 0$.
}
 
 
\medskip
\noindent
 Theorem A has broad implications for the arithmetic of $E$ over ring
 class fields of  quadratic fields, both imaginary and {\em real}.
% This is because, when   $\varrho_1$ and $\varrho_2$ are induced from finite order    characters of the same quadratic field $K$ with inverse central characters, their tensor  product decomposes as a  sum of  representations   induced from  {\em ring class characters} of $K$.
For any ring class   character $\psi$ of a quadratic field $K$ (of conductor relatively prime to $N_f=:N_E$),
 let $H/K$ denote the ring class field    cut out by  it, and  define 
$$
E(H)^{\psi} = \{ P\in E(H)\otimes L \mbox{ such that }  P^\sigma = \psi(\sigma) P \,\mbox{ for all }\,\sigma\in \Gal(H/K) \}.
$$
When $K$ is a real quadratic field, assume further that the pair $(E,K)$ satisfies the {\em non-vanishing
hypothesis} of Definition \ref{def:non-vanishing}. This  mild hypothesis
is expected to always hold, and
 is satisfied, for example, as soon as  $E$ has   quadratic twists $E'$  of both 
possible signs for which $L(E'/K,1)\ne 0$.

\medskip\noindent
 {\bf Corollary A1}. {\em If $L(E/K,\psi,1)\ne 0$ then $E(H)^{\psi}  = 0$.}

\medskip 
\noindent
When $K$ is imaginary quadratic, theorems of this sort
 were already established by a variety of approaches 
 (cf.~\cite{Ko}, \cite[Th.B]{BD-rigid-gz}, \cite[Thm.1.2.]{BD-jochnowitz}, \cite[Cor.4]{BD-MC} and 
 \cite[Th.A']{Ne3}, for example) 
  using Heegner points on modular or Shimura curves, combined with 
  $p$-adic techniques and/or
  congruences between modular forms.
Their common feature is that they  ultimately  rely on Kolyvagin's Euler system of CM points and on some
variant (either classical, or $p$-adic) of the Gross-Zagier formula \cite{GZ}. 
 The  approach followed in this article
 makes no use of CM points, which accounts for why
 it extends unconditionally to more general settings, including the case where $K$ is a real quadratic field. 
 
 

The  intimate connection between Corollary A1
 for $K$  real quadratic
and   the theory of {\em Stark-Heegner points} 
initiated in \cite{darmon_hpxh}  is suggested by 
  the articles  \cite{BDD} and \cite{LRV},   which  give
a  {\em conditional }  proof of Corollary A1 resting on the 
 algebraicity of
 Stark-Heegner points, essentially by
 replacing     CM points  with their conjectural real
 quadratic counterparts in  the proof of 
  \cite[Thm.1.2]{BD-jochnowitz}. 
Stark-Heegner points are still poorly understood, but
the  hope that the structures 
developed in 
the proof of Corollary A1 might  shed light on their algebraicity 
was an important motivation  for the present work. 

\vspace{0.1cm}

Specialising Theorem A to the case where the representation $\varrho=\varrho_1\otimes \varrho_2$ is irreducible establishes the Birch and Swinnerton-Dyer
conjecture in analytic rank zero,  in scenarios lying  beyond the scope  
of both  Heegner and Stark-Heegner points. For instance, in
 the case where the projective representations attached to 
 $\varrho_1$ and $\varrho_2$ cut out the same $A_5$-extension of $\Q$, but where
 $\varrho_1$ and $\varrho_2$ are not twists of each other,   Theorem A 
  leads to the following instance
 of the Birch and Swinnerton-Dyer over 
 quintic fields, which is spelled out in Theorem~\ref{thm:mainA5}.
 
 
\medskip\noindent
{\bf Corollary A2}. {\em Let $E$ be an elliptic curve over $\Q$, 
 let $K$ be a non-totally real quintic field with Galois group $A_5$,  and
assume that the discriminants of $E$ and $K$ are coprime. Then }
 $$\ord_{s=1} L(E/K,s) = \ord_{s=1} L(E/\Q,s) \ \  \Rightarrow \ \ \mbox{rank}\, E(K) = \mbox{rank}\, E(\Q).
 $$

 
\medskip 
\medskip
The second main theorem of this article is concerned with the case where 
$L(E,\varrho,s)$ vanishes at $s=1$ (and hence to order at least $2$).
 Note that $\varrho_1$ and $\varrho_2$ are both
 {\em regular}, i.e.,  there exists $\sigma\in G_\Q$ 
acting on each of these Galois representations  with distinct eigenvalues.

Fix an odd prime $p$ not dividing $N:= {\rm lcm}(N_f, N_g,N_h)$ at which $f$ is not Eisenstein and fix 
throughout an  embedding $\bar\Q \hookrightarrow \bar\Q_p$. 
Let $\Frob_p\in G_{\Q}$ denote the Frobenius element at $p$ induced by this embedding. 
Let $L_p$ denote the completion of $L$ in $\bar\Q_p$.
  Let $(\alpha_g,\beta_g)$ and $(\alpha_h,\beta_h)$ be the pairs of eigenvalues of 
  $\varrho_g(\Frob_p)$ and $\varrho_h(\Frob_p)$, respectively. It shall be assumed  throughout this article that
$$
E \, \mbox{ is ordinary at } p, \quad \mbox{and } \quad  \alpha_g \ne \beta_g, \quad \alpha_h \ne \beta_h. 
$$
Such a prime exists by the Chebotarev density theorem.  


Theorem B below relates the 
$p$-Selmer group attached to $E$ and $\varrho$ 
to the behaviour of the  Garrett-Hida $p$-adic $L$-functions attached to $f$, $g$, and $h$.
 In order to describe these $p$-adic $L$-functions more precisely,
  let $S^{\ord}_{\tilde\Lambda}(M,\psi)$ denote  the space of  ordinary 
     $\Lambda$-adic modular forms of tame level $M\geq 1$ and tame character $\psi$ with coefficients in a finite
     flat extension $\tilde\Lambda$ of  the Iwasawa algebra $\Lambda=\Z_p[[1+p\Z_p]]$.
%,  as described  in Definition  \ref{def:padic-fam} below. 
These spaces are locally free $\tilde\Lambda$-modules of finite rank, and are equipped with a natural 
     $\tilde\Lambda$-linear action of the Hecke operators. 

A Hida family $\hphi$ can be viewed concretely as a formal $q$-series with coefficients in 
     $\tilde\Lambda$ whose specialisations
   give rise to a $p$-adically coherent collection
   $\{\phi_x\}$  of overconvergent ordinary modular forms indexed by the  points  $x$ in 
  the rigid analytic space 
    $\tilde\Omega  := \Hom_{\cts}(\tilde\Lambda,\C_p)$. This space is equipped with a finite flat 
     morphism $\w: \tilde\Omega \lra \Omega$ to the weight space $\Omega := \Hom_{\cts}(\Lambda,\C_p)$, which is also equipped with a natural embedding
     $\Z\subset\Omega$.  When
     $k=\w(x) \in\Z$, the form 
     $\phi_x$ belongs to the space $S_k^{\oc}(M,\psi \omega^{1-k})$  of overconvergent
     modular forms of weight $k$, and  is classical when $k\ge 2$ by \cite[Ch.~7.3, Thm.~3]{Hi3}. When $k=1$, the form $\phi_x$ may be classical or not.

If $\hphi$ is a Hida family of eigenforms of tame level $M$ and tame character $\psi$, let
\begin{equation}
\label{eqn:hf-isotypic}
S^{\ord}_{\tilde\Lambda}(M,\psi)[\hphi] := \left\{ \hvarphi\in S^{\ord}_{\tilde\Lambda}(M,\psi) \quad \mbox{s.t.} \quad 
\left|\begin{array}{ll}
T_\ell \hvarphi = a_\ell(\hphi) \hvarphi,  & \forall \ell\nmid Mp, \\
U_p \hvarphi = a_p(\hphi) \hvarphi, & 
\end{array} \right. \right\}, 
\end{equation}
\begin{equation}
\label{eqn:hf-isotypic2}
S^{\ord}_{\tilde\Lambda}(M,\psi)^\vee[\hphi]=\left\{ \hgamma:  S^{\ord}_{\tilde\Lambda}(M,\psi) \ra \tilde\Lambda \quad \mbox{s.t. } \quad  \left|\begin{array}{ll}
\hgamma(T_\ell^* \hvarphi) = a_\ell(\hphi) \hgamma(\hvarphi),  & \forall \ell\nmid Mp, \\
\hgamma(U_p^* \hvarphi) = a_p(\hphi) \hgamma(\hvarphi) &  \quad  \, \hvarphi\in S^{\ord}_{\tilde\Lambda}(M,\psi) 
\end{array} \right. \right\}
\end{equation}
denote the $\hphi$-isotypic subspaces of $S^{\ord}_{\tilde\Lambda}(M,\psi)$ and of its $\tilde\Lambda$-linear dual, in which the latter is endowed with the adjoint action of the Hecke operators, 
as described in \eqref{eqn:adjoint-hecke} below, 
the adjoints being taken relative to Poincar\'e duality.
%The specialisation of an element $\hgamma\in S_{\tilde\Lambda}(M,\psi)^\vee$
%at a point $x$ with $k=w(x)\in \Z$ 
%is naturally a $\C_p$-linear functional $\gamma_x$ on
%$S_k^{\oc}(M,\psi \omega^{1-k})$.

Let
 $f_\alpha \in S_2(N_fp)$ denote the unique ordinary $p$-stabilisation of $f$, 
 and write  $g_\alpha$,  $g_\beta \in S_1(N_g p,\chi)$ (resp.  $h_\alpha, h_\beta \in S_1(N_hp,\chi^{-1})$) 
 for the pair of ordinary stabilisations of $g$ (resp. $h$).  By  \cite[Ch.~7.3, Thm.~3]{Hi3} combined with
  a recent result of Bella\"iche and Dimitrov \cite{BeDi},
there is a {\em unique} triple  of Hida families, denoted  by $(\hf,\hg,\hh)$, which 
  specialises to $(f_\alpha,g_\alpha,h_\alpha)$
   in weights $(2,1,1)$. 
     These Hida families    are 
   normalised eigenforms
        $$
   \begin{array}{lll}
   \hf \in S^{\ord}_{\Lambda_f}(N_f),  & \quad  \hg \in S^{\ord}_{\Lambda_g}(N_g,\chi), & \quad
    \hh\in S^{\ord}_{\Lambda_h}(N_h,\chi^{-1}),  
  %  \hf^* \in S_{\Lambda_f}(N_f), & \quad   \hg^* \in S_{\Lambda_g}(N_g,\chi^{-1}), &  \quad  \hh^* \in S_{\Lambda_h}(N_h,\chi),
   \end{array}
   $$
with coefficients in suitable finite flat extensions 
   $\Lambda_f$, $\Lambda_g$ and $\Lambda_h$ of $\Lambda$.

The Garrett-Hida $p$-adic $L$-functions   denoted
$$ \Lp^{f_\alpha}(\testhf^*, \testhg, \testhh), \qquad 
\Lp^{g_\alpha}(\testhf, \testhg^*, \testhh), \qquad
\Lp^{h_\alpha}(\testhf, \testhg, \testhh^*)
$$
depend on the choice of a triple
$(\testhf, \testhg, \testhh)$ of 
{\em $\Lambda$-adic test vectors}
\begin{equation}
\label{eqn:test-vectors}
\testhf \in S^{\ord}_{\Lambda_{f}}(N)[\hf], \qquad \testhg \in S^{\ord}_{\Lambda_{g}}(N,\chi)[\hg], \qquad \testhh\in
S^{\ord}_{\Lambda_{h}}(N,\chi^{-1})[\hh], 
\end{equation}
as well as  on a   triple of {\em dual test vectors}
\begin{equation}
\label{eqn:dual-test-vectors}
 \testhf^* \in S^{\ord}_{\Lambda_{f}}(N)^\vee[\hf] \qquad \testhg^* \in S^{\ord}_{\Lambda_{g}}(N,\chi^{-1})^\vee[\hg], \qquad \testhh^*\in
S^{\ord}_{\Lambda_{h}}(N,\chi)^\vee[\hh].
\end{equation}
% (or, equivalently, the dual of the $\hf^*$-isotypic quotient of  $S_{\Lambda_f}(N)$)
% Relative to the natural perfect pairing
% $$ \TT_{\Lambda_{f}}(N) \times S_{\Lambda_{f}}(N) \lra \Lambda_{f}$$
% between the $\Lambda$-adic Hida Hecke algebra of level $N$ and character $\chi$ and the corresponding space of $\Lambda$-adic modular forms, sending $(T,\phi)$ to $a_1(T(\phi))$, 
% the element $\testhf^*$  corresponds to the choice of an element in the 
%   ideal of  the total fraction field of $\TT_{\Lambda_f}$ generated by the idempotent attached to $\hf^*$. 
The   specialisations  $\testf_x$, 
$\testg_y$ and $\testh_z$   at points of weight $k,\ell,m\in\Z$ 
belong to the space  of  overconvergent oldforms of tame level $N$ and   weights 
 attached to $f_x$, 
$g_y$ and $h_z$ respectively, while $\testf_x^*$, $\testg_y^*$ and 
$\testh_z^*$  belong to the $\C_p$-linear duals $S_k^{\oc}(N,\omega^{1-k})^\vee[f_x^*]$,
$S_\ell^{\oc}(N,\chi^{-1}\omega^{1-\ell})^\vee[g_y^*]$ and
$S_m^{\oc}(N,\chi\omega^{1-m})^\vee[h_z^*]$.
 
  The space  
$S_{k}^{\noc}(N,\chi)$  of   nearly overconvergent 
 $p$-adic modular forms   is  described for instance in 
\cite[\S 2.4]{DR1}.  Lemma 2.7 of loc.cit.\,asserts that the image of 
Hida's ordinary projector 
$e_{\ord} := \lim U_p^{n!}$
 on 
$ S_k^{\noc}(N,\chi)$ is contained in the ordinary part of  $ S_k^{\oc}(N,\chi)$.
% Hida's classicality result  that an ordinary overconvergent modular form of weight $k\ge 2$ is  classical  thus implies that $e_{\ord}$ gives rise to  a map  $$ e_{\ord}: S_k^{\noc}(N,\chi) \lra S_k (Np,\chi)$$   whenever $k\in\Z^{\ge 2}$. 
Let
  $d = q \frac{d}{dq}$ be the Atkin-Serre operator on $p$-adic modular forms,  and let
  $\phi^{[p]}$ denote  the {\em $p$-depletion} of a classical or $p$-adic modular form $\phi$ of weight $k$,
  level $N$ and character $\chi$,
so that $d^t\phi^{[p]}$ is determined by the fourier expansion
$$ d^t \phi^{[p]}(q)  = \sum_{p\nmid n}  n^t a_n(\phi) q^n.$$
For  $t\ge 0$, this modular form belongs to the space $S_{k+2t}^{\noc}(N,\chi)$. In particular, 
the modular form $d^{t}  \testg_y^{[p]} \times \testh_z$ is a nearly overconvergent modular form of weight $2t+\ell+m$. 

With these basic notions and the
 $\Lambda$-adic test vectors in hand, the Garrett-Hida 
$p$-adic $L$-function  attached to 
$(\testhf^*, \testg,\testhh)$  is defined
 (as described in \cite[Def.~4.4]{DR1}, but  adopting a somewhat more flexible point of view)  to be the
 $p$-adic rigid-analytic function on $\Omega_f\times\Omega_g\times\Omega_h$
satisfying
\begin{equation}
\label{eqn:garrett-hida-f}
 \Lp^{f_\alpha}(\testhf^*, \testhg, \testhh)(x,y,z) 
 := \testf_x^*(e_{\ord}(d^{t}  \testg_y^{[p]} \times \testh_z)), \qquad t:= (k-\ell-m)/2,
 \end{equation}
as $(x,y,z)$ ranges over the  dense set of    points
 in $\Omega_f\times \Omega_g \times \Omega_h$   of  integral weights $(k,\ell,m)$, with 
  $k \equiv   \ell+m \pmod{2}$.  
  Viewing this as a $p$-adic $L$-function attached to the triple convolution of 
the Hida families $\hf$, $\hg$, and $\hh$ is justified by    fundamental results of Garrett, Harris-Kudla and Ichino
 relating the  quantity in \eqref{eqn:garrett-hida-f} to
 the central critical values 
 $$L(f_x,g_y,h_z,c), \qquad  c:= (k+\ell+m-2)/2,  \quad\mbox{ when } k,\ell, m \ge 1,  \qquad k \ge  \ell+m,$$
  up to a product of local factors which are non-zero for  a suitable choice of test vectors.    
  Identical 
  definitions apply to $\Lp^{g_\alpha}(\testhf,\testhg^*,\testhh)$ and $\Lp^{h_\alpha}(\testhf,\testhg,\testhh^*)$,
  with the  difference that their 
   regions of classical interpolation are defined by the inequalities $\ell \ge k+m$ and
  $m \ge k + \ell$ respectively.  
  

The point $(x_0,y_0,z_0)\in \Omega_f\times\Omega_g\times \Omega_h$ of weight $(2,1,1)$
for which $(f_{x_0}, g_{y_0}, h_{z_0}) = (f_\alpha,g_\alpha,h_\alpha)$ 
 thus lies within the region of interpolation  defining
 $\Lp^{f_\alpha}(\testhf^*,\testhg,\testhh)$, and its value at this point   is given by the formula 
 \begin{equation}
\label{eqn:garrett-hida-f11}
 \Lp^{f_\alpha}(\testf^*, \testg_\alpha, \testh_\alpha) 
 := \Lp^{f_\alpha}(\testhf^*,\testhg,\testhh)(x_0,y_0,z_0) 
 = \testf_{x_0}^*(e_{\ord}(\testg_\alpha^{[p]} \times \testh_\alpha))
 \end{equation}
 arising from \eqref{eqn:garrett-hida-f}.
Ichino's explicit form of Garret's formula asserts that the square of the expression on the right is a simple (non-zero, for a judicious choice of 
 test vectors)  multiple of the central critical value
  $L(E,\varrho,1)=L(f,g,h,1)$. In particular, although 
%the Garrett-Hida $p$-adic $L$-function 
 $\Lp^{f_\alpha}(\testhf^*,\testhg,\testhh)$ 
 depends crucially on the Hida families $\hf$, $\hg$ and $\hh$, and hence on the $p$-stabilisations of 
 $g$ and $h$ that were chosen to define these Hida families, the resulting {\em values} at the classical
 point $(x_0,y_0,z_0)$ do not depend in an 
 essential way on the choices  $\alpha_g$ and $\alpha_h$ of $U_p$-eigenvalues, or 
  on the concomitant choice of test vectors. It is therefore not too egregious an
 abuse of notations to set
$$    \Lp^{f_\alpha}(f,g,h) :=  
\Lp^{f_\alpha}(\testf^*, \testg_\alpha, \testh_\alpha)  
 \sim \Lp^{f_\alpha}(\testf^*, \testg_\alpha, \testh_\beta) 
  \sim \Lp^{f_\alpha}(\testf^*, \testg_\beta, \testh_\alpha) 
   \sim \Lp^{f_\alpha}(\testf^*, \testg_\beta, \testh_\beta), $$
   where 
     $\sim$ denotes an equality up to an algebraic factor (in the field generated by the
   Fourier coefficients of $f_\alpha$, $g_\alpha$, and $h_\alpha$) which can be made non-zero
   with a  good choice of test vectors.
   


% A classical eigenform $\phi$ (not necessarily new) of level $Np$   which is ordinary at $p$
%  gives rise to a system of  eigenvalues for the action of the Hecke operators acting 
%  on $S_k(Np,\chi)$. Let $S_k(Np,\chi)[\phi]$ denote the eigenspace
%  associated to this system of Hecke eigenvalues.
%  The Hecke algebra of level $Np$ also acts in a natural way on 
%  $S_k^{\oc}(N,\chi)$ but this action need not be semisimple. Let 
 % $S_k^{\oc}(N,\chi)[[\phi]]$ denote the generalised eigenspace
%  attached to $\phi$. 
%  The natural inclusion 
%  $$S_k^{\rm ord}(Np,\chi)\hookrightarrow S_k^{\oc, \ord}(N,\chi) $$
%  of the ordinary  classical into the overconvergent modular forms is   an isomorphism
 % if    $k\ge 2$, by 
 
% When either $k\in \Z^{\ge 2}$ or \eq } is satisfied, one
% therefore  obtains a $\phi$-isotypic projection  (factoring through   Hida's ordinary projection operator) to a space of classical modular forms:  \begin{equation}  \label{eqn:projection-bd}   e_{\phi}: S_k^{\noc}(N,\chi) \lra  S_k^{\oc}(N,\chi)[[\phi]] = S_k(Np,\chi)[\phi].   \end{equation}
 % If $\testphi$ is a test vector in $S_k(Np,\chi)[\phi]$, let $J_\testphi$ denote the linear functional on this space defined by the ratio   $$ J_\testphi(\xi) := {\langle \testphi,\xi \rangle}/{\langle \phi,\phi\rangle}$$  of Petersson scalar products. (This functional restricts to a $\bar\Q$-linear transformation on the subspace of modular forms with algebraic fourier coefficients, and is then extended by $\C_p$-linearity   via our fixed embedding $\bar\Q\lra \C_p$.)
% Any functional  $\testphi^* \in S_k(Np,\chi)^\vee$ on the space of 
% classical weight $k$ forms with the same Hecke eigenvalues as $\phi^*$ thus defines a functional
% $$ \testphi^*: S_k^{\noc}(N,\chi) \lra \C_p, \qquad \testphi^*(\xi) :=  \testphi^*(e_{\phi^*} e_{\ord}(\xi)).$$

The    value
\begin{equation}
  \label{eqn:garrett-hida-g}
 \Lp^{g_\alpha}(\testf, \testg_\alpha^*, \testh_\alpha)  :=
\Lp^{g_\alpha}(\testhf,\testhg^*,\testhh)(x_0,y_0,z_0)  =  {\testg_\alpha}^*(e_{\ord}(d^{-1}\testf^{[p]} \times \testh_\alpha))
 \end{equation}
 of   $\Lp^{g_\alpha}(\testhf, \testhg^*,\testhh)$ at the point 
$(x_0,y_0,z_0)$, which now lies outside the region of classical
interpolation, is  somewhat more subtle. 
%As explained in
 % \cite[Prop.~1.2]{DLR}, a fundamental  result of Bella\"{\i}che and Dimitrov 
  %implies that  
%the natural inclusion 
%\begin{equation}
%\label{eqn:target}
 %S_1(N p,\chi^{-1})[g_\alpha^*] \lra S_1^{\oc}(N,\chi^{-1})[[g_\alpha^*]]
 %\end{equation}
%of the $g_\alpha^*$-isotypic subspace in $S_1(N p,\chi^{-1})$ into the 
%associated generalised eigenspace in the space of  overconvergent modular forms of weight one 
%is an isomorphism,
 %provided that $g$ satisfies the condition 
  %\begin{equation}
 %\label{ -condition}
  %\begin{array}{l}
% \mbox{ $g$ is regular at $p$ and is not the theta series of a character } \\
 %\mbox{ of a real quadratic field in which the prime $p$ splits}.
 %\end{array}
 %\end{equation}
%Assuming \eqref{eqn:classicality-condition} from now on, the linear functional $\testg_\alpha^*$ %extends canonically
%to the target of  \eqref{eqn:target}. The 
 %$p$-adic $L$-value
 %$\Lp^{g_\alpha}(\testhf,\testhg^*,\testhh)(x_0,y_0,z_0)$
 %is  then well defined and described by 
%As shown in \cite[Prop.~2.5]{DLR}, 
  %referred to in loc.cit. as a {\em $p$-adic iterated integral},
  It depends  critically on the choice of stabilisation  of $g$, but 
\cite[Lemma 2.17]{DR1} shows 
that
$\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha)$ is left unchanged upon replacing $\testh_\alpha$ with the other $p$-stabilisation $\testh_\beta$ of $h$. Furthermore, a change in the choice of test vector (belonging to the same $(f,g_\alpha,h)$-isotypic subspaces for the action of the Hecke operators) 
has the effect of multiplying $\Lp^{g_\alpha}(\testf,\testg^*,\testh)$ by  explicit local zeta integrals at the  places dividing $N$, of exactly the same type as occur in Ichino's formula for the Garret-Rankin triple product.
One is hence
 justified in adopting the slightly abusive but simpler notations
\begin{equation}\label{Lpfgh} 
\Lp^{g_\alpha}(\testf,\testg^*,\testh) := 
\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha)  = \Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\beta),
\end{equation}
and likewise for the other $p$-stabilisations of $g$ and of $h$. 
   The four non-classical  $p$-adic $L$-values
\begin{equation}
\label{eqn:fourLps}
\Lp^{g_\alpha}(\testf,\testg^*,\testh), \quad \Lp^{g_\beta}(\testf,\testg^*,\testh), \quad 
\Lp^{h_\alpha}(\testf,\testg,\testh^*), \quad 
\Lp^{h_\beta}(\testf,\testg,\testh^*)
\end{equation}
ostensibly bear no simple direct relation to $L(f,g,h,1)$, or to each other.
When $L(f,g,h,1)=0$, they  should rather
be viewed as  different $p$-adic avatars of $L''(f,g,h,1)$.

Let $\Sel_p(E,\varrho) := H^1_{\fin}(\Q,V_p(E)\otimes V_\varrho\otimes_L L_p)
$ denote the
$\varrho$-isotypic component  of the Bloch-Kato Selmer group of $E/H$; cf.\,\eqref{defSelmer} for the precise definition of this group.
The following theorem shows that the above $p$-adic values convey non-trivial information about $\Sel_p(E,\varrho)$.
 
 
\medskip\noindent
 {\bf Theorem B}.   {\em 
If $L(E,\varrho,1)= 0$ and  $\Lp^{g_\alpha}(\testf,\testg^*,\testh)  \ne 0$ for some choice of test vectors,  then
 $$\mathrm{dim}_{L_p}\,\Sel_p(E,\varrho) \geq 2.$$
}
%Note that the condition on $\Lp^{g_\alpha}(f,g,h)$ in the statement  of theorem B
%should be understood as the statement $\Lp^{g_\alpha}(\testf, \testg^*_\alpha,\testh_\alpha) \ne 0$
%for {\em some } choice of triple $(\testf,\testg^*_\alpha,\testh_\alpha)$ of test vectors.
The same conclusion applies, of course, when any of the four 
$p$-adic $L$-values in \eqref{eqn:fourLps} is non-zero. 
Recent joint work with Alan Lauder \cite{DLR} describes
 an algorithm
 for calculating the expressions in
\eqref{eqn:garrett-hida-g}, and hence $\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha)$,
and uses this to verify  their non-vanishing   
 in many instances of analytic rank two, showing that Theorem B is not vacuous.
 
Theorem D below formulates a more precise version of Theorem B, in which
   two  canonical elements in $\Sel_p(E,\varrho)$ are 
 associated to the data $(f,g_\alpha,h)$, and shown to be linearly independent under the hypotheses of Theorem B. 
One of the main contributions of \cite{DLR}  and \cite{DR2} is the formulation of a conjecture expressing these two elements as explicit
$p$-adic linear combinations of global
points in the Mordell-Weil group attached to $E(H)^\varrho$. (Cf.~loc.cit.~ for the precise statement of
this conjecture and a summary of the
 experimental evidence that has been gathered in its support.)
 
Theorem  B  can be compared to the Theorem B of Skinner and Urban \cite{SU}, which 
asserts that the rank of the Selmer group of an elliptic curve is $\geq 2$ when the 
associated $L$-function vanishes to even order $\geq 2$.  
There is little overlap between both theorems, whose methods of proof are very different. 
The result of \cite{SU} has the virtue of requiring no non-vanishing hypotheses on an auxiliary $p$-adic $L$-function, while Theorem B  applies to a different class of Artin  representations, typified by the settings described in Corollaries A1 and A2. 



%\medskip
The proofs of Theorems A and B rest on a  system of global cohomology classes 
 for the Rankin convolution of three
modular forms of weights $(2,1,1)$, whose construction is one of the main contributions of the
present work. These classes arise from 
{\em generalised Gross-Kudla-Schoen cycles} 
 in the 
product 
 of three Kuga-Sato varieties  fibered over a
  classical modular curve, and (crucially)  from their
 {\em variation in Hida families}. 


The prequel \cite{DR1} to the present work initiated the authors' study of generalised
  Gross-Kudla-Schoen cycles, relating their images    under the $p$-adic Abel-Jacobi map 
to special values of  Garrett-Hida $p$-adic $L$-series attached to the 
triple convolution of three (Hida families of) cusp forms. 
The $p$-adic Abel-Jacobi map has the   feature---not present in its
 complex analogue,  or in related complex and $p$-adic heights---of factoring through 
 the restriction to
 $G_{\Q_p}$ of
  its $p$-adic \'etale counterpart. This turns out to be crucial in arithmetic applications
 relying on the theory of Euler systems.
 More precisely, 
  the \'etale
 Abel-Jacobi  images of generalised diagonal cycles
 give rise, after applying suitable projections, to distinguished global Galois 
 cohomology classes
   \begin{equation}
 \label{eqn:target-aj-et}
\kappa(f_x,g_y,h_z) \in  H^1(\Q, V_{f_x g_y h_z}(N)),
 \end{equation}
 where
 \begin{itemize}
 \item $f_x$, $g_y$ and $h_z$ are  classical specialisations  of weights $k$, $\ell$, and $m$ which are
 {\em balanced} in the sense that each weight is strictly smaller than the sum of the 
 other two;
 \item $V_{f_x g_y h_z}(N)$ is the Kummer self-dual twist of the direct sum of several copies of the tensor product of the  $p$-adic representations 
  $V_{f_x}$, $V_{g_y}$, $V_{h_z}$ of
 $G_\Q$ attached by Eichler-Shimura and Deligne  to these forms, 
  occuring in the middle cohomology of a Kuga-Sato variety $\cE_k$ (resp.\,$\cE_{\ell}$, $\cE_{m}$) of
  dimension  $k-1$ (resp.\,$\ell-1$,  $m-1$).
   \end{itemize}
The extensions of $p$-adic Galois representations
 associated to the classes $\kappa(f_x,g_y,h_z)$  arise {\em from geometry}; 
 namely, they are  realised in the $p$-adic \'etale cohomology of  an open subvariety
 of the product of Kuga-Sato varieties $\cE_{k,\ell,m}:=  \cE_k\times \cE_{\ell}\times \cE_{m}$,
which have good reduction at $p$. In particular, thanks to the work of Saito their restrictions  to a decomposition
 group at $p$ are known to be  {\em crystalline}. 
 
The main theorem of \cite{DR1} can be interpreted as a direct relationship between
  the Bloch-Kato  $p$-adic logarithms of the class $\kappa(f_x,g_y,h_z)$  and the
 special values of Garrett-Hida $p$-adic $L$-functions at points which lie
 {\em outside} their range of classical interpolation.
 The concluding paragraphs of the introduction to \cite{DR1}
   stressed
  the desirability of deforming this result {\em along $p$-adic families} and, in 
  particular, of $p$-adically interpolating the classes $\kappa(f_x,g_y,h_z)$ themselves
  as the triple  $(f_x,g_y,h_z)$  is made to vary
  over the classical, balanced
 specialisations of ordinary Hida families $\hf$, $\hg$ and $\hh$. Such 
 a $p$-adic interpolation would be described by a global cohomology 
 class
 \begin{equation}
  \hkappa(\hf,\hg,\hh) \stackrel{?}{\in} H^1(\Q, \hV_{\hf\, \hg\,\hh}(N)),
  \end{equation}
 where $\hV_{\hf\, \hg\,\hh}(N)$ is the direct sum of several copies of the tensor product of Hida's 
 $\Lambda$-adic representations $\hV_{\hf}$, $\hV_{\hg}$, $\hV_{\hh}$ attached to $\hf$, $\hg$, and $\hh$, twisted in such a way that
  for each triple $(f_x,g_y,h_z)$ of classical (balanced or unbalanced) specialisations of $(\hf,\hg,\hh)$ there are
 $G_\Q$-equivariant specialisation homomorphisms
  $$ \nu_{x y z}: \hV_{\hf\, \hg\,\hh}(N) \lra V_{f_x g_y h_z}(N).$$


The putative class $\hkappa(\hf,\hg,\hh)$
would be  uniquely determined by the requirement that its  specialisations at balanced triples
 $(f_x,g_y,h_z)$ should agree, up to normalisation by elementary fudge factors, with the 
 classes $\kappa(f_x,g_y,h_z)$ of \eqref{eqn:target-aj-et}. Guided by this ideal goal but falling somewhat short of it, 
Chapter \ref{sec:lambda-adic-class}  attaches a one-variable cohomology class  to a triple $(f,\hg,\hh)$
 formed by an eigenform $f$ of weight two and trivial nebentype and two Hida families $\hg$ and $\hh$ of tame character $\chi$ and $\chi^{-1}$, respectively. 
 This  $\Lambda$-adic class is described by:
\begin{itemize}
\item a finite flat algebra $\Lambda_{f\underline{g h}}$  over the Iwasawa algebra 
$\Lambda = \Z_p[[1+p\Z_p]]$;
\item   a $\Lambda_{f\underline{g h}}[G_\Q]$-module $\hV_{f\underline{g h}}(N)$ 
whose classical specialisations are of the form $V_{f g_y h_z}(N)$
 where the pairs $(g_y,h_z)$  range  over the  specializations of $\hg$ 
 and $\hh$ 
 at classical points  of  the {\em same} weight $\ell$
 and  nebentype character  at $p$;
\item a cohomology class $\kappa(f, \underline{g h})\in H^1(\Q, \hV_{f\underline{g h}}(N))$.
 \end{itemize}
 The class $\kappa(f,\underline{g h})$ 
gives rise to a collection of global classes
 in $H^1(\Q, V_{f g_y h_z}(N))$,
 varying $p$-adic analytically  as $(g_y,h_z)$ varies
 over pairs of specialisations of $\hg$ and $\hh$ with common 
  weight and nebenytpe character at $p$. In order to construct 
$\kappa(f,\underline{g h})$,
Section \ref{sec:twisted-diagonal} introduces a family $\{\Delta_{s}\}_{s\geq 1}$ of null-homologous
  {\em twisted diagonal cycles} on certain twists of finite quotients  
  of the three-folds $X_0(Np) \times X_1(Np^s) \times X_1(Np^s)$, satisfying an
explicit compatibility  under the natural push-forward maps as $s$ varies. The class $\kappa(f,\underline{g h})$ is then manufactured in \S \ref{sec:AJet}-\ref{sec:lambda-adic-class-hida} as the inverse limit of the cohomology classes one obtains
as the image of $\Delta_s$ under a suitable projection of the 
$p$-adic \'etale Abel-Jacobi map. 



 Although $\hkappa(f,\underline{gh})$ is constructed by interpolating 
geometric constructions attached only to the weight two specialisations
 of $\hg$ and $\hh$, it makes sense to consider its specialisations
\begin{equation}
\label{eqn:spec-classes}
\kappa(f,g_y, h_z) := \kappa(f, \underline{g h})_{y,z} \in H^1(\Q,V_{f g_y h_z}(N))
\end{equation} 
at any point $(y,z)$
attached  to a  pair $(g_y,h_z)$  of
classical modular forms  of arbitrary  weight $\ell \ge 1$
with inverse tame nebentype characters.
The relevance of this class to the proof of Theorems B and C
 arises  when $f$ is the weight $2$ newform
 associated to the elliptic curve $E$, and  $g_y=g_\alpha$ and $h_z=h_\alpha$ are (ordinary) $p$-stabilisations of the forms $g$ and $h$ associated to the Artin representations $\varrho_1$ and $\varrho_2$.

Since $g_\alpha \ne g_\beta$ and $h_\alpha \ne h_\beta$, the construction described above yields four relevant cohomology classes, denoted
\begin{equation}\label{classes}
 \kappa(f, g_\alpha, h_\alpha), \quad
\kappa(f, g_\alpha, h_\beta), \quad
\kappa(f, g_\beta, h_\alpha), \quad
\kappa(f, g_\beta, h_\beta) \quad \in    H^1(\Q,V_{f g h}(N)),
\end{equation}
where again $V_{f g h}(N)$ is just a direct sum of several copies of $V_p(E)\otimes V_\varrho \otimes_L L_p$.
These classes are called the {\em generalised Kato classes} attached to the triple $(f,g,h)$ of modular forms of weights 
$(2,1,1)$. Justifying this terminology is the fact that the classes of \eqref{classes}
 were  first constructed and studied by Kato in 
the special case where both $g$ and $h$ are weight one Eisenstein series, where they form the basis for his 
remarkable proof of the
Birch and Swinnerton-Dyer conjecture in analytic rank zero for $L$-functions of elliptic curves
twisted by Dirichlet characters.




The Kummer-self-dual representation
$V_{f g h}$ is  still crystalline, 
but there is no 
 a priori reason for the classes in \eqref{classes}
to be crystalline at $p$  because the
  weights $(2,1,1)$ of the triple $(f,g,h)$ are 
 {\em unbalanced}, with $f$ being the eigenform of  {\em dominant weight}
 in the sense of \cite{DR1} and thus    
 the points in weight space attached to the classes in \eqref{classes} lie outside 
  the range of ``geometric interpolation" 
  defining  $\kappa(f, \underline{g h})$.  


Theorem \ref{main2} of 
\S  \ref{sec:Lp}
 establishes  a reciprocity law  relating  the image  
of the classes in \eqref{classes}
under the Bloch-Kato {\em dual exponential map} 
 to   the central critical value $L(E,\varrho,1)$. It is the main 
 ingredient in the proof of Theorem A.
 
 
\medskip\noindent
{\bf Theorem C.} {\em The generalised Kato class $\kappa(f,g_\alpha,h_\alpha)$ is  crystalline at $p$
  if and only if $L(E,  \varrho,1) =  0$.}

 \medskip
When $L(E, \varrho,1) \ne 0$, Theorem C combined with the 
regularity of $V_{g}$ and $V_h$ at $p$ 
implies that the
cohomology classes in \eqref{classes} are linearly independent,
more precisely, that their natural images generate the
 ``singular quotient" 
of  the local cohomology at $p$. 
A standard argument involving the global 
Poitou-Tate exact sequence then  implies that the image
of the Selmer group of $V_{fgh}$ maps to zero in the local 
cohomology $H^1(\Q_p,V_{fgh})$ at $p$. 
 The fact that the Mordell-Weil group of  global points always injects into the 
 group of local points then shows  
the triviality of
 the $\varrho$-isotypical component of the Mordell-Weil group of $E$,
 thereby proving Theorem A.
 
 One also expects (as a consequence of the Shafarevich-Tate conjecture) that 
$\Sel_p(E,\varrho)$ is trivial when $E(H)_L^\varrho$ is, but the  proof of Theorem A
described above
 does not yield the  finiteness of the associated Selmer group or 
 Shafarevich-Tate group.
 The  Gross-Schoen diagonal cycles  can be viewed as a special instance of a broader 
 class of examples, namely the ``Euler systems of Garrett-Rankin-Selberg type"
 described in \cite{durham}, which also encompass Kato's Euler system of Beilinson elements and 
 the Euler system of Beilinson-Flach elements. 
In both these settings, finiteness results for
 the Shafarevich-Tate groups have been obtained by Kato \cite{kato}
 and Lei, Loeffler, and Zerbes  \cite{LLZ1}  
 respectively.
 It would be of great interest to extend   these
 authors' refinements  to the setting of diagonal cycles.  Recent work of 
 Yifeng Liu  makes striking progress in a closely related setting, 
where  diagonal cycles are replaced by twisted variants on the product of a modular curve
 and a Hilbert modular surface \cite{liu1} and on a Hilbert modular threefold \cite{liu2}, and
  the Selmer groups 
 that are studied are those attached to Galois representations of weights $(2,2,2)$
 (in the region of ``balanced weights", which can thus be treated without 
 recourse to the  $p$-adic deformation of Abel-Jacobi images  of cycles).
  

%\medskip
When $L(E,\varrho,1)=0$, 
 and hence $r_{\mathrm{an}}(E,\varrho) \geq 2$, 
Theorem C implies that the four classes in \eqref{classes} 
belong to the   Selmer
 group $\Sel_p(E,\varrho)$.
The submodule of $\Sel_p(E,\varrho)$ generated by these four classes   is expected to
  be non-trivial precisely when
  $r_{\mathrm{an}}(E,\varrho)=2$, i.e., when $L''(E,\varrho,1)\ne 0$. 
  The following result, whose 
  proof is described in Theorem \ref{theoremB} of Section
  \ref{sec:theoremsBD}, 
    involves
  $\Lp^{g_\alpha}(\testf,\testg^*,\testh)$ rather than $L''(f,g,h,1)$,
 and directly implies Theorem B above.

\medskip\noindent
 {\bf Theorem D}.  {\em  
  If $L(E,\varrho,1)=0$ and $\Lp^{g_\alpha}(\testf,\testg^*,\testh)  \ne 0$ for some choice of test vectors, there exist $G_\Q$-equivariant projections $j_\alpha, j_\beta: V_{f g h}(N) \ra V_p(E)\otimes V_\varrho$ such that the  classes
 $\kappa_{\alpha \alpha} = j_\alpha(\kappa(f,g_\alpha,h_\alpha))$ and $\kappa_{\alpha \beta} = j_\beta(\kappa(f,g_\alpha,h_\beta))$ 
 are linearly independent in $\mathrm{\Sel}_p(E,\varrho)$.}

\medskip
The canonical nature of the elements in
$\Sel_p(E,\varrho)$ described in Theorem D  is key to the formulation of the conjectures of \cite{DLR} 
and \cite{DR2}, which seem to admit no direct counterpart for the Selmer classes 
constructed in \cite[Thm.~B]{SU}. 
To further compare Theorem D with the results of \cite{SU}, 
note that   generalised Kato classes can
 only contribute to the pro-$p$ Selmer group $\Sel_p(E/\Q)$ 
 of $E$ over $\Q$ when $\varrho$ contains the trivial representation as a constituent,
which occurs precisely when $g$ and $h = \bar g$ are dual to each other, so that $\varrho = 1 \oplus {\rm Ad}^0(\varrho_g)$,
where $\Ad^0(\varrho_g)$ denotes the three dimensional {\em adjoint } of $\varrho_g$,
 consisting of the trace zero endomorphisms of $\varrho_g$. 
In the scenario where  $L(E/\Q,s)$ admits a double
zero at the center, the elliptic Stark conjectures of 
\cite{DLR} predict that all of the $p$-adic $L$-values in 
\eqref{eqn:fourLps}
vanish and that the
generalised Kato classes attached to $(f,g,\bar g)$ never generate the Selmer group or Mordell-Weil group
 of $E$.
However, they are expected to  generate a one-dimensional subspace of $\Sel_p(E/\Q)$ whenever $L(E,\Ad^0(\varrho_g),1)\ne 0$, consisting of the classes
whose restriction to $\Q_p$ lies in the kernel of the logarithm map $\log_p: E(\Q_p) \lra \Q_p$. 
The non-triviality  of such a class only   implies  that $\dim\Sel_p(E/\Q)\ge 1$, but it does imply 
 that ${\rm rank}(E(\Q)) \ge 2$
under the further (highly non-trivial) assumption
that $\sha(E/\Q)$ is finite.
See \cite[\S 4.5]{DR2} for a more detailed discussion of this intriguing scenario.

 
%\medskip
%\noindent{\bf Remark}.
%It is expected that the classes
%$\kappa(f,g_\alpha,h_\alpha)$ and 
%$\kappa(f,g_\alpha,h_\beta)$ may sometimes be non-trivial even
%when
%$L(E,\varrho,1)$  and $\Lp^{g_\alpha}(f,g,h)$ both vanish.
%A notable instance of this phenomenon occurs when $g = \bar h$
%(so that $\varrho_{gh}$ decomposes as a direct sum of the trivial representation
% and the adjoint $W_g$ of $g$) and 
%$L(E,s)$ has a double zero at $s=1$.
% It can then be shown that 
%$\Lp^{g_\alpha}(f,g,h)=0$, while the conjectures of \cite{DLR} suggest that
%$\kappa(f,g_\alpha,h_\alpha)$ and $\kappa(f,g_\alpha,h_\beta)$ generate a canonical
%one-dimensional subspace of the Selmer group of $E/\Q$   whenever
 %$L(E,W_g,1)\ne 0$.
 %\medskip\medskip
  
% The circle of ideas sketched above also sheds new light on Kato's % theorem \cite{kato} for twists of elliptic curves by Dirichlet characters. The latter can 
% be viewed as a ``degenerate instance" of the former  in which the
% weight one cusp forms $g$ and $h$  are replaced by weight one Eisenstein series, 
% and diagonal cycles are replaced by Beilinson elements in the second 
% $K$-groups of modular curves, attached to pairs of modular units whose logarithmic derivatives 
% are Eisenstein series of weight two.

% Viewed in this way, the so-called {\em Beilinson-Flach elements} in the first $K$-group of a product of two modular curves studied in \cite{BDR1} and, subsequently, in \cite{LLZ} occupy a position halfway between diagonal cycles and Kato's Beilinson elements and are germane to  the setting  where $g$ is  a cusp form and $h$ is an Eisenstein series.  In the forthcoming paper \cite{BDR2},  $p$-adic families of Beilinson-Flach elements are exploited to  prove Conjecture BSD$_0$ when $\varrho$ is an odd, irreducible, two-dimensional Galois  representation. This includes the Artin representations induced from ray class  characters of a real quadratic field which are of {\em mixed signature}  (i.e., even at one of of the infinite places, and odd at the other).  Note that there is no overlap between the main result of \cite{BDR2} and Corollary A1,  since ring class characters of a real quadratic field are either totally even or totally odd.

% The passage from Beilinson-Kato and Beilinson-Flach elements to Diagonal cycles which underlies this work greatly enhances the flexibility and scope of the Euler systems approach based on $p$-adic families of geometric objects. For instance, the cohomology classes
% considered by Beilinson-Kato and Flach rely crucially on modular units which exist only in the setting of classical modular curves over $\Q$, while diagonal cycles are present in a greater degree of generality (on the product of three Shimura curves over arbitrary totally real fields, for instance). 
 
 Underlying  the crucial  reciprocity law
 relating  the local behaviour  at $p$
 of the global classes in 
 \eqref{classes} to $p$-adic $L$-values
  is a precise relationship between the $p$-adic families
  $\kappa(f, \underline{g h})$  of global cohomology classes (more precisely, their 
  restriction to a decomposition group at $p$)  and the 
  Garrett-Hida $p$-adic $L$-functions alluded to above. 
  See Theorem \ref{pL-PR} for the precise statement.
 Such relationships 
 are part of  a long tradition of {\em explicit reciprocity laws} going  back   
 to the seminal work of Coates and Wiles, and   even further to 
 the work of Kummer, Iwasawa and 
 Kubota-Leopoldt on the arithmetic of cyclotomic fields. 
 For more on how the approach of  this paper
fits into the larger perspective
 of  Euler systems and explicit reciprocity laws, notably with Kato's work on
 Beilinson elements and 
the work of Bertolini and the authors on Beilinson-Flach elements,
the reader is invited to consult the survey \cite{durham}. 


\medskip
\noindent
{\em Acknowledgements}: The authors benefitted from many stimulating
exchanges
  with Massimo Bertolini, Antonio Lei, David Loeffler 
and Sarah Zerbes on the topics of this paper. 
They  are also grateful to the two anonymous referees for their 
 thorough proofreading and valuable feedback which led to significant improvements in the exposition. 
 (Of course, any flaws that remain in this regard are to be blamed on the authors alone.)
The first author was supported by an NSERC Discovery grant, and the second   by Grant MTM2012-34611.


\medskip
\medskip
 

 

\section{Global $\Lambda$-adic cohomology classes}
\label{sec:lambda-adic-class}

\subsection{Cycles, Chow groups, and correspondences}
\label{sec:cycles-and-correspondences}
  This section collects a few general facts and notations concerning algebraic cycles and correspondences for future reference. 

Given a smooth proper irreducible variety $W$ of dimension $d$ over a field $F$, 
 and any integer $c\geq 0$,
  let $\CC^c(W)(F)$ denote the group of codimension $c$ 
  cycles in $W$ defined over $F$ with coefficients in $\Z_p$, and let 
  $$\CH^c(W)(F) = \CC^c(W)_{/\sim}$$
   denote the Chow group 
  of rational equivalence classes of such cycles. The field $F$ is sometimes supressed from the notation when  this leads to no ambiguity.
  
  The standard conventions are employed
 for Tate twists: if $H$ is a $\Z_p[G_\Q]$-module and $k$ is an integer, 
then $H(k)=H\otimes \epsilon_{\cyc}^k$, where 
\begin{equation}
\label{eqn:epsilon-cyc}
\epsilon_{\cyc}: G_\Q\lra \Z_p^\times
\end{equation}
 denotes the $p$-adic cyclotomic character.
 
  
 If $\Delta$ is a cycle in $\CC^c(W)$, the same symbol 
 $\Delta$ will often be used as well to denote the class of this cycle in the Chow group $\CH^c(W)$. 
  There is a fundamental {\em  $p$-adic \'etale cycle class map}  over $F$
\begin{equation}
\label{eqn:cycle-class-map}
 \cl_F: \CH^c(W)(F)    \lra H^{2c}_{\et}(W ,\Z_p(c)),
 \end{equation}
 which is described in \cite[\S 23]{milne} when $F$ is algebraically closed. For more general
 fields $F$ (such as number fields), the cycle class map is
 described in \cite[eqn.~(0.3)]{jannsen}, and its target is to be interpreted as the 
  {\em  continuous
\'etale cohomology} of loc.cit. 
 After choosing an embedding of $F$ into an algebraic closure $\bar F$, the symbol ${\overline W}$ 
is used to denote the variety over $\bar F$ deduced from $W$ by extension of scalars, and 
 \begin{equation}
 \label{eqn:cycle-class-map-bis}
  \cl: \CH^c(W)(F) \lra H^{2c}_{\et}(\overline{W},\Z_p(c))
  \end{equation}
 denotes the
 natural map that is deduced from \eqref{eqn:cycle-class-map}.
 A cycle in $\CH^c(W)$ which is in the kernel of  
 $\cl$ is said to be {\em null-homologous}, and the subgroup of null-homologous cycles defined over 
 $F$  is denoted
 $\CH^c(W)_0(F)$. The Hochschild-Serre spectral sequence 
$$ H^{i}(F, H^j_{\et}(\overline{W}, \Z_p(c)) \Rightarrow H^{i+j}_{\et}(W,\Z_p(c))$$
of \cite{jannsen} relating 
continuous \'etale  cohomology with the continuous group cohomology of $G_F = \Gal(\bar F/F)$ 
 gives rise to the 
$p$-adic \'etale Abel-Jacobi map
\begin{equation}
\label{AJet-general} 
\AJ_{\et}: \CH^c(W)_0(F) \lra H^1(F,H^{2c-1}_{\et}(\bar{W},\Z_p)(c)),
\end{equation}
 as described, for instance, in the penultimate displayed equation of \cite{jannsen}.
 This map  plays a key role in the constructions of the global cohomology classes alluded
 to in the introduction.  
  
 A {\em correspondence} between two varieties $W_1$, $W_2$ of dimension $d$ is an element $T\in \mathrm{Corr}(W_1,W_2) :=\CC^d(W_1\times W_2)$. As indicated, the  symbol   $T$   is used to denote  both the cycle and
  its class in  $\CH^d(W_1\times W_2)$. Let $T^{\ad} \in \CC^d(W_2\times W_1)$ denote the transpose correspondence  from $W_2$ to $W_1$. 
An element $T\in \mathrm{Corr}(W_1,W_2)$ gives rise to a homomorphism from 
  $\CH^c(W_1)$  to $\CH^c(W_2)$ for any $c\geq 0$
  by the rule $T(\Delta) := \pi_{W_2,*}( T \cdot \pi_{W_1}^*(\Delta))$,
 where $\cdot$ denotes the intersection product on 
 $W_1\times W_2$,  the maps $\pi_{W_1}$ and $\pi_{W_2}$ denote the natural projections of $W_1\times W_2$ to its first and second factors,
and   $\pi_{W_1}^*$ and $\pi_{W_2,*}$ denote the induced pull-back and proper
 pushforward maps associated to 
 $\pi_{W_1}$ and $\pi_{W_2}$ respectively. 

 %obtained by applying to $T$ the proper push-forward map induced by the involution $W_1\times W_2 \ra W_2\times W_1$, $(x_1,x_2) \mapsto (x_2,x_1)$.

When $W=W_1=W_2$, set $\mathrm{Corr}(W):=\CH^d(W\times W)$. 
There is a natural identification 
$$H^{2d}(\overline{W}\times \overline{W},\Q_p)(d) =  \oplus_{i=0}^{2d} \End(H^i({\overline W},\Q_p))$$
arising from the K\"unneth decomposition combined with Poincar\'e duality, and the image of a correspondence $T\in \mathrm{Corr}(W)$ under the map $\cl$ of
 \eqref{eqn:cycle-class-map-bis} gives rise to a degree $0$ endomorphism of the cohomology of $W$, which is compatible with the
 action on Chow groups via the cycle class map.
Given a correspondence $T$ on $W$,   the same symbol is routinely used 
to describe   this induced endomorphism
on Chow groups and on cohomology. The transpose correspondence 
$T^{\ad}$ gives rise to the adjoint  endomorphisms  on cohomology
relative to  Poincar\'e duality.


\subsection{Modular curves}
\label{sec:modular-curves}
Let $M\geq 1$ be a positive integer and  $X_0(M)$ and $X_1(M)$ denote the 
classical modular curves over $\Q$ attached to the Hecke
congruence subgroups $\Gamma_0(M)$ and  $\Gamma_1(M)$, respectively.

Several distinct models over $\Q$ of $X_1(M)$ occur in the literature; in what follows, the affine 
curve $Y_{1}(M)$ obtained by removing the cusps from $X_1(M)$ is taken to be the coarse
moduli space of pairs $(A,i)$ where $A$ is an elliptic curve
  and $i: \mu_M \ra A$ is an isomorphism between the group scheme $\mu_M$ and a closed finite flat subgroup scheme of $A$.
This model of $X_1(M)$ is characterized as being the one whose field of rational functions $\Q(X_1(M))$ is the subfield of $\C(X_1(M))$ given by those functions whose $q$-expansion at the cusp $i\infty$ have rational coefficients.


Fix an integer $N\geq 3$ and an odd prime $p\nmid N$. For any $s\geq 1$,  let   $X_s:=X_{1}(Np^s)$. 
It will be convenient to view $X_s$ as classifying triples $(A,i_N,i_p)$
where $i_N: \mu_N \ra A$ and $i_p: \mu_{p^s} \ra A$ are embeddings 
of finite group schemes. It will also be useful to consider the curve $X^\dag_s/\Q$ which
 arises as the coarse moduli space associated to the problem of classifying 
 triples $(A,i_N,P)$ where $(A,i_N)$ is as above and $P\in A$ is a point of
  exact order $p^s$. The curves $X_s$ and $X^\dag_s$ are not naturally
 isomorphic over $\Q$, but are twists of each
  over the cyclotomic field of $p^s$-th roots of unity. 
  See below for more details.

%\medskip
The  curves $X_s$ are endowed with the following  additional structures:


\medskip\noindent
{\em Diamond operators}. 
Let
\begin{equation}
\label{eqn:proj-x1-x0}
j_s: X_s\lra X_0(Np^s)
\end{equation}
be the  forgetful map 
 sending 
a triple $(A,i_N,i_p)$ to $(A, C)$, where $C=\langle \mathrm{Im}(i_N), \mathrm{Im}(i_p) \rangle$ is the subgroup generated by the images of $i_N$ and $i_p$. It  is a Galois cover, whose 
Galois group is identified with the group 
$$G_s^{(N)} = \{ \langle d\rangle, d\in (\Z/Np^s\Z)^\times\} \simeq (\Z/Np^s\Z)^\times/\langle \pm 1\rangle$$ 
of diamond operators, acting on $X_s$ by the rule 
$\langle d\rangle(A,i_N,i_p):= (A,d\cdot i_N, d\cdot i_p)$. 
Let 
$$ \Gamma_s \subset G_s\subset G_s^{(N)}, \qquad \Gamma_s = 1+p(\Z/p^s\Z), \quad
G_s = (\Z/p^s\Z)^\times $$
be the natural subgroups, and write
\begin{equation}
\label{eqn:finite-group-rings}
 \Lambda_s := \Z_p[\Gamma_s], \qquad \LLambda_s := \Z_p[G_s], \qquad \LLambda_s^{(N)} := \Z_p[G_s^{(N)}]
 \end{equation} 
for the associated group rings with $\Z_p$-coefficients. 
The corresponding completed group rings are
\begin{equation}
\label{eqn:completed-group-rings}
\Lambda := \lim_{\leftarrow} \Lambda_s, \qquad \LLambda := \lim_{\leftarrow} \LLambda_s,
\qquad \LLambda^{(N)} := \lim_{\leftarrow} \LLambda_s^{(N)}.
\end{equation}
All of the group rings in \eqref{eqn:finite-group-rings} and \eqref{eqn:completed-group-rings}
act naturally on spaces of  modular forms on
$\Gamma_{1}(Np^s)$ as well as on the various cohomology groups attached to
 $X_s$. 
 

Given $a\in (\Z/N\Z)^\times$ and $b\in (\Z/p^s\Z)^\times$,  it will 
sometimes be convenient to denote by $\langle a;b\rangle$ the
 diamond operator $\langle d \rangle$, where $d\in (\Z/Np^s\Z)^\times$ is congruent to $a$ modulo $N$ and to $b$ modulo $p^s$. 
 This automorphism acts on $X_s$ by the rule $\langle a;b\rangle (A,i_N,i_p) =  (A,a\cdot i_N, b\cdot i_p).$

Let $\zeta_s$ be a primitive $p^s$-th root of unity and let 
$\epsilon_s: G_\Q\lra (\Z/p^s\Z)^\times$ denote the mod $p^s$ cyclotomic character, factoring through $\Gal(\Q(\zeta_s)/\Q)$. 
The curve $X^\dag_s$ introduced above is isomorphic to the twist of $X_s$ by the cocycle 
\begin{equation}\label{cocycle-twist}
\langle \epsilon_s^{-1}\rangle: G_\Q \lra \Aut(X_s), \quad \langle \epsilon_s^{-1}\rangle(\sigma) := \langle 1;\epsilon_s^{-1}(\sigma)\rangle,
\end{equation}
and therefore $X_s$ is isomorphic to $X^\dag_s$ over $\Q(\zeta_s)$.



 
\medskip\noindent
{\em Duality}.  The  (\'etale and de Rham) cohomology groups of
$X_s$  are endowed with   perfect pairings
\begin{equation}
\label{eqn:pairing-s}
 \langle \ , \ \rangle_s: H^1_{\et}(\bar X_s,\Z_p) \times H^1_{\et}(\bar X_s,\Z_p) \lra \Z_p(-1), 
\qquad
 H^1_{\dR}(X_s/\Q) \times H^1_{\dR}(X_s/\Q) \lra  \Q 
\end{equation}
arising from Poincar\'e duality. The symbol $\langle \ , \ \rangle_s$ is used to denote
both the \'etale and de Rham Poincar\'e pairing, relying on the context
will make it clear which is being used.
The de Rham pairing  is defined on classes of differentials of the
second kind by the usual formula
$$\langle \omega_1,\omega_2\rangle_s := \sum_{P\in X_s}  \mbox{res}_P( F_{\omega_1,P} \cdot \omega_2),$$
where  the sum is taken over the closed points $P$ of $X_s$, 
and $F_{\omega_1,P}$ denotes a local 
 primitive of $\omega_1$ at $P$.  




 \medskip\noindent
{\em Hecke operators}.  There is 
a commuting algebra of Hecke correspondences $T_n$  indexed by the integers prime
to $Np$ (defined for instance as in~\cite[Ch. 5]{diamond-shurman}). 
For all $s\ge 1$, write $\mathbb T_0(Np^s)$ and $\mathbb T_{1}(N p^s)$  for the $\Q$-algebra generated by these  Hecke operators, 
together with the diamond operators in the latter case, in the ring of correspondences of the curves $X_0(Np^s)$ and $X_s$ respectively.
These ``good" Hecke operators commute with their adjoints relative to the pairings
$\langle \ ,\ \rangle_s$ and, more generally, relative to the Poincar\'e duality on any modular curve. More precisely, 
\begin{equation}
 \label{eqn:adjoint-hecke}
 T_n^{\ad} = \langle n  \rangle T_n, \qquad \mbox{ for all } n \mbox{ with } \gcd(n,Np)=1.
\end{equation}

 
 

\medskip\noindent
{\em Atkin-Lehner automorphisms}.  Fix a norm-compatible collection $\{ \zeta_s\}_{s\geq 1}$ of primitive roots of unity of order $p^s$.
Associated to $\zeta_s$ 
is the Atkin-Lehner automorphism denoted $w_{s}$ which is 
defined on triples $(A,i_N,i_p)$ by the rule
$$ w_{s}(A,i_N,i_p) = (A/C_p, i_N,i'_p), \qquad C_p :=\langle i_p(\zeta_s)\rangle,
$$
where $i'_p(\zeta_s) = x' \pmod{C_p}$ and $x'$ is a point on $A$ of order $p^s$ satisfying 
$\langle i(\zeta_s),x'\rangle_{\mathrm{Weil}} = \zeta_s$.
Likewise, the choice of a primitive $N$-th root of unity $\zeta_N$  gives rise to a similar Atkin-Lehner
operation at $N$, which shall be denoted $w$, and is defined on any of the curves $X_s$. 
The automorphisms $w_s$ and $w$  do not commute with the diamond and Hecke operators,
and instead satisfy
\begin{align}
\label{eqn:commutation-w}
& w_{s} \langle a;b\rangle  =   \langle a; b^{-1} \rangle w_{s}, \qquad
w_{s} T_n   =   \langle 1;n \rangle T_n w_{s},
\qquad w_{s}^2 = \langle -p^s;1\rangle, \\
& w \langle a;b\rangle  =   \langle a^{-1}; b \rangle w, \qquad
w T_n   =   \langle n;1 \rangle T_n w,
\qquad w^2 = \langle 1; -N\rangle.
\end{align}
Moreover, the operators $w_{s}$ are   not defined over $\Q$, but only over $\Q(\zeta_s)$ and 
$\Q(\zeta_N)$ respectively.
More precisely,   for all $\sigma\in G_\Q$,
\begin{equation}
\label{eqn:galois-w}
 (w_{s})^\sigma 
= \langle 1;\epsilon^{-1}_s(\sigma)\rangle \, w_{s}, \qquad
 (w)^\sigma 
= \langle \epsilon^{-1}_N(\sigma); 1 \rangle \, w, 
 \end{equation}
 where $\epsilon_N$ denotes the mod $N$ cyclotomic character. The actions of $w$ and $w_s$ on differentials, functions, and  de Rham cohomology via
pullback will commonly  be denoted by $w$ and $w_s$ 
rather than by the more strictly correct (but more
notationally cumbersome) $w^*$ and $w_s^*$. 

 

 

\medskip\noindent
{\em The $U_p$ operator}. 
 The Hecke operator  $U_p$ on $X_s$ is realised by a correspondence 
  of bi-degree $p$, which   
 is given on geometric points by the rule 
\begin{equation}
\label{eqn:defUp}
 U_p(A,i_N,i_p) = \sum_{\varphi:A\lra A'} (A',\varphi \circ i_N,\varphi \circ i_p),
 \end{equation}
where  the sum is taken over the $p$ distinct isogenies $\varphi:A\lra A'$ of degree $p$
whose kernel is not equal to $p^{s-1} \cdot \mathrm{Im}(i_p)$. 
The Hecke operator $U_p$ acts on modular forms of weight two and its action on
differentials of the form $f(\tau)d\tau = \sum_{n\ge 1} a_n q^n \frac{dq}{q}$ 
 is given by
the familiar rule
\begin{equation}
\label{eqn:Up-on-differentials}
U_p(f(\tau) d\tau) = \frac{1}{p}\sum_{j=0}^{p-1} f\left(\frac{t+j}{p}\right) d\tau = 
\sum_{n \ge 1} a_{np} q^n.
\end{equation}
 Note that while  $U_p$ commutes with the diamond operators, it 
 does not commute with its adjoint $U_p^{\ad}$ relative to the Poincar\'e
 pairings $\langle \ , \ \rangle_s$, which is given by the formula 
 $$ U_p^{\ad} = w_{s} U_p w_{s}^{-1}.$$ 

A modular form (or cohomology class)  $\phi$ on $X_s$
  is said to be 
{\em ordinary} if there exists a polynomial $P(T)\in \bar\Q_p[T]$ such
 that $P(U_p)(\phi)=0$ and $P(0)$ is a $p$-adic unit.
If instead $\phi$ has a similar property relative to $U_p^*$, it is designated as {\em anti-ordinary}.

The ordinary and anti-ordinary projectors, which play a key role in the constructions of this article,
are defined as
\begin{equation}
\label{eqn:def-eord-eordstar}
 e_\ord := \lim_{n \rightarrow\infty} U_p^{n!}, \qquad
e_\ord^* := \lim_{n\rightarrow \infty} (U_p^*)^{n!}.
\end{equation}





\medskip\noindent
{\em Projective systems}:
The curves $X_s$  form projective systems in two different ways,
 relative to  the   collections of degeneracy maps 
\begin{equation}
\label{eqn:degeneracy}
\varpi_{1}, \varpi_2: X_{s+1} \ra X_s, \qquad \left\{\begin{array}{l}
 \varpi_{1}(A,i_N,i_p) = (A,i_N,p\cdot i_p),  \\
 \varpi_2(A,i_N,i_p) = (A, i_N, i_p)/C,  \quad C:= i_p(\mu_p), \end{array} \right.
\end{equation}
where the quotient of the triple $(A,i_N,i_p)$ by  the subgroup scheme $C$ is to be understood in the obvious sense.
The degeneracy maps
$\varpi_{1}$ and $\varpi_2$ are
 of degree $p^{2}$ if $s\ge 1$, and of degree $(p^2-1)$ if $s=0$.
  With respect to the analytic uniformisation by the upper-half plane, the map $\varpi_{1}$  corresponds to the natural map $\Gamma_{s+1}\cdot \tau \mapsto \Gamma_{s}\cdot \tau$, while
  $\varpi_2$ corresponds to  $\Gamma_{s+1}\cdot \tau \mapsto \Gamma_{s}\cdot (p\tau)$.
Both maps commute  with the good Hecke operators and the diamond operators,
and factor through the natural projection $\mu: X_{s+1} \lra X_s^{\flat}$, where 
 $X_s^\flat$ denotes the modular curve associated to $\Gamma_1(Np^s)\cap \Gamma_0(p^{s+1})$.
More precisely,  they  can be written as $\varpi_1 = \pi_1 \circ \mu$, 
$\varpi_2 = \pi_2\circ \mu$, and fit into 
the diagram below in which the vertical
map is a Galois cover (with Galois group $\Z/p\Z$, generated by suitable diamond operators) and
the two horizontal maps are non-Galois morphisms of degree $p$:
\begin{equation}
\label{eqn:factoring-the-maps}
 \xymatrix{
X_{s+1} \ar[d]_\mu \ar[dr]^{\varpi_1,\varpi_2}  &  \\
X_s^{\flat} \ar[r]_{\pi_1,\pi_2}  & X_s. }
\end{equation}
The Hecke operator $U_p$ acting on $\Omega_1(X_s)$ or on $H^1_{\dR}(X_s)$ 
is described by the standard formula
\begin{equation}
\label{eqn:def-Up}
 U_p = (\pi_2)_* \pi_1^*,
 \end{equation}
as can be verified directly from the formula for $U_p$ in terms of the coordinate $\tau$ given 
in \eqref{eqn:Up-on-differentials} (cf.~also, for example, \cite[Ex.\,7.9.3]{diamond-shurman}). It follows from this that
\begin{equation}
\label{eq-corr}
(\varpi_2)_* \varpi_1^* =  (\pi_2)_* \mu_* \mu^*  \pi_1^*   = p U_p.
\end{equation}
The  endomorphism in \eqref{eq-corr}
 is   the diagonal arrow in the  commutative diagram:
$$ 
\xymatrix{
H^1_{\dR}(X_s) \ar[r]^{\varpi_1^*} \ar[dr]_{p U_p} \ar[d]_{(\varpi_2)_*}  & H^1_{\dR}(X_{s+1}) \ar[d]^{(\varpi_2)_*} \\
H^1_{\dR}(X_{s-1}) \ar[r]_{\varpi_1^*} & H^1_{\dR}(X_s)},
$$
and it is apparent from this description that
$\pi_1^*$, $\varpi_1^*$,  $(\pi_2)_*$ and $(\varpi_2)_*$ 
commute with $U_p$, while
$\pi_2^*$, $\varpi_2^*$,  $(\pi_1)_*$ and $(\varpi_1)_*$ 
commute with the adjoint $U_p^{\ad}$. 
In particular, the Hecke operator $U_p$ acts naturally on the inverse limit 
of the (\'etale or de Rham) cohomology groups $H^1(X_s)$ taken relative to the maps
$(\varpi_2)_*$. Whenever inverse limits  of $H^1(X_s)$ are discussed, it should  always be understood that the  transition 
maps are given by the pushforward maps $(\varpi_2)_*$. 

 
 \medskip\noindent
 {\em Group ring-valued pairings}. 
 The action of the diamond operators on $X_s$ can be combined with Poincar\'e duality
  to
 define  group-ring valued pairings
\begin{equation}
\label{eqn:pairing-Gammas}
\langle \ , \ \rangle_{\Gamma_s} : H^1_{\et}(\bar X_s,\Z_p) \times H^1_{\et}(\bar X_s,\Z_p) \lra \Z_p(-1)[\Gamma_s],  \quad
 H^1_{\dR}(X_s/\Q) \times H^1_{\dR}(X_s/\Q) \lra  \Q[\Gamma_s] 
\end{equation}
 by the formula
 $$ \langle a, b \rangle_{\Gamma_s} = \sum_{\sigma\in \Gamma_s} \langle a^\sigma, b\rangle_s \cdot \sigma^{-1} 
 = \sum_{\sigma\in \Gamma_s} \langle a, b^{\sigma} \rangle_s \cdot \sigma.$$
 As before, the same symbols are reserved for the pairings arising from Poincar\'e duality on 
 de Rham or \'etale cohomology, leaving it to the context to determine which is being used.
The pairings $\langle \ , \rangle_{\Gamma_s}$ are $\Lambda_s$-linear  and anti-linear (with respect to the involution sending a group-like element 
to its inverse) 
in the first and second  argument
respectively.  

The pairings $\langle \ , \rangle_{\Gamma_s}$ and $\langle \ , \rangle_s$ are related by the formula, 
valid for all characters $\epsilon$ of $\Gamma_s$,
\begin{equation}
\label{eqn:specialise-pairing}
 \epsilon(\langle a, b \rangle_{\Gamma_s}) = \langle \theta_\epsilon a, b\rangle_s = 
\langle a, \theta_{\epsilon^{-1}} b \rangle _s = d_s^{-1}  \langle \theta_\epsilon a, \theta_{\epsilon^{-1}} b\rangle_s,
\end{equation}
where 
\begin{equation}
\label{eqn:def-theta-epsilon}
\theta_\epsilon := \sum_{\sigma\in \Gamma_s} \epsilon(\sigma) \cdot \sigma^{-1} \in \C_p[\Gamma_s].
\end{equation}
The first identity in \eqref{eqn:specialise-pairing}
 is a direct consequence of the definitions, while the second follows from the identity
 $\theta_\epsilon^2 = d_s \theta_\epsilon$. A convenient modification of the $\langle  \ , \ \rangle_{\Gamma_s}$ 
 pairings (in both their \'etale and de Rham incarnations) is given by setting
\begin{equation}
\label{eqn:twisted-pairing-Gammas}
 [a, b]_{\Gamma_s} := \langle a, w  w_{s} \cdot U_p^s \cdot b\rangle_{\Gamma_s}.
\end{equation}
All the Hecke operators on $X_s$, including $U_p$, are self-adjoint relative to this
 modified pairing, which is also $\Lambda_s$-linear relative to both its arguments.
 In addition, the pairings  $[\ ,\ ]_{\Gamma_s}$ have
   the   virtue of being compatible with the maps $\varpi_{2*}$ 
 and    the ring homomorphisms
$$p_{s+1}:\C_p[\Gamma_{s+1}] \lra \C_p[\Gamma_s]$$
induced from the  natural homomorphism $\Gamma_{s+1}\lra \Gamma_s$, in the following sense:
\begin{lemma}
\label{lemma:Gamma-s-compat}
For either $\Box\in\{ \et,\dR\}$, the diagram below commutes:
$$ \xymatrix{
H^1_{\Box}(X_{s+1}) \times H^1_{\Box}(X_{s+1}) \ar[rr]^-{[\ ,\ ]_{\Gamma_{s+1}}} \ar[d]_{\varpi_{2*}\times\varpi_{2*}} 
& & \C_p[\Gamma_{s+1}] \ar[d]_{p_{s+1}} \\
H^1_{\Box}(X_{s}) \times H^1_{\Box}(X_{s}) \ar[rr]^-{[\ ,\ ]_{\Gamma_s}} & & \C_p[\Gamma_{s}]     }
$$
\end{lemma}
\begin{proof} 
This  follows from a direct  calculation (in which the symbol $\langle\ ,\ \rangle_{X_s^\flat}$ 
refers to the Poincar\'e pairing on the curve $X_s^\flat$):
\begin{align*} 
p_{s+1}([ \eta_{s+1}, \phi_{s+1} ]_{\Gamma_{s+1}}) &=    
\sum_{\sigma\in\Gamma_s} \langle \sigma \mu_* \eta_{s+1}, \mu_*  w w_{s+1} U_p^{s+1} \phi_{s+1} \rangle_{X_s^\flat} \cdot \sigma^{-1}   \\
&=  
\sum_{\sigma\in\Gamma_s} \langle \sigma \mu_* \eta_{s+1},  w w_{s+1} U_p^{s}  \pi_1^* \pi_{2*} \mu_*\phi_{s+1} \rangle_{X_s^\flat} \cdot \sigma^{-1}  \\
&=  
\sum_{\sigma\in\Gamma_s} \langle \sigma \mu_* \eta_{s+1},  \pi_2^* w w_{s} U_p^{s}  \phi_{s} \rangle_{X_s^\flat} \cdot \sigma^{-1}   =    
\sum_{\sigma\in\Gamma_s} \langle \sigma   \pi_{2*}\mu_* \eta_{s+1},  w  w_{s} U_p^{s}  \phi_{s} \rangle_{X_s}  \cdot \sigma^{-1}\\
&=    
\sum_{\sigma\in\Gamma_s} \langle \sigma   \eta_{s},   w  w_{s} U_p^{s}  \phi_{s} \rangle_{X_s}   \cdot \sigma^{-1}
=
[ \eta_{s}, \phi_{s} ]_{\Gamma_{s}} .
\end{align*}
\end{proof}
 Define $$H^1_{\Box}(X_\infty) :=\invlim_s H^1_{\Box}(X_s),$$
 the inverse limit being taken relative to the maps $(\varpi_2)_*$.
  Lemma \ref{lemma:Gamma-s-compat}
 shows that the pairings $[\ , \ ]_{\Gamma_s}$ can be   packaged into  
$\Lambda$-adic pairings with values in the appropriate completed group rings 
\begin{align}
\label{eqn:ohta-pairing-deRham}
 [\ , \ ]_{\Gamma}:  & H^1_{\dR}(X_\infty) \times H^1_{\dR}(X_\infty) \lra \C_p[[\Gamma]], \\
 \label{eqn:ohta-pairing-etale}
 [\ , \ ]_{\Gamma}:  & H^1_{\et}(X_\infty,\Z_p) \times H^1_{\et}(X_\infty,\Z_p) \lra \Z_p[[\Gamma]](-1).
 \end{align}
 The \'etale incarnation of $[\ ,\ ]_{\Gamma}$ is even $G_\Q$-equivariant, after endowing 
 $\Z_p[[\Gamma]]$ with the tautological $G_\Q$-action whereby $\sigma\in G_\Q$ 
 acts as multiplication by the group-like element $\epsilon_{\cyc}(\sigma) \in \Z_p^\times$.
 

























\subsection{Twisted diagonal cycles}
\label{sec:twisted-diagonal}
For any   integer  $s \geq 1$ define
$$
W_{s,s} := X_0(Np) \times X_1(Np^s) \times X_1(Np^s) = X_0(Np) \times X_s \times X_s.
$$
This three-fold is equipped with a natural  action of the group 
$G_s^{(N)} \times G_s^{(N)}$    
via the diamond  operators acting on the second and third factors. 
 Let 
 $$\delta:X_s  \lra X_s\times X_s\times X_s, \qquad X_s^\flat \lra X_s^\flat \times X_s^\flat\times X_s^\flat$$
 denote the natural diagonal embeddings.



Fix a system $\{\zeta_s\}$ of  compatible $p^s$-th roots of unity and set 
\begin{equation}
 \label{eqn:def-Deltaf}
 \Deltaf_{s,s}  :=  
%(\pi_1^{s-1} \circ j_s   \circ w_{s}, \id ,w_{s}) (X_s) 
(j_1 \circ \varpi_{2}^{s-1} \circ w_{s}, \id ,w_{s})_* \delta_* (X_s) \in \CC^2(W_{s,s})(\Q(\zeta_s)),
 \end{equation}
 where $\varpi_{2}^{s-1} := \varpi_2 \circ \cdots \circ \varpi_2: X_s \ra X_1$ and $j_1: X_1 \ra X_0(Np)$  are the maps introduced in  \eqref{eqn:degeneracy}  and 
 \eqref{eqn:proj-x1-x0} respectively.
This codimension two cycle is just a  copy of  the curve $X_s$ embedded diagonally 
in the three-fold  product $X_s^3$, 
but with a twist by $w_{s}$  in the first and  third  factors, projected  to $W_{s,s}$ in the natural way. 
It is the twist by $w_{s}$  which causes  $\Deltaf_{s,s}$ 
to only be defined over
$\Q(\zeta_s)$ in general. 
The cycle $\Deltaf_{s,s}$ will be referred to as  the {\em twisted diagonal}  of level $s$. Let 
$$ 
(\id, U_p , \id): \CC^2(W_{s,s})\lra \CC^2(W_{s,s})
$$
be the endomorphism induced by the correspondence $U_p$  
acting on the middle factor of  the triple product $W_{s,s}$ and whose action on cycles
is spelled out in the proof of  Proposition \ref{Up-compf} 
below.  The correspondence $(\id,U_p,\id)$ is realised as a copy of
 $X_0(Np) \times X_s^\flat \times X_s$ embedded in $W_{s,s}^2$ via the map
 $$  (\id_1, \pi_1,\id_3, \id_1, \pi_2,\id_3): X_0(Np) \times X_s^\flat \times X_s \lra 
 (X_0(Np) \times X_s\times X_s) \times (X_0(Np) \times X_s \times X_s),$$
 where $\id_1$ and $\id_3$ refer to the identity maps on the first and third factors
  respectively in the direct
 product, and $\pi_1, \pi_2: X_s^\flat\lra X_s$ are defined on the second factor.
 A direct calculation 
 using the definition of the action of correspondences shows that 
 \begin{equation}
 \label{eqn:Up-pi1-pi2}
  (\id,U_p,\id)_*\delta_*(X_s) = (\pi_1,\pi_2,\pi_1)_* \delta_*(X_s^\flat), \qquad 
   (\id,U_p^*,\id)_*\delta_*(X_s) = (\pi_2,\pi_1,\pi_2)_* \delta_*(X_s^\flat).
  \end{equation}
 
 
The next proposition  studies the compatibilities  of  the cycles 
$\Deltaf_{s,s} $ for varying $s$ 
 under the maps   
\begin{equation}
\label{eqn:pi-ineq}
% \varpi_{11} = (\id, \ \varpi_1, \ \varpi_1), \quad
\varpi_{22}= (\id, \ \varpi_2, \ \varpi_2): W_{s+1,s+1}\lra W_{s,s}, \qquad  
s\geq 1
\end{equation}
induced by the degeneracy maps of \eqref{eqn:degeneracy}.  
%Denote by $\varpi_{22,*}$ the proper pushforward map on cycles induced by $\varpi_{22}$. 


\begin{proposition}
\label{Up-compf} 
For all $s \ge 1$, $\quad \varpi_{22,*}(\Deltaf_{s+1,s+1})  =
 p   \cdot (\id, U_p, \id)_*(\Deltaf_{s,s}) $ in  $ \CC^2(W_{s,s})$.
\end{proposition} 
\begin{proof}
The commutativity of the  diagram 
$$ 
\xymatrixcolsep{0.5in}
\xymatrix{ 
X_{s+1} \ar[r]^-\delta \ar[d]_\mu & X_{s+1}\times X_{s+1}\times X_{s+1} \ar[d]_{(\mu,\mu,\mu)} \ar[rr]^-{(j_1 \varpi_2^s w_{s+1},  \ \id, \  w_{s+1})} 
& & X_0(Np) \times X_{s+1} \times X_{s+1} \ar[d]^{(\id,\mu,\mu)} \\
X_s^\flat \ar[r]^-{\delta}  & X_s^\flat \times X_s^\flat\times X_s^\flat 
\ar[d]_{(\pi_1,\pi_2,\pi_1)}  \ar[rr]^-{(j_1 \varpi_2^{s-1} \pi_2 w_{s+1},  \ \Id,  \ w_{s+1})} & &
X_0(Np) \times X_s^\flat \times X_s^\flat  \ar[d]^{(\id,\pi_2,\pi_2)}  \\
  & X_s\times X_s\times X_s \ar[rr]^-{(j_1 \varpi_2^{s-1} w_{s},\ \id, \  w_{s})} & & X_0(Np)\times X_s\times X_s, 
}
$$
in which the composition of the rightmost vertical arrows is the map $\varpi_{22}$,
implies that
\begin{eqnarray*}
\varpi_{22*}(\Deltaf_{s+1,s+1}) &=& (\id,\pi_2,\pi_2)_* \circ  (\id,\mu,\mu)_* \circ (j_1\varpi_2^s w_{s+1},\id, w_{s+1})_* \circ \delta_*(X_{s+1}) \\
&=& (j_1 \varpi_2^{s-1}  w_{s},\id, w_{s})_* \circ (\pi_1,\pi_2,\pi_1)_* \circ \delta_* \circ \mu_*(X_{s+1}) \\
&=& p \cdot (j_1 \varpi_2^{s-1} w_{s},\id,  w_{s})_* \circ (\pi_1,\pi_2,\pi_1)_* \circ \delta_*  (X_{s}^\flat) \\
&=& p \cdot (j_1 \varpi_2^{s-1}  w_{s},\id, w_{s})_* \circ   (\id,U_p,\id)_*\delta_*(X_s),  
\end{eqnarray*}
where the last equality follows from \eqref{eqn:Up-pi1-pi2}. 
It follows that 
\begin{eqnarray*}
\varpi_{22*}(\Deltaf_{s+1,s+1}) &=& p \cdot (\id,U_p,\id)_*
 (j_1 \varpi_2^{s-1}  w_{s},\id, w_{s})_* \circ \delta_*(X_s) \\  
&=& p  \cdot (\id,U_p,\id)_* (\Deltaf_{s,s}), 
\end{eqnarray*}
as was to be shown.
\end{proof}

Proposition \ref{Up-compf} shows that, after  {\em formally } regularizing the 
cycles $\Deltaf_{s,s}$ by the rule 
 \begin{equation}
 \label{eqn:formally-regularising}
 \Deltaf_{s,s}^{\reg} :=\frac{1}{p^s} \cdot (\id,  U_p, \id)^{-s} (\Deltaf_{s,s}),
 \end{equation}
% (or, more accurately, their images under the \'etale Abel-Jacobi map)
  the family $\{ {\Deltaf}_{s,s}^{\reg}\}$ 
  becomes part of  a {\em projective system}
   under the push-forward maps  $\varpi_{22,*}$. 
%Anticipating   the discussion of  \S \ref{sec:limit},  the factor $(1\otimes   U_p^{-s} \otimes 1 )$ can be 
 %  dealt with by 
 %  projecting $\Deltaf_{s,s}$ to a group on which 
  % $(1\otimes  U_p \otimes 1)$ acts invertibly, namely,  the 
  %image of $\CH^2(W_{s,s})_0$ under the $p$-adic \'etale  Abel-Jacobi map  
  %of \eqref{AJet} below,
  % composed with Hida's {\em ordinary projector}.
  The factor $\frac{1}{p^s}$  arising in 
  \eqref{eqn:formally-regularising}
  is problematic, 
 introducing unwanted denominators which
%in the family of 
 % Galois cohomology classes obtained by piecing together the images
  %of $\Delta_{s,s}^{\reg}$ under the $p$-adic \'etale Abel-Jacobi map.
present a genuine obstruction to piecing the cycles $\Deltaf_{s,s}$
  into a ``two-variable $p$-adic family". This is what
  motivates the passage  to  a suitable quotient of $W_{s,s}$  by a group of 
   automorphisms of (essentially)  $p$-power order, which will now be explained.

Given $s\geq 1$, let $D_s$ denote the ``diagonal at $N$" and
 ``anti-diagonal at $p$" subgroup of $G_s^{(N)}\times G_s^{(N)}$ 
 consisting of elements of the form 
  $(\langle a; b\rangle,\langle a; b^{-1}\rangle)$ with $(a,b)\in (\Z/N\Z)^\times\times (\Z/p^s\Z)^\times$, and let 
$$
\pr_s: W_{s,s} \lra W_{s}  := W_{s,s}/D_s
$$
be the natural projection from $W_{s,s}$ to its quotient by the action of this 
group.

By \eqref{eqn:commutation-w},
the action of $D_s$ on $W_{s,s}$ preserves the cycle $\Deltaf_{s,s}$. Identifying
 $\Deltaf_{s,s}$ with the curve $X_s$, 
 the element $(\langle a; b\rangle, \langle a;b^{-1}\rangle) \in D_s$ acts as the diamond operator $\langle a;b\rangle$ on this curve. 
Let $\Deltaf_{s} \in \CC^2(W_{s})(\Q(\zeta_s))$
%%%%IMPORTANT: Define , \qquad\Delta_{s} \in \CC^2(W_{s})(\Q(\zeta_s)) 
be the natural image of $\Deltaf_{s,s}$  
 in the quotient $W_{s}$. Since $\pr_s$ induces a morphism of degree $d_s := \frac{1}{2} \varphi(N)(p-1)p^{s-1}$
 from
 $\Deltaf_{s,s}$ to
$\Deltaf_{s}$, the latter cycle is defined by the relation
\begin{equation}
\label{eqn:def-Deltafs}
 (\pr_s)_*(\Deltaf_{s,s}) =: d_s \cdot (\Deltaf_{s}) \mbox{ in } \CC^2(W_{s})(\Q(\zeta_s)).
 \end{equation}
%%%%IMPORTANT:  and likewise for $\Delta_{s}$.
Note that the curve $\Deltaf_{s}$ 
is isomorphic to an embedded copy of $X_0(Np^s)$ in the three-fold $W_{s}$,
but that the corresponding closed immersion $X_0(Np^s)\simeq \Deltaf_{s}\hookrightarrow W_{s}$
is only defined over $\Q(\zeta_s)$. 

By a slight abuse of notation, let 
\begin{equation}
\label{eqn:proj-Wrs}
 \varpi_{11}, \varpi_{22}: W_{s+1}=W_{s+1,s+1}/D_{s+1} 
 \lra W_{s}=W_{s,s}/D_s
\end{equation}
denote the natural projection map ``from level $ s+1$ to level $s$" induced from the maps 
$\varpi_{11}$ and $\varpi_{22}$ above, by passing to the quotient via $\pr_{s+1} $ and 
$\pr_s$.  

\begin{proposition}
\label{prop:Up-compat}
For all $s\ge 1$, $\quad \varpi_{22,*}(\Deltaf_{s+1})  = (\id,  U_p, \id)_*(\Deltaf_{s}).$
\end{proposition}
\begin{proof}   
 Proposition \ref{Up-compf} asserts 
 that $\varpi_{22,*}(\Deltaf_{s+1,s+1}) = 
 p \cdot  (\id,  U_p, \id)_*(\Deltaf_{s,s})$ in $\CC^2(W_{s,s}).$
 Applying the pushforward map induced by 
 $\pr_s:W_{s,s}\lra W_{s}$ to this identity, using the fact that
$\pr_s \circ \,\varpi_{22} = \varpi_{22} \circ \pr_{s+1},$ 
and that $\pr_s$ also commutes with $(\id, U_p, \id)$, one obtains
$$\varpi_{22,*} \pr_{s+1,*}(\Deltaf_{s+1,s+1}) =  p \cdot(\id,   U_p , \id)  \pr_{s,*}(\Deltaf_{s,s}).$$
The sought-for equality 
then follows from 
 \eqref{eqn:def-Deltafs}, because $\CC^2(W_s)$ is torsion free.
  \end{proof}
 It would also have been possible to adapt the proof of 
Proposition \ref{Up-compf} to the cycles $\Deltaf_s$ on the quotients $W_s$ to show directly
that Proposition \ref{prop:Up-compat} holds at the level of cycles. 
In reducing Proposition \ref{prop:Up-compat} to Proposition \ref{Up-compf},  it was expedient to work with the groups $\CC^2(W_{s,s})$ and $\CC^2(W_{s})$ of codimension $2$ cycles in $W_{s,s}$ and $W_s$, rather than the Chow groups $\CH^2(W_{s,s})$ and $\CH^2(W_s)$ of rational equivalence classes, because the latter are not necessarily torsion free. This is no longer necessary, and from now on   $\Deltaf_{s,s}$ and $\Deltaf_s$ will be identified with their classes in $\CH^2(W_{s,s})$ and $\CH^2(W_{s})$, respectively. 

These cycle classes can be made null-homologous
by applying  to them a suitable correspondence, denoted 
\begin{equation}
\label{eqn:def-epsilona}
\varepsilon_{s,s}:= (\varepsilon, \varepsilon_s, \varepsilon_s): 
X_0(Np) \times X_s \times X_s,
\end{equation}
where
\begin{itemize} 
\item
$\varepsilon$ is a correspondence on $X_0(Np)$ with coefficients in $\Z_p$ which annihilates 
$H^0(X_0(Np))$ and $H^2(X_0(Np))$ but fixes the $f$-isotypic subspace
$H^1(X_0(Np))[f]$ of the introduction (defined as an eigenspace only for the good Hecke operators 
$T_\ell$ with $\ell\nmid Np$).
To fix ideas, recall that the prime  $p$ is assumed to be a non-Eisenstein prime for $f$, meaning that $f$ is not congruent to an Eisenstein series modulo $p$.
 There is therefore an  auxiliary prime  $\ell\nmid Np$  for which  $\ell+1-a_\ell(f)$ lies in $\Z_p^\times$.  Set
$$ \varepsilon :=  (\ell+1-T_\ell)/(\ell+1-a_\ell(f)).$$
\item $\varepsilon_s$ is the correspondence on $X_s$ defined by setting $\varepsilon_{s}  := 2^{-1} \cdot (1- \langle -1;1\rangle)$,
where  the standard convention has been adopted  of
  identifying a morphism  $\varphi:V\lra W$ between varieties $V$ and $W$
  with the  correspondence on $V\times W$   associated to its graph. 
\end{itemize}
The {\em modified twisted diagonal cycle} of level $s$  is  then  defined by
\begin{equation}
 \label{eqn:def-Delta}
\Delta_{s,s}  :=  \varepsilon_{s,s}(\Deltaf_{s,s}) \in \CH^2(W_{s,s}).
\end{equation}
\begin{proposition}
\label{prop:null-homologous}
For all $s \geq 1$, the class of $\Delta_{s,s}$ lies in $\CH^2(W_{s,s})_0$.
 \end{proposition}
  \begin{proof} In order to lighten the notations in the proof,
  write $H^i(V)$ as a shorthand for the \'etale cohomology
  $H^i_{\et}(\overline{V},\Z_p)$ of $V$ with coefficients in $\Z_p$,
  and write
   $W_{s,s} = C_1 \times C_2 \times C_3$.
   This variety is a product of curves, whose integral cohomology is torsion-free.
  By the K\"unneth decomposition theorem (cf.~\cite[Thm.~22.4]{milne})
 $H^4(W_{s,s})$ can therefore be 
expressed 
as a sum of three terms   
\begin{equation*}
H^2(C_1)\otimes H^2(C_2\times C_3) \  + \   H^2(C_2)\otimes H^2(C_1 \times C_3)   \ +  \ H^2(C_3)\otimes H^2(C_1\times C_2).
\end{equation*}
By  \eqref{eqn:def-Delta}, $\cl(\Delta_{s,s}) = \varepsilon_{s,s}(\cl(\Deltaf_{s,s})).$ The correspondence $\varepsilon$ annihilates the first term in the above sum, while 
 $\varepsilon_s $   annihilates the second and third
  term,
%\eqref{eqn:kunneth-decomp-null}, 
since it annihilates
  $H^2(X_s)$.
 It follows that $\varepsilon_{s,s}$ annihilates the target of the cycle class map and 
 hence {\em a fortiori} that $\Delta_{s,s}$ is null-homologous.
\end{proof}
 \begin{remark}
 The element $\varepsilon_{s,s}$ has the virtue of annihilating $H^4_{\dR}(W_{s,s})$ while
 fixing much of the interesting part of the middle cohomology $H^3_{\dR}(W_{s,s})$. 
 More precisely, if  $\eta_\testf$ is any class in $H^1_{\dR}(X_0(Np))[f]$, and
  $\eta_\testg \in H^1_{\dR}(X_s)[g]$ and 
  $ \eta_\testh \in H^1_{\dR}(X_s)[h]$  are classes in the isotypic subspaces
   attached to eigenforms  $g$ and $h$ with nebentypus characters whose prime-to-$p$
  parts are  the odd Dirichlet characters $\chi$ and $\chi^{-1}$ respectively,
   then $\varepsilon_{s,s}(\eta_\testf\otimes \eta_\testg\otimes\eta_\testh) =  \eta_\testf \otimes \eta_\testg\otimes \eta_\testh$.
   \end{remark}
 
 

Recall that the group $G_s^{(N)}\times G_s^{(N)}$  acts freely on $W_{s,s}$,
and that this action descends to an action  
on $W_{s}$  factoring through the quotient  $(G_s^{(N)} \times G_s^{(N)})/D_s
\simeq G_s^{(N)}$.  More generally,   the linear endomorphism of 
$\CH^*(W_{s,s})$ induced by any element $\varepsilon \in \Z[G_s^{(N)}\times G_s^{(N)}]$  
descends to a linear map on $\CH^*(W_{s})$ which shall be  denoted by the same symbol.
With these notations,  letting $\Delta_{s}$ denote the natural image of $\Delta_{s,s}$ in the quotient $W_{s}$,
the classes of the cycles $\Delta_{s}$ and $\Deltaf_{s}$  
are related by
\begin{equation}
\label{def-Deltas}
\Delta_{s} = \varepsilon_{s,s}(\Deltaf_{s}) \in \CH^2(W_{s})_0(\Q(\zeta_s)).
\end{equation}

Moreover, similarly as above it again holds that
%\begin{equation}
%\label{eqn:def-Deltas}
% (\pr_s)_*(\Delta_{s,s}) = d_s \cdot (\Delta_{s}) \mbox{ in } \CH^2(W_{s})_0(\Q(\zeta_s)) \qquad \mbox{ and }
% \end{equation}
\begin{equation}
\label{Up-comp} 
(\pr_s)_*(\Delta_{s,s}) = d_s \cdot \Delta_{s}  \quad \mbox{ and } \quad \varpi_{22,*}(\Delta_{s+1})  = (\id,U_p,\id) _*(\Delta_{s}).
\end{equation} 

The first equality in \eqref{Up-comp} follows exactly as in \eqref{eqn:def-Deltafs}, 
while the second is a consequence of Proposition \ref{prop:Up-compat} because 
the correspondences $\varepsilon_{s+1,s+1}$ and  $\varepsilon_{s,s}$ commute with the diamond operators, with
  $(\id, U_p , \id)$, and   with the projections $\varpi_{22,*}$ in the sense that $\varepsilon_{s,s} \circ  \varpi_{22,*} =  \varpi_{22,*} \circ \varepsilon_{s+1,s+1}.$
 
 
The cycles $\Delta_{s,s}$ and $\Delta_s$
 are only defined over $\Q(\zeta_s)$; the following proposition describes the action of the Galois group $\Gal(\Q(\zeta_s)/\Q) = (\Z/p^s\Z)^\times$ on these cycles.

\begin{proposition}
\label{prop:Delta-Q-zeta-bis}
For all $\sigma\in G_\Q$, $\sigma(\Delta_{s,s}) = (\id,\id, \langle 1;\epsilon^{-1}_s(\sigma)\rangle) (\Delta_{s,s}).$
\end{proposition}
\begin{proof}
Equation  \eqref{eqn:galois-w}  
 implies that
\begin{equation}
\label{eqn:delta-flat-1.3} 
\sigma(\Deltaf_{s,s}) =(\id,\id, \langle 1;\epsilon^{-1}_s(\sigma)\rangle) (\Deltaf_{s,s}).
\end{equation}
Note the the correspondence $\varepsilon_{s,s}$ is defined over $\Q$ and hence commutes with the 
action of $G_\Q$ on cycles. It also commutes with the diamond operators and hence with the morphism
$(\id,\id, \langle 1;\epsilon^{-1}_s(\sigma)\rangle)$.
Applying it to  \eqref{eqn:delta-flat-1.3} and invoking
 equation  \eqref{eqn:def-Delta}, one obtains that $\sigma(\Delta_{s,s}) = (\id,\id, \langle 1;\epsilon^{-1}_s(\sigma)\rangle) (\Delta_{s,s}).$ Proposition 
 \ref{prop:Delta-Q-zeta-bis}
follows because $\pr_s$ is defined over $\Q$.
\end{proof}

In light of Proposition \ref{prop:Delta-Q-zeta-bis}, the varieties
 $W^\dag_{s,s}$ and $W^\dag_s$ are defined to be the twists of $W_{s,s}$  and $W_s$
respectively  by the cocycle  
$\sigma \in G_\Q \mapsto \langle \epsilon_s^{-1}\rangle(\sigma) = (1,1,\langle 1;\epsilon_s^{-1}(\sigma)\rangle)$. Note that $W^\dag_{s,s}=X_0(Np) \times X_s \times X^\dag_s$ and $W^\dag_s = W^\dag_{s,s}/D^\dag_{N,p}$, where 
$D^\dag_{N,p}$ is the subgroup of diamond operators acting ``diagonally at $N$" and 
``anti-diagonally at $p$" on $W^\dag_{s,s}$, defined in a similar way as 
 for $D_s$. 

 Thanks to Proposition \ref{prop:Delta-Q-zeta-bis}, the cycle $\Delta_{s}$ can be viewed as a 
 codimension two cycle in $W^\dag_s$ with coefficients in $\Z_p$ which is defined over $\Q$, and
 whose associated class in the Chow group
 gives rise to  a canonical element  
\begin{equation}
\Delta_s \in \CH^2(W^\dag_s)_0(\Q).
\end{equation}

\subsection{Global cohomology classes}
\label{sec:AJet}
Let
\begin{equation}
\label{AJet} 
\AJ_{\et}: \CH^2(W^\dag_{s})_0(\Q)  \lra H^1(\Q,H^3_{\et}(\bar{W}^\dag_{s},\Z_p)(2)),
\end{equation}
denote the $p$-adic \'etale Abel-Jacobi map arising from \eqref{AJet-general}. 
 Of crucial importance for the results of this paper are the images of the cycles
   $\Delta_{s}$
  under this map: 
\begin{equation}
  \label{eqn:def-kappas1}
 \kappa_{s}^{(1)}  :=  \AJ_{\et}(\Delta_{s}) \in H^1(\Q,
 H^3_{\et}(\bar{W}^\dag_{s},\Z_p)(2)) = H^1\big(\Q, H^3_{\et}(\bar W_{s}, \Z_p)(2 \, \langle \epsilon_s^{-1}\rangle)  \big). 
 \end{equation}


As was already observed, the $\Z_p$-modules $H^3_{\et}(\bar W^\dag_{s,s},\Z_p)$  and $H^3_{\et}(\bar{W}^\dag_{s},\Z_p)$
are equipped with a natural structure of modules over the rings  $\Z_p[G_s^{(N)}\times G_s^{(N)}] $ and 
$\Z_p[(G_s^{(N)}\times G_s^{(N)})/D_s] \simeq  \Z_p[G_s^{(N)}]$ 
  via the action of  the 
diamond operators. 


\begin{lemma}
For every $d\geq 0$, the group $H^d_{\et}(\bar{W}^\dag_{s,s},\Z_p)$ is free over $\Z_p$  and admits a canonical 
direct sum decomposition (the K\"unneth formula):
\begin{equation}
\label{kun}
H^d_{\et}(\bar{W}^\dag_{s,s},\Z_p) = 
\bigoplus_{{\begin{matrix}i+j+k=d\\0\leq i,j,k\leq 2\end{matrix}}} H^i_{\et}(\bar{X}_0(Np),\Z_p)\otimes H^j_{\et}(\bar{X}_{s},\Z_p) \otimes H^k_{\et}(\bar{X}^\dag_{s},\Z_p).
\end{equation}
\end{lemma}
\begin{proof} This follows from e.g.\,Theorem  22.4 of \cite{milne}, in light of the fact that the integral cohomology of smooth projective curves is torsion-free.
\end{proof}

The natural projection $\pr_s: W^\dag_{s,s}\lra W^\dag_{s}$ induces functorial maps between the corresponding \'etale cohomology groups in both directions
$$ \pr_s^*: H^3_{\et}(\bar W^\dag_{s},\Z_p) \lra H^3_{\et}(\bar W^\dag_{s,s},\Z_p)^{D_s},
\qquad
\pr_{s,*}: H^3_{\et}(\bar W^\dag_{s,s},\Z_p)_{D_s} \lra H^3_{\et}(\bar W^\dag_{s},\Z_p),$$
the second map being related  to the first by taking duals and invoking Poincar\'e 
duality.
%In the context of cohomology with $\Q_p$-coefficients these two maps are
These two maps become isomorphisms after tensoring with $\Q_p$, but working integrally requires  a bit more care. 

Recall the correspondence $\varepsilon_{s,s}$ on $W_{s,s}$ and $W_s$ introduced in \eqref{eqn:def-epsilona}. By an abuse of notation, continue to denote with the same symbol the correspondence on $W^\dag_{s,s}$ and $W^\dag_{s}$ defined exactly in the same way, taking into account that both $X_s$ and $X^\dag_s$ are equipped with a canonical action of diamond operators.

\begin{lemma} 
\label{lemma:kill-kernel-prs}
The kernels and cokernels of the
 maps $\pr_s^*$ and  
$\pr_{s,*}$ are  annihilated
by the element $\varepsilon_{s,s}$. 
\end{lemma} 
\begin{proof}
The Hochschild-Serre spectral sequence $H^p(D_s, H^q_{\et}(\bar W^\dag_{s,s},\Z_p)) \Rightarrow H^{p+q}_{\et}(\bar W^\dag_{s},\Z_p)$ shows 
that  $\pr_s^*$ sits in the middle of an
exact sequence
$$ \xymatrix{ 
\mathcal K \ar[r] & H^3_{\et}(\bar W^\dag_{s},\Z_p)   \ar[r]^{\pr_s^*}  & H^3_{\et}(\bar W^\dag_{s,s},\Z_p)^{D_s} \ar[r] & H^2(D_s, H^2_{\et}(\bar W^\dag_{s,s},\Z_p)),  }$$
  where the kernel $\mathcal K$ is canonically a 
   subquotient of
    $H^2(D_s, H^1_{\et}(\bar W^\dag_{s,s},\Z_p)) \oplus H^1(D_s,H^2_{\et}(\bar W^\dag_{s,s},\Z_p)).$ 
 Since  $\varepsilon_{s,s}$  
annihilates each term in the K\"unneth decomposition \eqref{kun} of 
$H_{\et}^d(\bar W^\dag_{s,s},\Z_p)$ when $0\le d \le 2$, the assertion of the lemma for $\pr_s^*$ follows.  The result 
for $\pr_{s,*}$ then    follows by taking duals. 
%$\Z_p$-linear duals.
\end{proof} 

Since the projector $\varepsilon_{s,s}$ of
\eqref{eqn:def-epsilona} is integrally defined over $\Z_p$,
Lemma \ref{lemma:kill-kernel-prs} 
makes it possible to define classes
\begin{equation}
 \label{eqn:from-kappas2-to-kappas3}  \kappa_{s}^{(2)} :=    \varepsilon_{s,s} \pr_{s,*}^{-1}(\varepsilon_{s,s} \kappa_{s}^{(1)}) \in H^1(\Q, H^3_{\et}(\bar W^\dag_{s,s},\Z_p)(2))_{D_s} = H^1(\Q, H^3_{\et}(\bar W_{s,s},\Z_p)(2\, \langle \epsilon_s^{-1}\rangle))_{D_s},
\end{equation}
where the first application of  $\varepsilon_{s,s}$ ensures the existence of the inverse image,
and the second  guarantees its uniqueness. Let 
\begin{equation}
\label{V1}
V_0(Np)= H^1_{\et}(\bar X_0(Np),\Z_p)(1), \qquad V_1(Np^s) = 
 H^1_{\et}(\bar{X}_{s},\Z_p)(1),
\end{equation}
denote the Tate modules of the jacobians of $X_0(Np)$ and $X_s$, 
and let
$$
 V_1(Np^s)^\twist =  H^1_{\et}(\bar{X}_{s}^\twist,\Z_p)(1) =  V_1(Np^s)(-1)(\langle \eps^{-1}_{s}\rangle).
 $$
 Define also
%\begin{equation}\label{V1*}
%V_1(Np^s)^* := H^1_{\et}(\bar{X}^\dag_{s},\Z_p) = V_1(Np^s)(-1)(\langle \eps^{-1}_{s}\rangle),
%\end{equation}
\begin{equation}
  V_{s,s} :=  V_0(Np) \otimes \left(V_1(Np^s) \otimes V_1(Np^s)^\twist\right),  \qquad
  V_s := V_0(Np) \otimes \left(V_1(Np^s) \otimes_{\LLambda_s} V_1(Np^s)^\twist\right).
 \end{equation}
The projection from $H^3_{\et}(\bar{W}^\dag_{s,s},\Z_p)$  
onto the $(1,1,1)$-component in the K\"unneth decomposition
\eqref{kun} induces a $G_\Q$-equivariant map on the $D_s$-coinvariants:
$$ \pr_{111}: H^3_{\et}(\bar{W}^\dag_{s,s},\Z_p)(2)_{D_s} 
\lra   (V_{s,s})_{D_s}  = V_s.$$
  The classes $\kappa_{s}^{(3)}$ are now defined by setting
\begin{equation}
\label{eqn:def-hVrs}
 \kappa_{s}^{(3)} := \pr_{111}(\kappa_{s}^{(2)}) \in 
H^1(\Q,  V_{s}).
\end{equation}

 
By a slight abuse of notation, the symbol $\varpi_{22,*}$ is also used to 
denote the proper push-forward maps 
$\varpi_{22,*}: V_{s+1} \lra  V_{s}$
induced  on \'etale cohomology by the projections $\varpi_{22}$ of 
\eqref{eqn:proj-Wrs}. 
Denote also by  $(1\otimes   U_p\otimes 1)$ the endomorphism acting on
$V_{s}$ via the action of $U_p$ 
on the second factor in \eqref{eqn:def-hVrs}.
\begin{proposition}
\label{prop:compatibility-ddot}
For all $s\ge 1$, $\quad \varpi_{22,*}(\kappa_{s+1}^{(3)}) = (1\otimes U_p\otimes 1) ( \kappa_{s}^{(3)}).$
 \end{proposition}
 \begin{proof} The classes $\kappa_{s}^{(3)}$ arise from the cycles $\Delta_s$ by applying succesively the \'etale Abel-Jacobi map, the inverse of the push-forward map $\pr_{s *}^{-1}$ induced by the projection $\mathrm{pr}_s$, the correspondence $\varepsilon_{s,s}$ constructed using diamond operators and the projection $\mathrm{pr}_{111}$ arising from the K\"unneth decomposition. It is immediate to verify that $\varpi_{22 *}$ commutes with these four maps. Since in addition $\varpi_{22 *}$ commutes with $(\id, U_p, \id)$ as proved in the paragraph following \eqref{eq-corr}, the result follows from \eqref{Up-comp}.
 \end{proof}
 In order to  make the classes $ \kappa_{s}^{(3)}$ fit into families that are compatible under the natural projections, and take values in manageable $\LLambda:=\lim\LLambda_s$-modules
 (locally free of finite rank, say) it is desirable to cut down the modules
 $V_s$ by applying Hida's ordinary projector $e_{\ord}= \lim U_p^{n!}$ to them.
 Write 
    \begin{equation*}
  V^{\ord}_1(Np^s) :=  e_{\ord} V_1(Np^s), \qquad V^{\ord}_1(Np^s)^\twist :=  e_{\ord} V_1(Np^s)^\twist, 
 \end{equation*}
    \begin{equation*}
   V_{s}^{\ord} :=   V_0(Np) \otimes (V^{\ord}_1(Np^s) \otimes_{\LLambda_s}V^{\ord}_1(Np^s)^\twist)
   \end{equation*}
  \begin{equation}
   \label{eqn:from-kappas-to-kappas4} 
  \mbox{  and set }   \qquad \kappa_{s} := (\id, U_p, \id)^{-s} (1\otimes e_\ord\otimes e_\ord)  \kappa_{s}^{(3)}
   \in H^1(\Q, V_s^{\ord}).
   \end{equation}
 Note that \eqref{eqn:from-kappas-to-kappas4} is well-defined because $U_p$ acts invertibly on 
 $V^{\ord}_1(Np^s)$. Define the $\LLambda^{(N)}[G_\Q]$-modules
\begin{equation}
\label{eqn:Hida-infty-ord}
\hV^{\ord}_1(Np^\infty)  := \invlim_s V^{\ord}_1(Np^s), \quad \hV^{\ord}_1(Np^\infty)^\twist := \invlim_s V^{\ord}_1(Np^s)^\twist \quad \mbox{ and }
\end{equation}
\begin{equation} 
\hV_{\infty}^{\ord}  := \invlim_{s} V_s^{\ord} \simeq  V_0(Np) \otimes \left(\hV^{\ord}_1(Np^\infty)\otimes_{\LLambda} \hV^{\ord}_1(Np^\infty)^\twist\right)
\end{equation}
where the inverse limits are taken relative to the   maps induced  from $\varpi_2$ 
and $\varpi_{22}$ by covariant functoriality
on the \'etale cohomology of the towers of curves $X_s$ and of threefolds $W_s$. 

Since $U_p$ commutes with $\varpi_{2*}$, 
% as justified at the end of \S \ref{sec:modular-curves}, 
it
follows from Proposition \ref{prop:compatibility-ddot} that the classes
 $\kappa_{s}$
  form  a system of global classes 
satisfying the crucial compatibility $\varpi_{22,*} (\kappa_{s+1}) = \kappa_{s}$. Hence the classes $\kappa_{s} $ can be pieced together into the $\Lambda$-adic class
 \begin{eqnarray}
 \label{eqn:def-kappa-infty}
 \kappa_{\infty} &:= &  \invlim_{s} \kappa_{s} \in H^1(\Q, \hV_{\infty}^{\ord}).
\end{eqnarray}


The next two sections explain how  the $\LLambda^{(N)}[G_\Q]$-
module
 $\hV_{\infty}^{\ord}$ 
realises  one-parameter
 families of $p$-adic Galois representations interpolating the Kummer self-dual twists of the triple tensor product of the Galois representations associated to $f$, $g$ and $h$, as  $(g,h)$ 
 range  over certain  pairs of   classical specialisations of common weight $\ell$
 of   
 {\em Hida families} of tame level $Np$. The classes $\kappa_{\infty}$  
  likewise will give rise to $p$-adic families of Galois
   cohomology classes with values in these representations. 
   
  
\subsection{Modular forms and Galois representations in families}
\label{sec-Hida}

\subsubsection{Classical modular forms and Galois representations}\label{subsubsec}
Let $S_k(M)_R = S_k(\Gamma_1(M))_R$ denote the space of classical modular forms of weight $k\geq 1$ and level $M\geq 1$ with coefficients in some ring $R$. %with respect to the congruence subgroup $\Gamma_1(R)$. 

If $\chi$ is a Dirichlet character mod $M$, let $S_k(M,\chi)_R$ denote the subspace of $S_k(M)_R$ on which diamond operators act through $\chi$. Let
\begin{equation}\label{modform}
\phi = q+ \sum_{n\geq 2} a_n(\phi)q^n \ \ \in \ \  S_k(N_\phi p^s,\chi)_{\C}
\end{equation}
be a normalized cuspform of weight $k\geq 1$, level $N_\phi p^s$, $s\geq 1$ and character 
$\chi: (\Z/N_\phi p^s\Z)^\times \ra \C^\times$. Here $p$ is taken to be a prime number not dividing $N_\phi$.
Write $\chi=\chi_{0} \chi_p$ as the product of Dirichlet 
characters of conductor dividing $N_\phi$ and $p^s$, respectively. 
  
Assume $\phi$ is an eigenform (with respect to all good Hecke operators $T_\ell$, $\ell \nmid N_\phi p$) and let $K_\phi$ denote the finite extension of $\Q_p$ generated by the fourier coefficients $a_n(\phi)$ of $\phi$.
% together (if $\chi_p=1$) with the roots of the $p$-th Hecke polynomial $x^2-a_p(\phi)x+\chi(p)p^{k-1}$. 
Write $\cO_\phi$ for its valuation ring. 
Eichler-Shimura, Deligne and Serre associated to $\phi$ a two-dimensional representation 
$$
\varrho_\phi: G_\Q \lra \GL(V_\phi)\simeq \GL_2(\cO_\phi) 
$$
which is unramified at all primes not dividing $N_\phi  p$ and satisfies 
 $\Tr(\varrho_\phi(\mathrm{Frob}_\ell)) = a_\ell(\phi)$ for 
 any such $\ell \nmid N_\phi p$, where $\Frob_\ell$ denotes 
 an arithmetic Frobenius element at $\ell$. 

If $\phi$ is ordinary at $p$, then the Hecke 
polynomial $T^2-a_p(\phi)T+\chi(p)p^{k-1}$ has at least one root  
which is a $p$-adic unit. If $k\geq 2$ or $\chi_p\ne 1$, this root is unique. Fix such a root, call it $\alpha_\phi$, and let
 $\psi_{\phi}:G_{\Q_p}\lra \cO_{\phi}^\times$ 
 be  the unramified character such that 
 $\psi_\phi(\mathrm{Frob}_p) = \alpha_p(\phi)\in \cO_\phi^\times$. 
 By \cite[Theorem 2]{W}, the restriction of $V_\phi$ to a 
 decomposition group at $p$ takes the form 
\begin{equation}
  \label{eqn:local-triangular-class} 
{V_{\phi}}_{|G_{\Q_p}}:  \left(
 \begin{array}{cc} \psi_{\phi}^{-1} \chi  \epsilon_{\cyc}^{k-1}  & * \\ 0  & \psi_{\phi}
  \end{array}\right)
  \end{equation}
on a suitable basis. Here $\epsilon_{\cyc}=\invlim_s \epsilon_s$ is the cyclotomic character
introduced in 
\eqref{eqn:epsilon-cyc}. 
\begin{definition}
\label{def:Vfpm}
We let $V_{\phi}^+$ denote the one-dimensional $G_{\Q_p}$-submodule of $V_{\phi}$ on which $G_{\Q_p}$ acts via $\psi_{\phi}^{-1} \chi  \epsilon_{\cyc}^{k-1}$, and write $V_{\phi}^-$ for the unramified quotient on which $G_{\Q_p}$ acts via $\psi_{\phi}$. 
\end{definition}

 When $k\geq 2$, the representation $V_{\phi} $ arises as the $\phi$-isotypic
  component of $H^{k-1}_{\et}(\bar{\cE}_k,\cO_\phi)(k-1)$, 
  where $\cE_k$ is the $(k-1)$-dimensional Kuga-Sato variety 
  fibered over $X_s$. In particular, when $\phi$ has weight
   $2$, $V_\phi$ is a quotient of $V_1(N_\phi p^s)\otimes \cO_\phi$ 
   and  $\varpi_\phi: V_1(N_\phi p^s)\otimes \cO_\phi \ra V_\phi$ denotes the canonical projection. 

If $N$ is any multiple of $N_\phi$,  let $V_{\phi}(Np^s)$ denote the 
$\phi$-isotypical component of $V_1(N p^s)\otimes \cO_\phi$, which is isomorphic 
to a direct sum of several copies of $V_\phi$, and write
 $\varpi_\phi: V_1(N p^s)\otimes \cO_\phi \ra V_\phi(Np^s)$ for the resulting projection. 
 Define $V_{\phi}(Np^s)^+$ and $V_{\phi}(Np^s)^-$ analogously as above, and likewise for $V^{\ord}_1(Np^s)^+$ and $V^{\ord}_1(Np^s)^-$.
  It will also be convenient to introduce the representation
\begin{equation}\label{Vphi*}
V_{\phi}^* := V_{\phi} \otimes \chi_p^{-1}\epsilon_{\cyc}^{1-k}, \quad \quad
 {V^*_{\phi}}_{|G_{\Q_p}}:   \left(
 \begin{array}{cc} \psi_{\phi}^{-1} \chi_{0}    & * \\ 0  & \psi_{\phi}
  \epsilon_{\cyc}^{1-k} \chi_p^{-1}
  \end{array}\right),
\end{equation}
whose restriction to $G_{\Q_p}$ contains an \'etale submodule (instead of having an \'etale quotient like $V_\phi$). If $k=2$, note that $V_{\phi}^*\simeq V_\phi(-1)(\langle \eps^{-1}_{s}\rangle) \simeq V_\phi(-1)(\chi^{-1}_p)$, which is consistent with \eqref{V1}.  Geometrically, the Galois representation $V_\phi^*$ is naturally realised in the 
{\em anti-ordinary quotient} of $H^1_{\et}(\bar{X}^\dag_{s},\Z_p)$. 

\subsubsection{$\Lambda$-adic modular forms}
\label{subsec:L}
 A  Dirichlet character $\epsilon$ of conductor dividing  $p^s$, 
regarded either  as a character of $\Gamma_s$ or as a locally constant character of
$\Gamma = \invlim \Gamma_s \simeq 1+p\Z_p$, gives rise to a natural 
 ring homomorphism, denoted 
$\epsilon: \C_p[[\Gamma]] \lra \C_p$,  by extending $\epsilon$ to a linear homomorphism on
the group ring $\C_p[\Gamma_s]$. 

Let $\omega: \Z_p^\times \ra \mu_{p-1}$ denote the Teichm\"uller character, so that $z \omega^{-1}(z)\in 1+p\Z_p$ for any $z\in \Z_p^\times$.
\begin{definition}
\label{def:L}
A  $\Lambda$-adic modular cuspform  of tame level $N$ 
and character $\chi: (\Z/N\Z)^\times \ra \C_p^\times$ is a formal $q$-series
$$ \hphi := \sum_n a_n(\hphi) q^n \in \Lambda[[q]]$$
with the property that, for all finite order Dirichlet characters $\epsilon: \Gamma \lra \Gamma_s\lra \C_p^\times$,
the specialisation
$$ \epsilon(\hphi) := \sum_n \epsilon(a_n(\hphi)) q^n \in \C_p[[q]]$$
is the $q$-expansion of a modular cuspform of weight $2$, level $Np^s$ and character
 $\chi \epsilon \omega^{-1}$. 
\end{definition}
Write $S_{\Lambda}(N,\chi)$ for the space of such $\Lambda$-adic modular forms, 
and $S_{\Lambda}(N)$ for the direct sum of the spaces $S_{\Lambda}(N,\chi)$ as
 $\chi$ ranges over all characters mod $N$.
 

 The natural projection of $\hphi$ to $\Z_p[\Gamma_s][[q]]$, denoted $\hphi_s$, can be written as
 $$\hphi_s = \sum_{\sigma\in\Gamma_s}  \phi_s^\sigma \cdot \sigma^{-1} \in \Z_p[\Gamma_s][[q]]$$
where $\phi_s \in S_2(Np^s) = \Omega^1(X_s)$ is a classical modular form that 
will be  referred to as the {\em level $s$ specialisation} of  
 $\hphi$.
%To be concrete, $\psi_s = \sum_{\epsilon: \Gamma_s \ra \C_p^\times} \epsilon(\hphi_s).$
 With these notations, note that $\epsilon(\hphi) = \epsilon(\hphi_s) = \theta_{\epsilon}\phi_s.$
Of course, the system $\{\phi_s\}_{s\geq 1}$ of classical modular forms  determines
$\hphi$ completely, and vice-versa. 

While the forms  $\phi_s$ fail to be compatible 
under pushforward by either of the transition  maps $\varpi_1$ or $\varpi_2$, they 
do  satisfy 
$$ \mu_*(\phi_{s+1}) = \pi_1^*(\phi_s), \qquad  (\varpi_1)_*(\phi_{s+1}) = p \phi_s, \qquad
(\varpi_2)_*(\phi_{s+1}) = U_p \phi_s, $$
where the first relation is merely a restatement of the fact that the forms $\hphi_s$ are compatible under the natural projections $\Z_p[\Gamma_{s+1}] \lra \Z_p[\Gamma_s]$, and the second and third follow directly from the first in light of 
\eqref{eqn:def-Up}.
In particular, the systems of modular forms $\{ p^{-s} \phi_s\}$ and $\{U_p^{-s} \phi_s\}$
are compatible under the pushforward maps $(\varpi_1)_*$ and $(\varpi_2)_*$, respectively. It is thus natural to define the maps
$$ 
\mho_s: S_2(Np^s) \ra H^1_{\dR}(X_s) \quad \mbox{and} \quad \mho_\infty: S_\Lambda(N) \ra H^1_{\dR}(X_\infty)$$
given by
$\mho_s(\phi) = U_p^{-s} \omega_\phi$ and $\mho_\infty(\hphi) = \{ U_p^{-s} \omega_{\phi_s}\}_s.$






For any integrally closed subring $\cO$ of $\cO_{\C_p}$, set
$$ H^1_{\dR}(X_\infty)_{\cO} := \left\{ \underline{\eta} \in H^1_{\dR}(X_\infty) \mbox{ such that } [ \eta, \mho_\infty(\underline{\phi}) ]_{\Gamma} 
\in \cO[[\Gamma]]] \mbox{ for all } \underline{\phi} \in S_\Lambda(Np) \right\}.$$
Recall the group ring element
$\theta_\epsilon \in \C_p[\Gamma_s]$ introduced in \eqref{eqn:def-theta-epsilon}.
The next lemma records the behaviour of the $\Lambda$-adic pairing $[ \ ,\ ]_{\Gamma}$ and
the inclusion map $\mho_\infty$ under specialisation.
\begin{lemma}
\label{Gammas}
 Let
 $\underline{\eta} = \{\eta_s\} $
 and $\hphi = \{\phi_s\}$ be elements of 
$H^1_{\dR}(X_\infty)_\cO$ and $S_{\Lambda_{\cO}}(N)$. For all Dirichlet 
characters $\epsilon$ of conductor $p^{s'}$ with $1\le s'\le s$,  
$$\epsilon([\underline{\eta},\mho_\infty(\hphi)]_{\Gamma}) =  \langle  \eta_s, w w_{s}\theta_\epsilon(\phi_s) \rangle_{s} = 
\frac{1}{d_s} \langle  \theta_\epsilon(\eta_s), w w_{s}\theta_\epsilon(\phi_s)\rangle_{s}.$$
\end{lemma}
\begin{proof} The first equality follows directly from the definitions, since
\begin{align*}
\epsilon([\underline{\eta},\mho_\infty(\hphi)]_{\Gamma}) &= 
\epsilon\left( \sum_{\sigma\in \Gamma_s} \langle \sigma \eta_s, w w_s U_p^s U_p^{-s} \phi_s \rangle_s \cdot \sigma^{-1} \right) = \epsilon\left( \sum_{\sigma\in \Gamma_s} \langle  \eta_s, w w_s  \sigma \phi_s \rangle_s \cdot \sigma^{-1} \right)\\
&  = \langle \eta_s, w w_s \theta_\epsilon(\phi_s) \rangle_s = \langle  w^{-1} w_s^{-1} \eta_s,   \theta_\epsilon(\phi_s) \rangle_s. 
\end{align*}
The second identity is a direct consequence of the first in light of the fact that 
$\theta_{\epsilon}$ and $\theta_{\epsilon^{-1}}$ are adjoint to each other relative to Poincar\'e duality,
that $\theta_{\epsilon^{-1}} w w_s = w w_s \theta_{\epsilon}$, and that
$\theta_\epsilon^2 = d_s \theta_\epsilon$. 
\end{proof}
Note that the formula of Lemma \ref{Gammas}
 does not depend on the value of $s \ge s'$, since
\begin{eqnarray*}
 \langle w_{s+1} \eta_{s+1},\epsilon(\hphi_{s+1}) \rangle_{s+1} &=&
   \langle w_{{s+1}}\eta_{s+1}, \varpi_1^* \epsilon(\hphi_s)\rangle_{s+1} = 
\langle \varpi_{1*} w_{{s+1}} \eta_{s+1},\epsilon(\hphi_s) \rangle_{s}  \\
&=&   \langle  w_{s} (\varpi_{2*}  \eta_{s+1}),\epsilon(\hphi_s) \rangle_{s} = 
 \langle  w_{s} (\eta_{s}),\epsilon(\hphi_s) \rangle_{s}.
\end{eqnarray*}

\subsubsection{Hida families}\label{subsec:Hi}
Let $\cO$ be a finite extension of $\Z_p$ containing the values of
 all Dirichlet characters of conductor $Np$. 
%introduced in  \S \ref{sec:twisted-diagonal}. 
For any $\Z_p$-module $M$, 
  write $M_\cO = M\otimes_{\Z_p} \cO$. Assuming that $p$ does not divide $\varphi(N)=|(\Z/N\Z)^\times|$, the
 semi-local ring $\LLambda_{\cO}^{(N)}$ 
decomposes as a sum over the characters 
$\psi:(\Z/Np\Z)^\times \lra \cO^\times$:
$$ \LLambda_\cO^{(N)} = \oplus_{\psi}   \Lambda^{\psi}_\cO, \qquad  \Lambda^{\psi}_{\cO} := e_{\psi} (\LLambda_\cO) \simeq \Lambda_\cO,$$
where 
$e_{\psi}$ is the $\cO$-algebra  homomorphism defined on group-like elements by
$$ 
e_{\psi}(\langle a; b\rangle) = \psi(\langle a;b\rangle) \langle 1; b \omega^{-1}(b)\rangle \in \Lambda_{\cO}.
$$



 The {\em weight space} $\Omega$ is defined  to be the rigid analytic space 
 underlying the formal spectrum of $\Lambda$, so that for any complete
  $\Z_p$-algebra $R$, the set of $R$-rational points of $\Omega$ is
\begin{equation*}
\Omega(R)=\Hom (\Lambda,R) 
= \Hom_{\cts}(\Gamma,R^\times).
 \end{equation*}
An element of $\Omega$ is said to be  {\em classical} if it 
corresponds to  a character of $\Gamma$, denoted $\nu_{k,\epsilon}$,
 of the form
$$ \nu_{k,\epsilon}(n) = n^{k-1} \epsilon(n),  \qquad \mbox{ for all } n\in 1+p\Z_p, $$
for some integer $k$ and some continuous  character $\epsilon$  of finite order 
  (hence,   conductor $p^s$  for some $s\geq 1$,  factoring through a primitive
   character of $\Gamma/\Gamma^{p^{s-1}}\simeq \Z/p^{s-1}\Z\subset (\Z/p^s\Z)^\times$).

More generally, if $\tilde\Lambda$  is a finite flat extension of $\Lambda$,    let 
$\tilde\Omega := \Hom(\tilde\Lambda,\C_p)$ denote the space of
continuous algebra homomorphisms 
and  
 write $\mathrm{w}: \tilde\Omega \ra \Omega$ for the natural  ``weight map"
  induced by pull-back from the structure map $\Lambda \hookrightarrow\tilde\Lambda$.
A point in $\tilde\Omega$ is said to be classical if its image under $\mathrm{w}$ is classical.

Generalising Definition \ref{def:L} slightly, 
a  {\em $\Lambda$-adic ordinary  modular form   of tame level $N$ and  
character $\chi: (\Z/N\Z)^\times \ra \cO^\times$  with coefficients in  
$\tilde\Lambda$}  is
a pair $(\tilde\Omega_{\cl}, \hphi)$,
where
\begin{enumerate}
\item[(i)]  
$\tilde\Omega_{\cl}$ is a  dense  subset of the classical points in 
 $\tilde\Omega$ for the rigid analytic topology;
   \item[(ii)] $\hphi := \sum \underline{a}_n q^n\in \tilde\Lambda[[q]]$
is an element of $S_{\tilde\Lambda}(N,\chi) :=
S_\Lambda(N,\chi)\otimes_{\Lambda} \tilde \Lambda$, such that
for  all  $x \in \tilde\Omega_{\cl}$ of weight  $\mathrm{w}(x)=\nu_{k,\epsilon}$,
with $k\ge 2$ and character $\epsilon$ of conductor $p^s$, 
the power series
$$ \phi_x:= \sum_{n=1}^\infty \underline{a}_n(x) q^n \in \C_p[[q]], \quad \underline{a}_n(x):=x(\underline{a}_n),
$$
is the $q$-expansion of a classical ordinary cusp form  in  
$S_{k}(N  p^{s}, \chi  \epsilon    \omega^{1-k};\C_p)$. 
\end{enumerate}
The $\tilde\Lambda$-module of $\Lambda$-adic ordinary modular forms of tame level $N$ and  
character $\chi$ is denoted $S^{\ord}_{\tilde\Lambda}(N,\chi)$. 
\begin{definition}
\label{def:padic-fam} 
An element $\hphi\in S^{\ord}_{\tilde\Lambda}(N,\chi)$ whose classical
specializations $\phi_x$ are eigenforms (that is to say,  
simultaneous eigenvector for the good Hecke operators $T_\ell$ with $\ell \nmid Np$)
 is called a $\Lambda$-{\em adic eigenform} or a {\em Hida family}.
\end{definition}
%In the above definition $\underline{a}_n$ are elements in $\tilde\Lambda$, and by $\underline{a}_n(x)$ we mean $\underline{a}_n(x)=x(\underline{a}_n)$.
If  the subset $\tilde\Omega_{\cl} \subset \tilde\Omega$ can be clearly inferred from the context, 
such a form  will simply  be denoted as $\hphi$, suppressing
 the datum of $\tilde\Omega_{\cl}$ from the notation.      

 If in addition the specialisation $\phi_x$ of a $\Lambda$-adic eigenform $\hphi$
does not share the same eigenvalues with any eigenform of level $N_0 p^r$ with $N_0\mid N$, $N_0<N$ and $r\leq s$, then
$\hphi$ is called a $\Lambda$-{\em adic newform}.
Note that a form $\hphi$ as in Definition \ref{def:padic-fam} above is in particular a  $\Lambda$-adic modular form in the weaker sense of Definition \ref{def:L}, and hence the material of \S \ref{subsec:L} applies to it.

% We next turn to recall Hida's work on the explicit construction of  $\Lambda$-adic eigenforms and their associated Galois representations.
Recall the Hecke algebras $\TT_1(N)$ and $\TT_{1}(N p^s)$ that were introduced in 
Section \ref{sec:modular-curves}, and let  
$$ \TT_{1}(N p^\infty) := e_{\ord}\left( \invlim_s (\TT_{1}(N p^s)\otimes \Z_p)\right).$$
It follows from a theorem of Hida (cf.~\cite{Hi86}, \cite{Emerton}),   
that the   algebra $\TT_{1}(Np^\infty)$ is locally free
 of finite rank as a  $\LLambda^{(N)}$-module,
 and the algebra
 $$\TT_{1}^{\chi}(Np^\infty) = e_{\chi \omega^{-1}} \TT_{1}(Np^\infty)_\cO = 
 \TT_{1}(Np^\infty)\otimes_{e_{\chi\omega^{-1}}} \Lambda_\cO$$
 is therefore free of finite rank over the local ring $\Lambda_\cO$. 
 Let $\cL$ denote the fraction field of $\Lambda_\cO$, and write 
 $$\TT_{1}^{\chi}(Np^\infty)_{\cL }  := 
 \TT_{1}^{\chi} (Np^\infty)\otimes_{\Lambda_\cO} {\cL }.$$
 It is a finite-dimensional \'etale $\cL$-algebra, hence decomposes as a  product of 
 finite extensions $\cL_j$ of $\cL$:
 $$ \TT_{1}^{\chi}(Np^\infty)_{\cL}  = \oplus_{j=1}^t  \cL_j.$$
 Furthermore, for each $1\le j \le t$, the integral closure of the  
  image of $\TT_{1}^{\chi}(Np^\infty)$ in $\cL_j$, denoted $\Lambda_{j}$,  is a finite flat extension of $\Lambda_{\cO}$.
  The  $\Lambda$-adic newforms $\hphi$ of  character $\chi$ (of conductor dividing $N$) and primitive tame level $N_{\phi}$ (with $N_{\phi}$ ranging over the divisors of $N$ that are 
  multiples of the conductor of $\chi$) are in bijection with the  resulting homomorphisms
  $$ \eta_j: \TT_{1}^{\chi}(Np^\infty) \lra \Lambda_{j}.$$
   If $\hphi$ is one such $\Lambda$-adic eigenform, let $\Lambda_\phi$ denote the corresponding finite flat extension  of $\Lambda$ and corresponding homomorphism $\eta_\phi: \TT_{1}(N_0p^\infty)_\cO \ra \TT_{1}^{\chi}(Np^\infty) \ra \Lambda_\phi$.
Let 
\begin{equation}
\label{eqn:def-epsilon-cyc}
\uepsilon: G_\Q\lra \Lambda^\times
\end{equation}
be the $\Lambda$-adic cyclotomic character characterised by
$\nu_{k,\epsilon} \circ \uepsilon = \epsilon \cdot \epsilon_{\cyc}^{k-1} \cdot  \omega^{1-k}.$
\begin{theorem}[Hida]
\label{Hida-thm} 
  Let $\hphi$ be a  $\Lambda$-adic newform  of tame level $N_\phi$ and tame character $\chi$. The
    modules
\begin{equation}\label{defVV} 
\hV_{\hphi} := \hV^\ord_1(N_\phi p^\infty)\otimes_{\TT_{1}(N_\phi p^\infty)_\cO, \eta_\phi} \Lambda_\phi,
  \qquad 
  \hV_{\hphi}^*  :=  \hV_{\hphi}\otimes \uepsilon^{-1}
\end{equation}
 are both locally free of rank two over $\Lambda_\phi$. 
%{\bf The isomorphism $\Lambda^{\cyc}(1) \simeq \hV_{\hphi}\otimes \uepsilon^{-1}$ is indeed crucial in order for our $\Lambda$-adic cohomology class to project to the right place. But I already told you I don't trust this isomorphism much: do you really think the Tate twist (1) is correct there? Tracing the definitions I find that $\Lambda^{\cyc} \simeq \hV_{\hphi}\otimes \uepsilon^{-1}$ because $\sigma\in G_\Q$ acts on $\Lambda^{\cyc}$ by the group-like element $[\epsilon_{\cyc}^{-1}(\sigma)]$ which specializes via $\nu_k$ to $\langle \epsilon_{\cyc}^{1-k}(\sigma)\rangle$, exactly as $\uepsilon^{-1}$.}
  \end{theorem}
   The representations $\hV_{\hphi}$ and  $\hV_{\hphi}^*$ 
will be referred to  respectively as   the {\em ordinary } and {\em anti-ordinary
$\Lambda$-adic representation}  of $G_\Q$ attached  to the 
Hida family $\hphi$. Their  restrictions  to a decomposition group at $p$ are reducible
  and can be written in upper triangular form as
  \begin{equation}
  \label{eqn:local-triangular} 
  (\hV_{\hphi})_{|G_{\Q_p}}:  \left(
  \begin{array}{cc} \Psi_{\hphi}^{-1} \chi \uepsilon  & * \\ 0  & \Psi_{\hphi} 
  \end{array}\right),
\qquad
 (\hV_{\hphi}^*)_{|G_{\Q_p}}:  \left(
  \begin{array}{cc} \Psi_{\hphi}^{-1} \chi   & * \\ 0  & \Psi_{\hphi} \uepsilon^{-1}
  \end{array}\right),
  \end{equation}
  where $\Psi_{\hphi}: G_{\Q_p}\lra \Lambda_{\phi}^\times$ is the unramified 
  character defined by $\Psi_{\hphi}(\mathrm{Frob}_p) = a_p(\hphi) \in \Lambda_\phi^\times$. As in Definition \ref{def:Vfpm}, 
   let $\hV_{\hphi}^+$ denote the one-dimensional $G_{\Q_p}$-submodule of $\hV_{\hphi}$ on which $G_{\Q_p}$ acts via $\Psi_{\hphi}^{-1} \chi \uepsilon$, and write $\hV_{\hphi}^-$ for the unramified quotient on which $G_{\Q_p}$ acts via $\Psi_{\hphi} $.

  
   More generally, if $N$ is any multiple of $N_\phi$,  set
 $$ \hV_{\hphi}(N) := \hV^\ord_1(N  p^\infty)\otimes_{\TT_{1}(N  p^\infty)_\cO, \eta_\phi} \Lambda_\phi,
 \qquad 
  \hV_{\hphi}^*(N) :=  \hV_{\hphi}^*(N)\otimes \uepsilon^{-1}, $$
and write 
  $\varpi_{\hphi}$ and $\varpi_{\hphi}^*$  for the canonical projections 
  $$\varpi_{\hphi}: \hV^\ord_1(N  p^\infty) \lra \hV_{\hphi}(N), \qquad \varpi_{\hphi}^*: \hV^\ord_1(N  p^\infty)^\twist  \lra \hV_{\hphi}^*(N).$$
  The quotient $\hV_{\hphi}(N)$   is (non-canonically) isomorphic to a finite direct sum of the $\Lambda$-adic representations $\hV_{\hphi}$, and likewise for $\hV_{\hphi}^*(N)$. If   $x \in\tilde\Omega_{\cl}$ is a classical point of weight  $\mathrm{w}(x)=\nu_{k,\epsilon}$,
with $k\ge 2$ and character $\epsilon$ of conductor $p^s$, it gives rise to   $G_\Q$-equivariant specialization homomorphisms (which are all denoted by $x$)
$$
x: \hV_{\hphi} \lra V_{\phi_x}, \qquad  x: \hV_{\hphi}^* \lra  V_{\phi_x}^*, \qquad 
x: \hV_{\hphi}(N) \lra V_{\phi_x}(Np^s), \qquad  x: \hV_{\hphi}^*(N) \lra  V_{\phi_x}^*(Np^s).
$$
where $V_{\phi_x}$ and $V_{\phi_x}^*$ are  the Galois representations associated in \eqref{Vphi*} to the eigenform $\phi_x$. 




\subsection{The $\Lambda$-adic class  attached to two Hida families} 
\label{sec:lambda-adic-class-hida}

Let $f\in S_2(N_f)$ be the newform associated to the elliptic curve $E$ in the introduction and let  $\hg$ and $\hh$ be two
 $\Lambda$-adic newforms of tame level $N_g$ and $N_h$  and tame  character $\chi$ and $\chi^{-1}$, respectively.  The prime $p$ arising as the residual characteristic of $\Lambda$ is taken here not to divide neither $N=\mathrm{lcm}(N_f,N_g,N_h)$ nor $\varphi(N)$. Moreover, as we assumed in \eqref{eqn:def-epsilona}, we also take $p$ to be a non-Eisenstein prime for $f$.

% Let $\testf\in S_2(Np)[f]$ be a test vector of level $Np$ and let $\testhg$ and $\testhh$ be  test vectors of level $N$ attached respectively to $\hg$ and $\hh$ as in \eqref{eqn:Lambda-adic-test-vector}. 

Recall the $\Lambda$-adic representation $\hV_{\infty}^{\ord}:=  V_0(Np) \otimes \left(\hV^{\ord}_1(Np^\infty)\otimes_{\LLambda} \hV^{\ord}_1(Np^\infty)^\twist\right)$, and 
define
$$ \hV_{f\underline{gh}} := V_f \otimes (\hV_{\hg}\otimes_{\Lambda} \hV_{\hh}^*), \qquad
\hV_{f \underline{g h}}(N) :=  V_{f}(Np) \otimes (\hV_{\hg}(N)\otimes_{\Lambda} \hV_{\hh}^*(N)),
$$
  which are  modules over the ring
$$\Lambda_{f \underline{g h}}:=\cO_f \otimes (\Lambda_\hg\otimes_\Lambda \Lambda_\hh),$$
a finite flat extension of the   Iwasawa 
algebra $  \Lambda$. The rigid analytic space $\Omega_{f \underline{g h}}$ attached to the formal spectrum of 
$\Lambda_{f \underline{g h}}$  is  a finite
covering of the  weight space $\Omega$.
  The 
``classical points" in $\Omega_{f \underline{g h}}$   correspond to 
pairs  of classical points of the form $(y,z)\in \Omega_{\hg} \times \Omega_{\hh}$
satisfying $\w(y)=\w(z)$, that is to say, having the same weight and the same nebentypus at $p$. 

Let $g_y$ and $h_z$ denote as in the previous section the specialisations of $\hg$ and $\hh$ at the points $y$ and $z$, respectively. Following similar notations as above, let $K=K_{f g_y h_z}$ denote the finite extension of $\Q_p$ generated by $K_f$, $K_{g_y}$ and $K_{h_z}$ and let $\cO =\cO_{f g_y h_z}$ denote the valuation ring of $K$. 
The specialisation   $V_{f g_y h_z}$ of  $\hV_{f \underline{g h}}$ at the 
 classical  point $(y,z)$ of weight $\ell$ and nebentype character $\epsilon$ is a 
 Galois representation of rank $8$ over $\cO$, and there are  natural identifications
\begin{equation}\label{VfghN} 
V_{f g_y h_z} = V_{f} \otimes V_{g_y}\otimes V^*_{h_z}, \qquad
V_{f   g_y h_z}(N) :=  V_{f}(Np) \otimes V_{g_y}(Np^s)\otimes V_{h_z}^*(Np^s).
\end{equation}
The reader will note that this Galois representation is pure 
of weight $-1$, since a geometric 
Frobenius element at $\ell$ acting on $V_f$ has eigenvalues of complex absolute value
$\ell^{-1/2}$, while its eigenvalues on 
 $V_{g_y}\otimes V_{h_z}^*$ have complex absolute value $1$. 
Note that 
$V_{f \underline{g h}}$ interpolates only the {\em Kummer self-dual } Tate twists of the tensor product of the representations
$V_{f}$, $V_{g_y}$ and $V_{h_z}$. 





The canonical projections $\varpi_{f}$,
$\varpi_{\hg}$, and $\varpi^*_{\hh}$ associated to $f$, $\hg$ and $\hh$ give rise to a surjective $\Lambda $-module homomorphism
\begin{equation}\label{projector}
\varpi_{f, \hg, \hh}: \hV_{\infty}^{\ord} \lra \hV_{f \underline{g h}}(N).
\end{equation}
\begin{definition}
\label{def:kappa-fgh}
The one-variable $\Lambda$-adic cohomology class attached to the triple $(f,\hg,\hh)$ 
is the class
$$ \kappa(f, \underline{g h}) :=  
\varpi_{f, \hg, \hh}(\kappa_{\infty})\in H^1(\Q,\hV_{f \underline{g h}}(N)).$$
\end{definition}
This class  takes values in the Galois
representation $\hV_{f \underline{g h}}(N)$,  
which is (non-canonically) isomorphic to a finite sum of copies of the representation
$\hV_{f\underline{gh}}$.


 \section{Local $\Lambda$-adic cohomology classes}
  
  
 \subsection{Local Galois representations}
 
Let $f\in S_2(N_f)$ be a newform and let $p\nmid N_f$ be a prime at which $f$ is ordinary. Let  $\hg$ and $\hh$ be two
 $\Lambda$-adic newforms of tame level $N_g$ and $N_h$ and character $\chi$ and $\chi^{-1}$, respectively.  Set $N=\mathrm{lcm}(N_f,N_g,N_h)$ as usual. 
This chapter analyses the restrictions  to $G_{\Q_p}$  of the Galois representation $\hV_{f \underline{g h}}$ and of the $\Lambda$-adic cohomology class  $\kappa(f, \underline{g h})$. 
 
\begin{lemma}
\label{lemma:4-step-filtration} 
The Galois representation $\hV_{f \underline{g h}}$ is endowed with a four-step filtration
$$ 0 \subset  \hV_{f \underline{g h}}^{++} \subset \hV_{f \underline{g h}}^+ \subset \hV_{f \underline{g h}}^- 
\subset \hV_{f \underline{g h}}$$ by
$G_{\Q_p}$-stable $\Lambda_{f \underline{g h}}$-submodules 
of ranks  $0$, $1$, $4$, $7$ and $8$ respectively. 
The Galois group  
 $G_{\Q_p}$ acts 
 on the   successive quotients for this filtration    as a direct sum of  $\Lambda_{f \underline{g h}}$-adic
 characters. More precisely:
$$
 \begin{matrix}
 \hV_{f \underline{g h}}^{++} &\simeq &  \Lambda_{f \underline{g h}}(\heta^{+}),&  &\\
 \hV^+_{f \underline{g h}}/\hV^{++}_{f \underline{g h}} &=&  \hV_{f \underline{g h}}^f \oplus
   \hV_{f \underline{g h}}^g \oplus  \hV_{f \underline{g h}}^h &\simeq &  \Lambda_{f \underline{g h}}(\heta_f^{+}) \oplus 
 \Lambda_{f \underline{g h}}(\heta_{\hg}^{+}) \oplus \Lambda_{f \underline{g h}}(\heta_{\hh}^{+}), \\
  \hV^-_{f \underline{g h}}/\hV^+_{f \underline{g h}} &=&  \hV_{f \underline{g h}}^f \oplus
   \hV_{f \underline{g h}}^g \oplus  \hV_{f \underline{g h}}^h &\simeq &  \Lambda_{f \underline{g h}}(\heta_f^{-}) \oplus 
 \Lambda_{f \underline{g h}}(\heta_{\hg}^{-}) \oplus \Lambda_{f \underline{g h}}(\heta_{\hh}^{-}), \\
 \hV_{f \underline{g h}}/ \hV_{f \underline{g h}}^- &\simeq& \Lambda_{f \underline{g h}}(\heta^{-}), & &  
 \end{matrix}
$$
 where the characters   arising on the right 
 are described in the following table:
$$ \begin{array}{c|c|l}
 \hline
 \mbox{\em Subquotient} & \mbox{\em Rank} & \ \  \ \ \ \ \ \ \  \mbox{\em Galois action} \\ \hline 
 \hV_{f \underline{g h}}^{++} &  1 &  \heta^{+} = (\psi_f \Psi_{\hg}\Psi_{\hh})^{-1}   \times \epsilon_{\cyc}\uepsilon     \\ \hline
  \hV_{f \underline{g h}}^+/ \hV_{f \underline{g h}}^{++}   &  3 & 
 \heta_f^{+} =  \psi_{f}    \Psi_{\hg}^{-1} \Psi_{\hh}^{-1} 
 \times \uepsilon    \\
    & &  \heta_{\hg}^{+} = \psi_{f}^{-1}\Psi_{\hg}   \Psi_{\hh}^{-1} \times \chi^{-1} \times \epsilon_{\cyc}     \\
  & & \heta_{\hh}^{+} :=   \psi_{f}^{-1} \Psi_{\hg}^{-1} \Psi_{\hh}  \times \chi\times  \epsilon_{\cyc} 
 \\   \hline
  \hV_{f \underline{g h}}^-/ \hV_{f \underline{g h}}^{+}   &  3 & 
  \heta_{\hh}^- =  \psi_{f}  \Psi_{\hg}   \Psi_{\hh}^{-1} \times  \chi^{-1}
   \\
 & & \heta_{\hg}^- =   \psi_{f} \Psi_{\hg}^{-1}  \Psi_{\hh}   \times  \chi \\ 
 & & 
\heta_f^- =    \psi_{f}^{-1}  \Psi_{\hg}  \Psi_{\hh}    \times
 \epsilon_{\cyc} \uepsilon^{-1} 
  \\ \hline
  \hV_{f \underline{g h}}/ \hV_{f \underline{g h}}^{-}   &  1 &  \heta^- =  
    \psi_{f}   \Psi_{\hg}    \Psi_{\hh}    \times \uepsilon^{-1}.
    \end{array}
    $$
 \end{lemma}
 \begin{proof} The lemma follows directly from \eqref{eqn:local-triangular-class}, \eqref{defVV} and \eqref{eqn:local-triangular}.
 \end{proof}
 
Since $\hV_{f \underline{g h}}(N)$ is a finite sum of copies of the representation
$\hV_{f\underline{gh}}$, in view of the above lemma  it also admits a natural filtration 
$$ 0 \subset  \hV_{f \underline{g h}}(N)^{++} \subset \hV_{f \underline{g h}}(N)^+ \subset \hV_{f \underline{g h}}(N)^- 
\subset \hV_{f \underline{g h}}(N) $$
of $\Lambda_{f \underline{g h}}[G_{\Q_p}]$-modules,
and
\begin{eqnarray}\label{3decom}
\hV_{f \underline{g h}}^+(N)/\hV^{++}_{f \underline{g h}}(N) & =: &
  \hV_{f \underline{g h}}^f(N) \oplus
   \hV_{f \underline{g h}}^g(N) \oplus  \hV_{f \underline{g h}}^h(N).
\end{eqnarray}
For each $\phi=f,g,h$,  the subquotient 
$\hV_{f \underline{g h}}^\phi(N)$ is a direct sum of finitely many
 copies of $\hV_{f \underline{g h}}^\phi$, on which $G_{\Q_p}$ acts via the 
 character  $\heta_{\hphi}^{+}$. 
 
 Let
\begin{equation}
\label{eqn:def-kappap-f-gh}
\kappa_p(f,\underline{g h}):= \res_p(\kappa(f, \underline{g h})) \in H^1(\Q_p,\hV_{f \underline{g h}})
\end{equation} 
 denote the natural image of the global class
 $\kappa(f, \underline{g h})$ in the local cohomology at $p$, and set
$$ \xi_{f\underline{gh}} = (1-\alpha_f a_p(\hg) a_p(\hh)^{-1} \chi^{-1}(p)) (1-\alpha_f a_p(\hg)^{-1} a_p(\hh) \chi(p)).$$
 \begin{proposition}
 \label{prop:cristalline-lambda}
 The class $\xi_{f\underline{gh}} \cdot \kappa_p(f,\underline{g h})$ belongs to the natural 
 image of $H^1(\Q_p, \hV_{f \underline{g h}}^{+}(N))$ 
 in $H^1(\Q_p, \hV_{f \underline{g h}}(N))$ under the map induced from the inclusion
 $ \hV_{f \underline{g h}}^+(N) \hookrightarrow
\hV_{f \underline{g h}}(N)$.
  \end{proposition}
  \begin{proof} 
 % Recall that  the $\Lambda$-algebra structure of  $\Lambda_{f\underline{gh}}$ gives rise to a natural  inclusion  $\Lambda\subset \Lambda_{f\underline{gh}}$.
 %  Let $\theta_s$ be the ideal of $\Lambda_{f\underline{gh}}$ 
 Recall the natural projection $\Lambda\lra \Lambda_s$ to the finite group ring of level $p^s$. 
 The Galois representation $\hV_{f\underline{gh}}(N) \otimes_\Lambda \Lambda_s$ is realised in the middle
 $p$-adic \'etale cohomology of
  the quotient $W_s$ of
 $X_0(Np)\times X_s \times X^\dag_s$, and the
  natural  image of $\kappa_p(f,\underline{g h})$ in 
  $H^1(\Q_p, \hV_{f\underline{gh}}(N)\otimes_\Lambda\Lambda_s)$ 
    is proportional to the image  of the  cycle $\Delta_s$ 
      under the 
 $p$-adic \'etale Abel-Jacobi map. The three-fold $W_s$ 
 is smooth and proper over $\Q_p$, and   the purity conjecture for the monodromy 
 filtration is known to hold by the work of Saito (cf.\,\cite{Sa}, \cite[(3.2)]{Ne2}). 
 By Theorem 3.1 of \cite{Sa}, it follows that the
  natural  image of $\kappa_p(f,\underline{g h})$ belongs to 
 $H^1_{\fin}(\Q_p, \hV_{f \underline{g h}}(N)\otimes_\Lambda \Lambda_s)$,  
where
\begin{equation}\label{Hfin}
H^1_{\fin}(\Q_p, \hV_{f  \underline{gh}}(N)\otimes_\Lambda \Lambda_s) \subset H^1(\Q_p, \hV_{f \underline{gh}}(N)\otimes_\Lambda\Lambda_s)
\end{equation}
denotes the {\em finite} submodule in the sense of  Bloch-Kato, as defined in \cite[\S 3]{BK} or \cite[(2.2)]{Ne2}.
Lemma 1.15  implies that $\hV_{f \underline{gh}}(N)\otimes_\Lambda \Lambda_s$ is
 an ordinary $p$-adic representation in the sense of Greenberg (cf.\,\cite[(4)]{Gr}, \cite{Fl}). Hence the finite subspace may be described concretely as
$$H^1_{\fin}(\Q_p, \hV_{f \underline{gh}}(N)\otimes_\Lambda \Lambda_s) = \ker\left(H^1(\Q_p, \hV_{f \underline{gh}}(N)\otimes_\Lambda\Lambda_s) \lra 
 H^1(I_p,(\hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N)) \otimes_\Lambda\Lambda_s)\right),$$ 
 where $I_p$ denotes the inertia group at $p$. (See \cite[Lemma 2, p.\,125]{Fl} for a proof
  of this useful fact. Note that 
  $H^1_{\fin}(\Q_p, \hV_{f \underline{gh}}(N)) = H^1_{\mathrm{g}}(\Q_p,  V_{f \underline{gh}}(N))$
   in this case, by \cite[(3.1) and (3.2)]{Ne2}.)
By passing to the inverse limit and using the fact that
$\invlim_s  \hV_{f\underline{gh}}(N)\otimes_\Lambda \Lambda_s = \hV_{f\underline{gh}}(N)$, 
one concludes that 
$$ \kappa_p(f,\underline{g h}) \in \ker\left(H^1(\Q_p, \hV_{f \underline{gh}}(N)) \lra 
 H^1(I_p, \hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N))\right).$$
 The kernel of the   restriction map $H^1(\Q_p, \hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N))\lra 
H^1(I_p,\hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N))$
is naturally a quotient of $H^1(\Q_p^{\rm ur}/\Q_p,  (\hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N))^{I_p})$.

The local Galois representation $ (\hV_{f \underline{gh}} /\hV_{f \underline{gh}}^{+})$ 
 lies in the middle of an exact sequence
 $$ 0 \lra \hV_{f\underline{gh}}^{f-} \oplus \hV_{f\underline{gh}}^{g-} \oplus \hV_{f\underline{gh}}^{h-} 
 \lra (\hV_{f \underline{gh}} /\hV_{f \underline{gh}}^{+}) \lra (\hV_{f \underline{gh}} /\hV_{f \underline{gh}}^{-})\lra 0, $$
 which induces an isomorphism $\hV_{f\underline{gh}}^{g-} \oplus \hV_{f\underline{gh}}^{h-} 
 \lra (\hV_{f \underline{gh}} /\hV_{f \underline{gh}}^{+})^{I_p}$
 by Lemma \ref{lemma:4-step-filtration}.
 The term 
 $$ 
 H^1(\Q_p^{\rm ur}/\Q_p,   \hV_{f \underline{gh}}^{g-}  \oplus
  \hV_{f \underline{gh}}^{h-})   
= (\Lambda_{f \underline{gh}}(\heta_{\underline{g}}^-) \oplus 
 \Lambda_{f \underline{gh}} (\heta_{\underline{h}}^-))/(\Frob_p-1),
$$
  is annihilated by $\xi_{f\underline{gh}}$, as can be read off from the table in Lemma
\ref{lemma:4-step-filtration}, and the same is therefore  true of 
$ H^1(\Q_p^{\rm ur}/\Q_p, (\hV_{f \underline{gh}} /\hV_{f \underline{gh}}^{-})^{I_p})$ as
well as of
$ H^1(\Q_p^{\rm ur}/\Q_p, (\hV_{f \underline{gh}}(N) /\hV_{f \underline{gh}}^{-}(N))^{I_p})$.
It follows that 
$\xi_{f\underline{gh}}  \cdot \kappa_p(f,\underline{g h})$ belongs to the kernel of the natural map
to $H^1(\Q_p, \hV_{f \underline{gh}}(N)/\hV_{f \underline{gh}}^{+}(N))$.
\end{proof}

It will be convenient, for the discussion that follows {\em and in the remainder of this article}  to replace the
ring $\Lambda_{f \underline{gh}}$ and all modules over it (such as 
$\hV_{f \underline{gh}}$ and its various subquotients) with  their localisations by the multiplicative set
generated by $\xi_{f\underline{gh}}$. Since the 
 element  $\xi_{f\underline{gh}}$  is non-zero at all classical points of weight $\ell \ge 1$, the passage to this localisation is
 harmless and still makes it permissible to specialise the 
 $\Lambda$-adic class $\kappa(f, \underline{g h})$
 at these points. With this convention in place, the following   is immediate:
 \begin{corollary}
 \label{cor:cristalline-lambda}
 The class $\kappa_p(f,\underline{g h})$ belongs to the
 image of $H^1(\Q_p, \hV_{f \underline{g h}}^{+}(N))$ 
 in $H^1(\Q_p, \hV_{f \underline{g h}}(N))$ under the map induced from the inclusion
 $ \hV_{f \underline{g h}}^{+}(N) \hookrightarrow
\hV_{f \underline{g h}}(N)$.
  \end{corollary}

The decomposition \eqref{3decom} induces a corresponding decomposition of the local cohomology group
  $H^1(\Q_p, \hV_{f \underline{g h}}^+(N)/\hV^{++}_{f \underline{g h}}(N))$ into a direct sum of three contributions.
Write 
\begin{equation}\label{local-contributions}
 \kappa_p^f(f,\underline{g h}) \in  H^1(\Q_p,\hV_{f \underline{g h}}^f(N)),\quad   \kappa_p^g(f,\underline{g h}) \in H^1(\Q_p, \hV_{f \underline{g h}}^g(N)), \quad \kappa_p^h(f,\underline{g h}) \in H^1(\Q_p,  \hV_{f \underline{g h}}^h(N))
\end{equation}
%\begin{eqnarray*}
 % \kappa_p^f(f,\underline{g h}) &\in & H^1(\Q_p, \hV_{f \underline{g h}}^f(N)), \\
 % \nonumber
%    \kappa_p^g(f,\underline{g h}) &\in&H^1(\Q_p, \hV_{f \underline{g h}}^g(N)), \\
 %   \nonumber
 % \kappa_p^h(f,\underline{g h}) &\in& H^1(\Q_p,  \hV_{f \underline{g h}}^h(N))
%  \end{eqnarray*}
for the  projections of the local class $\kappa_p(f,\underline{g h})$ in each direct summand.
  
  The reader will note that the $\Lambda_{f \underline{g h}}$-adic characters $\heta_g^+$ and $\heta_h^+$
   agree with the cyclotomic character when restricted  to the inertia group at $p$, 
   and thus are not ``truly varying", while the character
  $\heta_f^+$ is somewhat more interesting, since it  involves the $\Lambda$-adic cyclotomic character $\uepsilon$. 
  



 
\subsection{Classical specializations}

Let $(y,z)\in \Omega_g \times\Omega_h$ be a classical point of weight  $\ell$ and character $\epsilon$. The goal of this section is to analyze the specialization 
$$ \kappa(f, g_y, h_z) := \kappa(f, \underline{g h})_{(y,z)} \in H^1(\Q,V_{f   g_y h_z}(N))$$
 of the global $\Lambda$-adic class
$\kappa(f, \underline{g h})$ at $(y,z)$, with special emphasis
 on its local behaviour at $p$.

A one-dimensional character $\Upsilon: G_{\Q_p}\lra \C_p^\times$ is said to be 
{\em of Hodge-Tate weight $-j$} if it is equal to a finite order character times
the $j$-th power of the cyclotomic character. The following is an immediate corollary of Lemma 
\ref{lemma:4-step-filtration} of the previous chapter.
\begin{corollary}
\label{cor:4-step-filtration}
The Galois representation $V_{f g_y h_z}$   is endowed with a four-step $G_{\Q_p}$-stable
 filtration
  $$ 0\subset V_{f g_y h_z}^{++} \subset V_{f g_y h_z}^+ \subset V_{f g_y h_z}^- 
\subset V_{f g_y h_z},$$
and the action of $G_{\Q_p}$ on its successive quotients is via the 
one-dimensional representations
described in the following table:
$$ \begin{array}{c|l|c|r|r}
 \hline
\mbox{\em Subquotient} & \qquad \qquad \mbox{\em Galois action}   & \mbox{\em Hodge-Tate}  &   \ell =2     &  \ell =1\\ 
                                       &   &    \mbox{\em \ \ \  \ \ weights} & &  \\
\hline 
 V_{f g_y h_z}^{++} & \Upsilon^+(y,z) :=  (\psi_f\psi_{g_y}\psi_{h_z})^{-1} \times     \epsilon \times \omega^{1-\ell} \times \epsilon_{\cyc}^{ \ell}   & -\ell  & -2 &   -1  \\ \hline
 V_{f g_y h_z}^+/ V_{f g_y h_z}^{++}   & \Upsilon_f^+(y,z) := \psi_f \psi_{g_y}^{-1} \psi_{h_z}^{-1} \times  \epsilon \times  \omega^{1-\ell} \times  \epsilon_{\cyc}^{\ell-1}  & 1-\ell &  -1 & 0 \\
& \Upsilon_{g_y}^+(y,z) :=  \psi_f^{-1} \psi_{g_y} \psi_{h_z}^{-1} \times \chi^{-1}  \times \epsilon_{\cyc} & -1  & -1 & -1  \\
 &  \Upsilon_{h_z}^+(y,z) := \psi_f^{-1} \psi_{g_y}^{-1} \psi_{h_z} \times \chi\times \epsilon_{\cyc} & -1  & -1 & -1 \\ 
 \hline
 V_{f g_y h_z}^-/ V_{f g_y h_z}^{+}      
  & \Upsilon_{h_z}^-(y,z):=  \psi_f\psi_{g_y} \psi_{h_z}^{-1} \times  \chi^{-1}    & 0 & 0 & 0 \\ 
 &\Upsilon_{g_y}^-(y,z) :=   \psi_f  \psi_{g_y}^{-1} \psi_{h_z} \times  \chi & 0 & 0 & 0 \\
&  \Upsilon_f^-(y,z) :=  \psi_f^{-1} \psi_{g_y} \psi_{h_z} \times \epsilon^{-1} \times \omega^{\ell-1} \times \epsilon_{\cyc}^{2-\ell}  & \ell-2 & 0 & -1 \\ \hline
    V_{f g_y h_z}/ V_{f g_y h_z}^{-}   & \Upsilon^-(y,z)  := \psi_f\psi_{g_y}\psi_{h_z} \times \epsilon^{-1} \times \omega^{\ell-1}\times  \epsilon_{\cyc}^{1-\ell} & \ell-1  & 1 & 0 \\
\hline
    \end{array}
    $$
 \end{corollary}



 
\subsubsection{Weight two specializations}
\label{sec:w2}
When $\ell=2$, the
   classes $\kappa(f,g_y,h_z)$ are then,
 by their very construction, directly related to the \'etale Abel-Jacobi images of twisted diagonal cycles.
 Recall that $\alpha_{g_y}$ denotes the eigenvalue of
 $U_p$ acting on the ordinary modular form $g_y$.

 

 
 \begin{proposition}
\label{prop:specialisation-wt-2}
Assume that  the classical points $(y,z)$ have weight $\ell=2$ and character $\epsilon$ of conductor $p^s$. 
 Then 
$$ \kappa(f, g_y,h_z) =  \alpha_{g_y}^{-s}    
\varpi_{f g_y h_z}(\AJ_{\et}(\Delta_s)) \in H^1(\Q,V_{f g_y h_z}(N)).$$ 
\end{proposition}
\begin{proof}
By Definition \ref{def:kappa-fgh} 
 and equation \eqref{eqn:def-kappa-infty}, $\kappa(f,g_y,h_z) = \varpi_{f g_y h_z}
 \circ \varpi_{f \hg \hh} (\kappa_\infty) = \varpi_{f g_y h_z}(\kappa_s).$
But since $\varpi_{f g_y h_z} (1\otimes U_p \otimes 1) = \alpha_{g_y} \varpi_{f g_y h_z}$ and
the projection $\varpi_{f g_y h_z}$ factors through the ordinary projection $e_{\ord}\otimes e_{\ord}$,   equation
\eqref{eqn:from-kappas-to-kappas4} 
implies that
$\kappa(f,g_y,h_z) = \alpha_{g_y}^{-s}  \varpi_{f g_y h_z}(\kappa_s^{(3)}).$ The projection $\varpi_{f g_y h_z}$ also factors through the K\"unneth projection 
$\pr_{111}$ of equation 
\eqref{eqn:def-hVrs}, and thus
$\kappa(f,g_y,h_z) = \alpha_{g_y}^{-s}  \varpi_{f g_y h_z}(\kappa_s^{(2)}).$ Now, using equation
 \eqref{eqn:from-kappas2-to-kappas3} 
and the fact that $\varpi_{f g_y h_z} \varepsilon_{s,s} = 
  \varpi_{f g_y h_z}$,  it can be seen that
$\kappa(f,g_y,h_z) = \alpha_{g_y}^{-s}     \varpi_{f g_y h_z}(\kappa_s^{(1)}).$
Proposition \ref{prop:specialisation-wt-2} now follows from the definition of $\kappa_s^{(1)}$ given in 
\eqref{eqn:def-kappas1}. 
\end{proof}

 
 
 \begin{corollary}
 \label{cor:cristalline-wt2}
 For all classical $(y,z)\in\Omega_g\times\Omega_h$ of weight two and 
 character $\epsilon$ of conductor $p^s$, the class $\kappa_p(f,g_y,h_z)$ 
 belongs to 
 the
  image of $H^1(\Q_p(\zeta_s),  V_{f g_y h_z}^{+}(N))$ 
  in $H^1(\Q_p(\zeta_s),  V_{f g_y h_z}(N))$ under the map induced from the inclusion
  $   V_{f g_y h_z}^+(N) \hookrightarrow
  V_{f g_y h_z}(N)$.
\end{corollary}
\begin{proof} This follows by specialising Corollary \ref{cor:cristalline-lambda} 
to the point $(y,z)$. (It could also be deduced from Proposition \ref{prop:specialisation-wt-2}
in light of the known properties of \'etale Abel-Jacobi images of algebraic cycles.)
\end{proof}


 
  

   
%\subsubsection{Higher weight specializations}

%Consider now the case where the classical point $(y,z)\in \Omega_g\times\Omega_h$ is of weight $\ell\ge 2$.
%The following statement extends Corollary \ref{cor:cristalline-wt2} to these
%higher weight specializations:
%\begin{proposition}
% \label{prop:cristalline-wtl}
% For all classical $(y,z)\in\Omega_g\times\Omega_h$ of weight  $\ell \ge 2$ and 
% character $\epsilon$,  the  class $\kappa_p(f,g_y,h_z)$
% belongs to  
% $$ H^1_{\fin}(\Q_p,   V_{f g_y h_z}(N)) := \ker\left(H^1(\Q_p,  V_{f g_y h_z}(N)) \lra H^1(\Q_p,
% V_{f g_y h_z}(N)/ V_{f g_y h_z}^{+}(N))\right).$$
%\end{proposition}
%\begin{proof}
%This statement follows directly from its 
%$\Lambda_{f \underline{g h}}$-adic counterpart 
%stated in Corollary \ref{cor:cristalline-lambda}.
%\end{proof}
 
% \begin{remark} 
% Theorem  \ref{main1} 
% of  chapter \ref{sec:Lp}
% takes 
% Prop.~\ref{prop:cristalline-wtl}  further by showing that for a point $(y,z)$ of weight 
% $(\ell,1)$ with $\ell \equiv 1 \pmod{p-1}$, 
% the  class $\kappa_p(f,g_y,h_z)$ arises from a geometric construction, namely,  (up to $\bar\Q^\times$ multiples) 
% from the image  of the generalized diagonal cycles of
% \cite{DR1} in the product $X_0(Np) \times \cE_\ell(N) \times \cE_\ell(N)$. 
% \end{remark} 



\subsubsection{Weight one specializations}\label{subsec:w1}

The richest arithmetic phenomena arise when studying the specialisation of $\kappa(f, \underline{g h})$ 
at a point  $(y,z)$ of weight $\ell=1$ and trivial nebentype character at $p$, which will be assumed in this section. 
%As shown in \cite{mazur-wiles-an}, 
The specializations of $\hg$ and $\hh$ at $y$ and $z$ need not be classical in general; let us assume however that they are: the departure point of
 this section is the assumption that there exist newforms 
 $g\in S_1(N_g,\chi)$ and $h\in S_1(N_h,\chi^{-1})$
such that 
$$
g_y = g_\alpha \in S_1(N_g p,\chi), \quad h_z = h_\alpha \in S_1(N_h p,\chi^{-1}).
$$
Recall from the introduction that associated to $g$ and $h$ there exist Artin representations $\varrho_g$ and $\varrho_h$ with coefficients in a finite extension $L/\Q$. Setting $\varrho=\varrho_g\otimes \varrho_h$, we have $V_{gh}:=V_g\otimes V_h \simeq V_\varrho \otimes_L L_p$.


After labelling the two (distinct, by the hypothesis imposed in the introduction) 
roots of the $p$-th Hecke polynomial $x^2-a_p(g)+\chi(p)$ as $\alpha_g$ and $\beta_g$, 
%TESTVECTORSwrite
%$\testg_\alpha(q) = \testg(q) -\beta_g\testg(q^p)\in  S_1(Np,\chi)$ for the $p$-stabilization of $\testg$ %on which $U_p$ acts with eigenvalue $\alpha_g$;  a similar convention is adopted
%for $\testg_\beta$, $\testh_\alpha$ and $\testh_\beta$.
one may define the global cohomology class
$$
 \kappa(f,g_\alpha, h_\alpha) := \kappa(f,\underline{g h})_{y,z}\in H^1(\Q,V_{f g h}(N)) $$
with values in the Galois representation $V_{f g h}(N) = V_f(Np)\otimes V_{g}(Np) \otimes V_h(Np) \simeq (V_f \otimes V_{g h})^a$ for some $a\ge 1$. Let also $\kappa_p^f(f,g_\alpha, h_\alpha)\in H^1(\Q_p,  V^f_{f g h}(N))$  denote the associated local cohomology class arising likewise from \eqref{local-contributions}.

% When the weight two modular eigenform $f$ is attached to an elliptic curve $E$, one has
%$$
%V_{f g h}(N) \simeq  (V_p(E) \otimes V_{g h})^a
%$$
%,where $V_p(E)=H^1_{\et}(\bar E,\Z_p)(1) \simeq V_f$ is the (Kummer self-dual) $p$-adic Tate module of $E$ and $V_{g h}:=V_g\otimes V_h$ is an Artin representation of determinant one. The class 
% $\kappa(f,g_\alpha,h_\alpha)$ then becomes  relevant for the study of the natural $G_\Q$-equivariant refinement of the
 %Birch and Swinnerton-Dyer conjecture attached to the elliptic curve $E$ and the Artin representation $V_{g h}$. 
 

The weight one setting distinguishes itself in
(at least) two further respects. Firstly, recall the running assumption that $\alpha_g \ne \beta_g$, $\alpha_h \ne \beta_h$. Let us denote 
$\hg_\alpha$, $\hg_\beta$, $\hh_\alpha$, $\hh_\beta$ the Hida families passing through $g_\alpha$, $g_\beta$, $h_\alpha$, $h_\beta$, respectively.
One thus obtains four (a priori distinct) 
global classes
\begin{equation}
\label{eqn:classes-wt1}
 \kappa(f,g_\alpha, h_\alpha), \quad
 \kappa(f, g_\alpha, h_\beta), \quad
 \kappa(f, g_\beta, h_\alpha), \quad
 \kappa(f,g_\beta, h_\beta) \quad \in H^1(\Q,V_{fgh}(N)).
 \end{equation}

Note that for primes $\ell\ne p$, it follows from \cite[(2.5) and (3.2)]{Ne2} that $H^1(\Q_\ell,V_{fgh}(N))=0$  and hence a fortiori the restriction to $G_{\Q_\ell}$ of the above cohomology classes are all trivial. This is in general not true for $\ell=p$, as we shall see later in Theorem \ref{main2}.

Since $V_{g h}$ is an Artin representation which is unramified at $p$,  and for which both $V_g$ and $V_h$ are
regular, the local cohomology 
group 
$H^1(\Q_p,V_{f}(Np)\otimes V_{gh}(N))$ 
decomposes into a direct sum of
four vector spaces over $L_p = K_{ f g h}$:
\begin{eqnarray*}
 H^1(\Q_p,V_{f}(Np)\otimes V_{gh}(N))  &=&  
H^1(\Q_p,V_{f}(Np)\otimes V^{\alpha\alpha}_{g h}(N)) 
\oplus H^1(\Q_p,V_{f}(Np) \otimes V^{\alpha\beta}_{g h}(N))  \\ 
& & \quad \oplus H^1(\Q_p,V_{f}(Np)\otimes V^{\beta\alpha}_{g h}(N)) 
\oplus H^1(\Q_p,V_{f}(Np)\otimes V^{\beta\beta}_{g h}(N)),
\end{eqnarray*}
where $V^{\alpha\alpha}_{g h}(N) := V_g(N)^{\alpha_g}\otimes V_h(N)^{\alpha_h}$ denotes the $G_{\Q_p}$-stable subspace in $V_{gh}(N)$ formed from the tensor product of the two eigenspaces in $V_g(N)$ and $V_h(N)$ on which $\Frob_p$ acts with eigenvalue $\alpha_g$ and $\alpha_h$, respectively,
and  likewise for the other spaces. 
\begin{proposition}
\label{prop:the-clincher}
The local class $\kappa_p(f,g_\alpha,h_\alpha)$ belongs to the kernel of the natural homomorphism
$$ H^1(\Q_p,V_{f}(Np) \otimes V_{gh}(N)) \lra H^1(\Q_p,V_{f}(Np) \otimes V^{\alpha\alpha}_{g h}(N)).$$
\end{proposition}
\begin{proof}
This follows from Corollary \ref{cor:cristalline-lambda}
 after observing that the specialisation of $\hV^+_{f\underline{gh}}$ 
at $(y,z)$ is contained in  
$$  V_{f}(Np) \otimes  \left( V^{\alpha\beta}_{g h}(N)   
 \oplus   V^{\beta\alpha}_{g h}(N)  
\oplus   V^{\beta\beta}_{g h}(N)\right).
$$
\end{proof}



Although the Galois representation $V_{fgh}$, which is 
independent of the choice of stabilisations of $g$ and $h$,
 is crystalline under our running assumptions, the same need not be
true of the extension classes in \eqref{eqn:classes-wt1}. 
This can   be seen by observing
 that the Hodge-Tate weights attached to the specializations of the characters $\heta_f^+$ 
 and $\heta_f^-$, which in weight $\ell \ge 2$ are $<0$ and $\ge 0$ respectively,
acquire Hodge-Tate weights that are $\ge 0$ and $<0$ respectively when $\ell=1$, as summarized 
in the table of Corollary \ref{cor:4-step-filtration}. The  local condition $H^1_{\fin}(\Q_p, V_{f g h}(N))$  at $p$ defining the Selmer group
of $V_{fgh}(N)$ 
  is given by 
\begin{equation}\label{H1f}
H^1_{\fin}(\Q_p, V_{fgh}(N)) = \ker\left( H^1(\Q_p,V_{fgh}(N))\lra H^1(\Q_p, V_{f}(Np)^- \otimes V_{gh}(N))\right),
\end{equation}
where $V_f(Np)^-$ is the unramified quotient of $V_f(Np)$ of Definition 
\ref{def:Vfpm} and $V_{gh}(N) = V_g(N)\otimes V_h(N)$.
Let 
$$ \partial_p: H^1(\Q, V_{fgh}(N))  \lra H^1(\Q_p, V_{f}(Np)^-\otimes V_{gh}(N)) $$
be the natural projection to the ``singular quotient" of the cohomology group at $p$, whose kernel 
is    $H^1_{\fin}(\Q_p, V_{fgh}(N))$.
\begin{proposition}
\label{prop:later}
The class $\partial_p(\kappa(f,g_\alpha,h_\alpha))$ belongs to
 $H^1(\Q_p, V_{f}(Np)^-\otimes V_{g h}^{\beta\beta}(N))$, and is non-zero if and only  
 $
  \kappa_p^f(f,g_\alpha, h_\alpha)\ne 0$.
\end{proposition}
\begin{proof}  The second row in the  table in Corollary 
\ref{cor:4-step-filtration}
shows that the specialisation of $\heta_f^+$ at   $(y,z)$ is the unramified character
that sends $\Frob_p$
to 
$\alpha_f (\alpha_g\alpha_h)^{-1} = \alpha_f \beta_g \beta_h$ and hence the local class $\kappa_p^f(f,g_\alpha, h_\alpha)$ belongs to $H^1(\Q_p, V_{f}(N)^-\otimes V_{g h}^{\beta\beta}(N))$. 
Since
$$\hV_{f\underline{gh}}^+/ (\hV_{f\underline{gh}}^+ \cap  (V_f^{+} \otimes (\hV_{\hg} \otimes_{\Lambda} \hV^*_{\hh}))) = \cO(\heta_f^+),$$
it follows from \eqref{3decom} that $\partial_p(\kappa(f,g_\alpha,h_\alpha)) = \kappa_p^f(f,g_\alpha, h_\alpha)\in H^1(\Q_p, V_{f}(N)^-\otimes V_{g h}^{\beta\beta}(N))$. 
\end{proof}
 Note that  analogous statements hold for   the three other classes attached in \eqref{eqn:classes-wt1}   to the other $p$-stabilisations of $g$ and $h$.  In particular, if the four associated singular parts    are non-zero,  
 they are necessarily linearly independent over $L_p$, 
 and therefore generate the target of $\partial_p$. 


\subsection{Dieudonn\'e modules and the Bloch-Kato logarithm}\label{blo}
 
Let $\mathbf{B}_{\dR}$ denote Fontaine's ring of de Rham $p$-adic periods.
Given a  representation $V$ of $G_{\Q_p}$ with coefficients in a finite extension $K/\Q_p$, 
the {\em de Rham Dieudonn\'e module} of $V$ is defined as
$$
D_{\dR}(V) := (V \otimes \mathbf{B}_{\dR})^{G_{\Q_p}}.
$$
It is a finite-dimensional $K$-vector space 
 equipped  with a descending exhaustive filtration $\Fil^j D_{\dR}(V)$ by $K$-vector subspaces.
%Let $M\geq 1$ be an integer and let  $\phi\in S_2(M)$ be a (not necessarily new) eigenform with respect to the good Hecke operators $T_\ell$, $\ell\nmid M$.  Let $V_\phi$ and $V_\phi(M)$ denote the %Galois representations associated to $\phi$ and $M$ in \S \ref{sec-Hida} with coefficients in $K:=K_\phi$; recall that the latter is a direct sum of several copies of the former. These representations are de %Rham and 
Falting's comparison theorem (cf.\,e.g.\cite{Fal2}) asserts that there is an isomorphism of filtered  modules
\begin{equation}
\label{eqn:faltings-comparison}
D_{\dR}(V_0(Np)) \simeq  H^1_{\dR}(X_0(Np)/\Q_p)(1), \qquad
D_{\dR}(V_1(Np^s)) \simeq  H^1_{\dR}(X_s/\Q_p)(1).
%, \quad D_{\dR}(V_\phi(M)) \simeq H^1_{\dR}(X_1(M)/\Q_p)\otimes_{\Q_p} K(1)[\phi].
\end{equation}
For $V=V_0(Np)$ and $V_1(Np^s)$, the filtration on $D_{\dR}(V)$ is given by
$$
\Fil^{-1} D_{\dR}(V) = D_{\dR}(V) \supsetneq \Fil^0 D_{\dR}(V) \supsetneq \Fil^1 D_{\dR}(V)=0
$$
and \eqref{eqn:faltings-comparison} identifies $\Fil^0 D_{\dR}(V)$ with $\Omega^1(X_0(Np)_{/\Q_p})$ and $\Omega^1(X_{s/\Q_p})$ respectively.



Let $\mathbf{B}_{\cris} \subset \mathbf{B}_{\dR}$ denote Fontaine's ring of crystalline $p$-adic periods. The {\em crystalline Dieudonn\'e modules} associated to the Galois representations $V_f$ and $V_f(Np)$ are
$$
D_f :=  (\mathbf{B}_{\cris}\otimes V_f)^{G_{\Q_p}}, \quad D_f(Np) :=  (\mathbf{B}_{\cris}\otimes V_f(Np))^{G_{\Q_p}}.
$$
These modules are vector spaces over $\Q_p$ equipped with a Hodge filtration and a $\Q_p$-linear action of a Frobenius automorphism $\Phi$, with eigenvalues $\alpha_f$ and $\beta_f$. As filtered modules, forgetting the operator $\Phi$, there is canonical isomorphism $D_f(Np)\simeq D_{\dR}(V_f(Np))$ and \eqref{eqn:faltings-comparison} identifies $D_f(Np)$ with $H^1_{\dR}(X_0(Np)/\Q_p)(1)[f]$. Via this identification, the regular differential form $\omega_{\testf}$ associated to a test vector $\testf \in S_2(Np)[f]$ gives rise to an element of $ \Fil^0 D_f(Np)$. Likewise, given  a dual test vector $ \testf^* \in S_2(Np)^\vee[f]$, let $\eta_{\testf^*}\in D_f(Np)$ denote the unique class such that $\Phi(\eta_{\testf^*}) = \alpha_f\cdot \eta_{\testf^*}$ and
\begin{equation}
\label{eqn:define-eta-f-star}
\langle    \eta_{\testf^*} ,  w_1 \omega\rangle_{X_0(Np)} = \testf^*(\omega) \qquad \mbox{ for all } \, \omega\in e_{\ord}\Omega^1(X_0(Np)).
\end{equation}

As in \eqref{subsubsec},
let $\phi\in S_2(Np^s,\chi_\phi)$ be an eigenform of weight two, level $Np^s$ 
and nebentype $\chi_\phi = \chi_0 \chi_p$. 
Assume that $\phi$ is {\em primitive at $p$} in the sense of \cite[\S 3]{mazur-wiles-mc}, i.e., the $p$-part of the primitive level of $\phi$ and the primitive conductor of $\chi_p$ are both $p^s$. 
The Galois representation $V_\phi$ (as well as $V_\phi(Np^s)$, of course)
is potentially cristalline; more precisely, it becomes cristalline 
 when restricted to $G_{\Q_p(\zeta_s)}$.
 The crystalline Dieudonn\'e modules
$$
 D_{\phi} :=  (\mathbf{B}_{\cris}\otimes V_{\phi})^{G_{\Q_p(\zeta_s)}}, \qquad  D_{\phi}(Np^s) :=  (\mathbf{B}_{\cris}\otimes V_{\phi}(Np^s))^{G_{\Q_p(\zeta_s)}}
$$
are vector spaces over $K=K_\phi$ equipped with 
\begin{itemize}
\item a Hodge filtration by $K\otimes_{\Q_p} \Q_p(\zeta_\phi)$-submodules of $D_\phi \otimes_{\Q_p} \Q_p(\zeta_\phi)$, such that \eqref{eqn:faltings-comparison} identifies $D_{\phi}(Np^s)$ with $H^1_{\dR}(X_s/\Q_p(\zeta_s))\otimes K(1)[\phi]$;
\item a $K$-linear action of a Frobenius operator $\Phi$, with eigenvalues $\alpha_\phi=a_p(\phi)$ and $\beta_\phi=\chi_0(p)\overline{a_p(\phi)}$.
\end{itemize}
%The {\em ordinary unit root subspace} $H^1_{\dR}(X_{s/\Q_p(\zeta_s)})^{\ord,\ur}$ is defined in the same way
% as above. 
Given a dual test vector $\phi^*\in S_2^{\ord}(Np^s)^\vee[\phi]$, let us denote by 
$\eta_{\phi^*}\in D_\phi$ the unique class satisfying $\Phi(\eta_{\phi^*}) = \alpha_\phi\cdot \eta_{\phi^*}$ and
\begin{equation}
\label{eqn:define-eta-phistar}
\langle \eta_{\phi^*},   w_s \omega\rangle_s = \phi^*(\omega) \qquad \mbox{ for all }  \, \omega \in \Omega^1(X_s)^{\ord}.
\end{equation}
If $\hphi^* := \{ \phi_s^*\}_{s\geq 1}$ is a collection of elements  $ \phi^*_s\in S_2^{\ord}(Np^s)^\vee$ which are compatible under the maps $(\varpi_1^*)^\vee$ dual to the pullback by $\varpi_1$, then the corresponding 
elements $\eta_{\phi_s^*}$ in $H^1_{\dR}(X_{s/\Q_p(\zeta_s)})$ are compatible under the transition maps 
$\varpi_{2*}$, and hence describe an element
 of $H^1_{\dR}(X_\infty)$.

The weight two specialisations of the $\Lambda$-adic test vectors $\testhg \in S^{\ord}_{\Lambda_{g}}(N,\chi)[\hg]$ and $\testhh\in
S^{\ord}_{\Lambda_{h}}(N,\chi^{-1})[\hh]$ in the introduction provide natural examples of eigenforms $\phi$ that are primitive at $p$. Namely, let $\epsilon$ be a Dirichlet character of conductor $p^s$ and let $(y,z) \in \Omega_g \times_\Omega \Omega_h$ be a classical point of weight $(2,\epsilon)$. Assuming that $\epsilon \ne \omega$, the eigenforms $g_y$, $h_z$ and $\testg_y\in S_2(Np^s,\chi \epsilon \omega^{-1})$ and $\testh_z \in S_2(Np^s, \chi^{-1} \epsilon \omega^{-1})$ are primitive at $p$. Let
$$D_{f g_y h_z}(N) :=  (\mathbf{B}_{\cris}\otimes_{\Q_p} V_{f g_y h_z}(N))^{G_{\Q_p(\zeta_s)}} \simeq D_f(Np)\otimes D_{g_y}(Np^s)\otimes D_{h_z}(Np^s)(-1)$$
denote the Dieudonn\'e module associated to the Galois representation $V_{f g_y h_z}(N)$ introduced in \eqref{VfghN}. 

%This is a vector space over $K$ equipped with 
%\begin{itemize}
%\item a  filtration by $K\otimes_{\Q_p} \Q_p(\zeta_s)$-submodules of $D_{f g_y h_z}(N) \otimes_{\Q_p} \Q_p(\zeta_s)$ such that 
%$$D_{f g_y h_z}(N)\otimes_{\Q_p} \Q_p(\zeta_s) \simeq D_{\dR}(V_{f g_y h_z}(N))\otimes_{\Q_p} \Q_p(\zeta_s)\quad \mbox{ as filtered modules;}$$

%\item a $K$-linear action of a Frobenius automorphism $\Phi$, whose .

%\end{itemize}

Let $H^1_{\exp}(\Q_p(\zeta_s),V_{f g_y h_z}(N)) \subseteq H^1_{\fin}(\Q_p(\zeta_s),V_{f g_y h_z}(N)) \subseteq H^1(\Q_p(\zeta_s),V_{f g_y h_z}(N))$ be the subspaces introduced in \cite{BK}  by Bloch and Kato and let 
\begin{equation*}
 \log_p : H^1_{\exp}(\Q_p(\zeta_s),V_{f g_y h_z}(N)) \lra D_{f g_y h_z}(N)/(\Fil^0(D_{f g_y h_z}(N)) + D_{f g_y h_z}(N)^{\Phi=1})
 \end{equation*}
denote the Bloch-Kato logarithm map of loc. cit. 
%where
%\begin{equation}\label{ab}
%\alpha_{g_y} = {a}_p({g_y}), \quad \beta_{g_y} =  \chi(p)\overline{a_p({g_y})}, \quad \alpha_{h_z} = {a}_p({h_z}), \quad \beta_{h_z} =  \chi^{-1}%(p)\overline{a_p({h_z})}.
%\end{equation}

Since the eigenvalues of the Frobenius operator $\Phi$ acting on $D_{f g_y h_z}(N)$ are $\alpha_f \alpha_{g_y} \alpha_{h_z},\ldots, \beta_f \beta_{g_y} \beta_{h_z}$, they
all are algebraic numbers of complex absolute 
value $p^{3/2}$, and it follows from \cite[Corollary 3.8.4]{BK} that 
$D_{f g_y h_z}(N)^{\Phi=1}=0$ and $H^1_{\exp}(\Q_p(\zeta_s),V_{f g_y h_z}(N))
 = H^1_{\fin}(\Q_p(\zeta_s),V_{f g_y h_z}(N))$. 
  Bloch-Kato's logarithm map may therefore be recast as a map
\begin{equation}
\label{eqn:bloch-kato-logarithm-bis}
 \log_p : H^1_{\fin}(\Q_p(\zeta_s),V_{f g_y h_z}(N)) \lra D_{f g_y h_z}(N)/\Fil^0(D_{f g_y h_z}(N)) \simeq \Fil^0(D_{f g_y h_z}(N))^\vee,
 \end{equation}
where the latter isomorphism is induced by Poincar\'e duality.



The $p$-adic syntomic Abel-Jacobi map is then defined  as the composition
\begin{equation}
\label{AJpsyn}
\xymatrixcolsep{0.4in}
\xymatrix{
\AJ_p: \CH^2(W_s)_0(\Q_p(\zeta_s))  \ar[rr]^-{\varpi_{f g_y h_z} \circ \AJ_{\et}}
&& H^1_{\fin}(\Q_p(\zeta_s),V_{f g_y h_z}(N)) \ar[r]^-{\log_p}  & \Fil^0(D_{f g_y h_z}(N))^\vee. }
\end{equation}

 \subsection{Ohta's periods and $\Lambda$-adic test vectors}\label{sec:Ohta}



Let $\hat{\Z}_{p}^{\mathrm{ur}}$ denote the completion of the ring of integers in the maximal
unramified extension $\hat\Q_p^{\mathrm{ur}}$ of $\Q_p$. Recall the convention whereby the symbol $\hat\otimes$ denotes 
the completed tensor product, so that, for example, $\Lambda\, \hat\otimes\,\hat\Z_{p}^{\mathrm{ur}}$ 
is  isomorphic to the power series ring $\hat\Z_p^{\mathrm{ur}}[[T]]$. 

Define $D(V_1^{\ord}(Np^s)^-) := (V_1^{\ord}(Np^s)^- \otimes \hat\Z_p^{\ur})^{G_{\Q_p}}$. Since the modules $V_1^{\ord}(Np^s)^-$ are unramified, they are admissible for the period ring 
${\hat\Z_p}^{\ur}$ and hence $D_{\dR}(V_1^{\ord}(Np^s)^-) =  D(V_1^{\ord}(Np^s)^-)$. Define also
$$
 \hD(\hV_1^{\ord}(Np^\infty)^-) := (\hV_1^{\ord}(Np^\infty)^-\hat\otimes \hat\Z_p^{\ur})^{G_{\Q_p}} = \invlim_s D(V_1^{\ord}(Np^s)^-).
$$

Recall that the comparison theorem of \eqref{eqn:faltings-comparison} leads to a natural inclusion
 $$ S_2^{\ord}(Np^s) = e_{\ord} \Omega^1(X_s) \lra e_{\ord}H^1_{\dR}(X_s) = D_{\dR}(V_1^{\ord}(Np^s)).
 $$
Write 
$$ \cmp_s: S_2^{\ord}(Np^s) \lra D(V_1^{\ord}(Np^s)^-)$$
for the natural map obtained by composing this inclusion
with the natural projection
$$ D_{\dR}(V_1^{\ord}(Np^s)) \lra  D(V_1^{\ord}(Np^s)^-)$$
arising from the $G_{\Q_p}$-invariant filtration on $V_1^{\ord}(Np^s)$. 


 
 We begin by recalling a theorem of Ohta \cite{ohta}, as refined subsequently by Wake \cite{Wake}
 which provides a $p$-adic interpolation of the maps $\cmp_s$, relating  the space
  $S_\Lambda^{\ord}(N)$ of ordinary $\Lambda$-adic modular forms to the Galois
   representation $\hV_1^{\ord}(Np^\infty)$.
 \begin{proposition}[Ohta]
 There is a canonical isomorphism
 $$  \cmp_{\infty}: S_\Lambda^{\ord}(N) \lra   \hD(\hV_1^{\ord}(Np^\infty)^-)$$
 characterised by the commutativity of  the diagram
 $$ \xymatrix{
 S_\Lambda^{\ord}(N) \ar[r]^-{\cmp_{\infty}} \ar[d]_{\mho_s} & \hD(\hV_1^{\ord}(Np^\infty)^-) \ar[d]\\
 S_2^{\ord}(Np^s) \ar[r]^-{\cmp_s}   & D(V_1^{\ord}(Np^s)^-) }
 $$
 where the left  vertical arrow sends the ordinary $\Lambda$-adic form
 $\hg = \{g_s\}$ to $\mho_s(g_s) =  U_p^{-s} g_s$,  and the right vertical arrow
 denotes the natural specialisation map of level $p^s$.
 \end{proposition}
 The reader will recall that the system $\{ \mho_s(g_s)\}$ of ordinary 
 modular forms on $X_s$ are compatible under  the pushforward  maps
 $\varpi_{2*}$, in harmony with the fact that the inverse limit defining  $\hD(\hV_1^{\ord}(Np^\infty)^-))$ is also taken relative to these maps.
 The statement differs slightly from the one in Ohta's work,
 which focusses on the anti-ordinary part of the \'etale cohomology of the tower $X_s$,
 in which the inverse limits are taken relative to the maps $\varpi_{1*}$. Working with the ordinary part obviates the application of the automorphism $w_{s}$ to the specialisation at level $s$ which occurs in Ohta's result.
 
 Since the $U_p$ operator (and hence, the ordinary projector, as well) are self-adjoint relative
 to the  twisted $\Lambda$-adic
  Weil pairing $[ \ , \ ]_{\Gamma}$  of 
  \eqref{eqn:ohta-pairing-etale}, this pairing
induces  a pairing on the ordinary parts, 
which shall be denoted by the same symbol:
 \begin{equation}
\label{eqn:ohtaB}
 [ \, ,  ]_{\Gamma}: \hV_1^{\ord}(Np^\infty) \times \hV_1^{\ord}(Np^\infty) \lra \Lambda(\uepsilon). 
 \end{equation}
The submodule 
$ \hV_1^{\ord}(Np^\infty)^+ $ is isotropic for this pairing, which therefore induces a natural $\Lambda(\uepsilon)$-valued duality between $ \hV_1^{\ord}(Np^\infty)^+ $ and the quotient  $ \hV_1^{\ord}(Np^\infty)^-$.
 (This duality is the one described, for instance, in 
Theorem B of  \cite{ohta}.) 
Combining $[\ ,\ ]_\Gamma$ with Ohta's map $\cmp_\infty$ yields  
 canonical $G_{\Q_p}$-equivariant $\Lambda$-linear maps,  
as described in \cite[Theorem 1.3]{Wake}:
\begin{align}
 \label{ohtawake}
\underline{c}^+: & S_{\Lambda}^{\ord}(N)   \lra \Hom_\Lambda(\hV_1^{\ord}(Np^\infty)^+,  \Lambda(\uepsilon) \ {\hat \otimes} \  {\hat  \Z_p}^{\ur}),   \\ 
\nonumber
  \underline{c}^-:  &  S_{\Lambda}^{\ord}(N)^\vee  \lra\Hom_\Lambda(\hV_1^{\ord}(Np^\infty)^-,  \Lambda  \ {\hat \otimes} \  {\hat  \Z_p}^{\ur}),
 \end{align}
 which are given by the explicit formulae
 \begin{align}
 \label{eqn:explicit-for-c+}
 \underline{c}^+(\hphi)(\hxi) & = [\mho_\infty(\hphi), \hxi]_{\Gamma}  
   =  \left\{ \langle U_p^{-s} \phi_s,  w w_s U_p^s \xi_s  \rangle_{\Gamma_s} \right\}_{s\ge 1}  \\
   \nonumber
   &  =  \left\{ \langle  \phi_s,   w w_s  \xi_s \rangle_{\Gamma_s} \right\}_{s\ge 1} = 
     \left\{ -\langle  \xi_s,   (w w_s)^{-1}  \phi_s \rangle_{\Gamma_s} \right\}_{s\ge 1}.
         \\
    \label{eqn:explicit-for-c-}
 \underline{c}^-(\hphi^*)(\hxi)  & =   [  \hphi^*, \hxi]_\Gamma 
     = \left\{  -\langle \xi_s,  (w w_s)^{-1} U_p^s  \eta_{\phi_s^*} \rangle_{\Gamma_s} \right\}_{s\ge 1}.
  \end{align}
After tensoring over $\Lambda$ with $\Lambda_g$ and taking  $\hg$-isotypic parts in \eqref{ohtawake}, one deduces $\Lambda_g$-linear $G_{\Q_p}$-equivariant homomorphisms
 \begin{align}
 \label{ohtawakeg+}
 \underline{c}_{g}^+: & S_{\Lambda_g}(N)[\hg]   \lra \Hom_{\Lambda_g}(\hV_g^+(N),  \Lambda_g(\uepsilon) \ {\hat \otimes} \  {\hat  \Z_p}^{\ur}),   \\ 
 \label{ohtawakeg-}
  \underline{c}_{g}^-:  &  S_{\Lambda_g}(N)^\vee[\hg]   \lra\Hom_{\Lambda_g}(\hV_g^-(N),  \Lambda_g  \ {\hat \otimes} \  {\hat  \Z_p}^{\ur}),
 \end{align}
 which are described by the rules, deduced from \eqref{eqn:explicit-for-c+} and 
 \eqref{eqn:explicit-for-c-},
 \begin{align}
 \label{eqn:explicit-for-c+g}
 \underline{c}_g^+(\testhg)(\hxi) &   
    =  \left\{ -\langle \xi_s,  (w w_s)^{-1}   \testg_s \rangle_{\Gamma_s} \right\}_{s\ge 1}, \\
    \label{eqn:explicit-for-c-g}
 \underline{c}_g^-(\testhg^*)(\hxi)  &  
   = \left\{  \alpha_{\hg_s}^s  \langle \xi_s,  (w w_s)^{-1}  \eta_{\testg_s^*} \rangle_{\Gamma_s} \right\}_{s\ge 1}.
  \end{align}



 If  $\tilde\Lambda$ is any complete Noetherian $\Z_p$-algebra and  $\Psi: G_{\Q_p} \lra  \tilde \Lambda^\times$ is  an unramified character, it is convenient to define
 $$
\tilde\Lambda\{\Psi\} := (\tilde\Lambda \ {\hat \otimes} \  {\hat  \Z_p}^{\ur})^{\Psi}
$$
to be the $\tilde\Lambda$-submodule
of $\tilde\Lambda\ {\hat \otimes} \  {\hat  \Z_p}^{\ur}$ 
on which $G_{\Q_p}$ acts via the character $\Psi$. 
As explained in \cite[Prop.~3.3]{LZ},
 $\tilde\Lambda\{\Psi\}$ is non-canonically isomorphic to $\tilde\Lambda(\Psi) $
as  a $\tilde\Lambda[G_{\Q_p}]$-module. The  advantage of working with $\tilde\Lambda\{\Psi\}$ is that one has a {\em canonical} identification
\begin{equation}
\label{eqn:the-whole-point-of-introducing-this-notation-in-the-first-place}
 \hD( \tilde\Lambda\{\Psi\}) = (((\tilde\Lambda\ \hat{\otimes} \ {\hat\Z_p^{\ur}})^{\Psi}) \  \hat  \otimes \  {\hat\Z_p^{\ur}})^{G_{\Q_p}} = \tilde\Lambda,
 \end{equation}
 the last equality being simply induced from  the multiplication on the coefficients. 
 
The  $G_{\Q_p}$- equivariance of the   maps \eqref{ohtawakeg+} and 
 \eqref{ohtawakeg-} above shows that 
 $\underline{c}_g^+(\testhg)$  and 
  $\underline{c}_g^-(\testhg^*)$ give rise to $G_{\Q_p}$-equivariant 
  $\Lambda_g$-linear maps
 $$
\underline{c}_g^+(\testhg):  \hV_g(N)^+\lra \Lambda\{\Psi_{\hg}^{-1}\}(\uepsilon), 
\qquad 
\underline{c}_g^-(\testhg^*): 
 \hV_{g}(N)^-\lra \Lambda\{\Psi_{\hg}\},$$
with similar remarks applying, of course, to the elements
$\underline{c}_h^+(\testhh)$ and $\underline{c}_h^-(\testhh^*)$. 
The maps $ \underline{c}_g^+(\testhg)$ and $\underline{c}_g^-(\testhg^*)$ 
also admit simpler variants associated to the fixed forms $\testf$ and $\testf^*$,
 $$
{c}_f^+(\testf):  V_f(Np)^+\lra \Q_p\{\psi_{f}^{-1}\}(\epsilon_{\cyc}), 
\qquad 
\underline{c}_f^-(\testf^*): 
 V_{f}(Np)^-\lra \Q_p\{\psi_{f}\}. $$




 
 Recall from Lemma \ref{lemma:4-step-filtration} that there are canonical isomorphisms of $\Lambda_{f \underline{ gh}}[G_{\Q_p}]$-modules
\begin{eqnarray*}
\hV_{f \underline{g h}}^f(N) &\simeq & 
V_f(Np)^-\otimes \hV_{\hg}(N)^+ \otimes_{\Lambda} \hV_{\hh}(N)^+(\uepsilon^{-1}),\\ 
\hV_{f \underline{g h}}^g(N) &\simeq &V_f(Np)^+\otimes \hV_{\hg}(N)^- \otimes_{\Lambda} \hV_{\hh}(N)^+(\uepsilon^{-1}),
\\ 
 \hV_{f \underline{g h}}^h(N) &\simeq &V_f(Np)^+\otimes \hV_{\hg}(N)^+ \otimes_{\Lambda} \hV_{\hh}(N)^-(\uepsilon^{-1}).
\end{eqnarray*}
Choose a triple of $\Lambda$-adic test vectors
\begin{equation}
\label{eqn:test-vectors2}
\testf \in S_2(Np)[f], \qquad \testhg \in S^{\ord}_{\Lambda_{g}}(N,\chi)[\hg], \qquad \testhh\in
S^{\ord}_{\Lambda_{h}}(N,\chi^{-1})[\hh], 
\end{equation}
and   a    triple of   dual test vectors 
\begin{equation}
\label{eqn:dual-test-vectors2}
 \testf^* \in S_2(Np)^\vee[f], 
 \qquad \testhg^* \in S^{\ord}_{\Lambda_{g}}(N,\chi^{-1})^\vee[\hg], \qquad \testhh^*\in
S^{\ord}_{\Lambda_{h}}(N,\chi)^\vee[\hh]
\end{equation}
as in the introduction. 

We can then define the 
  local $\Lambda$-adic cohomology classes 
\begin{eqnarray}
\nonumber
  \kappa_p^f(\testf^*,\underline{\testg\testh}) := c_f^-(\testf^*) \otimes \underline{c}_g^+(\testhg) \otimes \underline{c}_h^+(\testhh)(\kappa_p^f(f,\underline{gh})) &\in & H^1(\Q_p, \Lambda_{f \underline{g h}}\{ \psi_{f}    \Psi_{\hg}^{-1} \Psi_{\hh}^{-1} \}(\uepsilon)), \\
 \label{eqn:ohta-classes}
    \kappa_p^g(\testf,\underline{\testg^*\testh}) :=   c_f^+(\testf) \otimes \underline{c}_g^-(\testhg^*) \otimes \underline{c}_h^+(\testhh)(\kappa_p^g(f,\underline{gh})) &\in&H^1(\Q_p, \Lambda_{f \underline{g h}}\{ \psi_{f}^{-1}\Psi_{\hg}   \Psi_{\hh}^{-1}  \chi^{-1}  \}(\epsilon_{\cyc})), \\
    \nonumber
  \kappa_p^h(\testf, \underline{\testg\testh^*}) := c_f^+(\testf) \otimes \underline{c}_g^+(\testhg) \otimes \underline{c}_h^-(\testhh^*)(\kappa_p^h(f,\underline{gh})) &\in& H^1(\Q_p, \Lambda_{f \underline{g h}}\{ \psi_{f}^{-1} \Psi_{\hg}^{-1} \Psi_{\hh}    \chi \}(\epsilon_{\cyc})).
  \end{eqnarray}
%  by setting
 % \begin{eqnarray*}
 % \kappa_p^f(\testf^*,\underline{\testhg\testhh}) & := & 
 % \eta_{\testf^*} \otimes \underline{c}_g^+(\testhg) \otimes \underline{c}_h^+(\testhh)
 %  \    (\kappa_p^f(f,\underline{gh})), \\
 %    \kappa_p^g(\testf,\underline{\testhg^*\testhh}) & := & 
 % \omega_{\testf} \otimes \underline{c}_g^-(\testhg^*) \otimes \underline{c}_h^+(\testhh) 
 %  \   (\kappa_p^g(f,\underline{gh})), \\
  %  \kappa_p^h(\testf,\underline{\testg\testh^*}) & := & 
 % \omega_{\testf} \otimes \underline{c}_g^+(\testhg) \otimes \underline{c}_h^-(\testhh^*)  
 % \   (\kappa_p^h(f,\underline{gh})). 
 % \end{eqnarray*}
The modules in which the above classes take values are non-canonically isomorphic 
to $\Lambda_{f \underline{g h}}(\heta_{\hf}^{+})$, $\Lambda_{f \underline{g h}}(\heta_{\hg}^{+})$ and $\Lambda_{f \underline{g h}}(\heta_{\hh}^{+})$, respectively, but  elements of the former 
incorporate a choice of period (in the style of \cite{LZ})
for the relevant unramified Galois characters.

As a piece of notation, we let
 \begin{eqnarray}
\label{eqn:specialised-ohta-classes}
  \kappa_p^f(\testf^*,\testg_y,\testh_z) &\in & H^1(\Q_p, K_x\{ \psi_{f}    \Psi_{g_y}^{-1} \Psi_{h_z}^{-1} \}(\epsilon^{\ell -1}_{\cyc} \epsilon  \omega^{1-\ell})), \\
  \nonumber
    \kappa_p^g(\testf,\testg_y^*,\testh_z) &\in&H^1(\Q_p, K_x\{ \psi_{f}^{-1}\Psi_{g_y}   \Psi_{h_z}^{-1} \times \chi^{-1}  \}(\epsilon_{\cyc})), 
%\\    \nonumber  \kappa_p^h(\testf,\testg_y,\testh_z^*) &\in& H^1(\Q_p, K_x\{ \psi_{f}^{-1} \Psi_{\hg}^{-1} \Psi_{\hh}  \times \chi \}(\epsilon_{\cyc}))
  \end{eqnarray}
denote the local  cohomology classes obtained by specialising the $\Lambda$-adic classes of  \eqref{eqn:ohta-classes}  at a classical point $x:= (y,z) \in \Omega_g \times_\Omega \Omega_h$ of weight $(\ell,\epsilon)$ for some positive integer $\ell \geq 1$ and character $\epsilon$ of conductor $p^s$, corresponding to a ring homomorphism $x: \Lambda_{f\underline{gh}} \lra K_x$
 to a finite extension $K_x$ of $\Q_p$.




Our most immediate goal  is now to relate the images of 
 the classes $\kappa_p^f(\testf^*,\underline{\testg\testh})$ and $ \kappa_p^g(\testf,\underline{\testg^*\testh})$ 
under Perrin-Riou's $\Lambda$-adic logarithm map
 to the restriction to $\Omega_{f \underline{g h}}$ of the Garrett-Hida $p$-adic $L$-functions $\Lp^{f_\alpha}(\testhf^*, \testhg, \testhh)$ and
$\Lp^{g_\alpha}(\testhf, \testhg^*, \testhh)$,
respectively. (A   similar relation exists   between $\Lp^{h_\alpha}(\testhf, \testhg, \testhh^*)$ and
 $\kappa_p^h(\testf,\underline{\testg\testh^*})$, since    $g$ and $h$ play  symmetric roles in all the constructions;
 the details have been omitted because the latter relation plays no role
  in the proof of the  main theorems of the article.)

The family $\kappa_p^f(\testf^*,\underline{\testg\testh})$ and the $p$-adic $L$-function $\Lp^{f_\alpha}(\testhf^*, \testhg, \testhh)$ play a crucial role in the proofs of Theorems A and C, while 
$\kappa_p^g(\testf,\underline{\testg^*\testh})$ and $\Lp^{g_\alpha}(\testhf, \testhg^*, \testhh)$ 
are the main ingredient in the proofs of Theorems B and D. It is worth remarking
 that the family of cohomology classes $\kappa_p^f(\testf^*,\underline{\testg\testh})$, whose underlying representation space involves the  $\Lambda$-adic cyclotomic character,
is of greater arithmetic complexity than the family $\kappa_p^g(\testf, \underline{\testg^*\testh})$,
whose restriction to the inertia group at $p$ is equal to $\epsilon_{\cyc}$, and whose  specialisations  are therefore
independent of $(y,z)$ when restricted to inertia.




 \subsection{The logarithm of weight two specialisations}
 \label{subsec:log}
 
Let $\epsilon\ne \omega$ be a Dirichlet character of conductor $p^s$ and let $x:= (y,z) \in \Omega_g \times_\Omega \Omega_h$ be a classical point of weight $(2,\epsilon)$. Put $K=K_x=K_{f g_y h_z}$ as above and let $\kappa_p^f(\testf^*,\testg_y,\testh_z)$ and $\kappa_p^g(\testf,\testg_y^*,\testh_z)$ be the local cohomology classes introduced in \eqref{eqn:specialised-ohta-classes}. Since the Hodge-Tate weights of the target 
  Galois representations are all $-1$, the Bloch-Kato $p$-adic 
logarithm maps induce isomorphisms 
\begin{eqnarray}\label{logs}
 \log_p&:&H^1(\Q_p,K_x\{ \psi_{f}    \Psi_{g_y}^{-1} \Psi_{h_z}^{-1} \}(\epsilon_{\cyc}  \epsilon  \omega^{-1}))  \lra K  \\ \nonumber
 \log_p &:& H^1(\Q_p, K_x\{ \psi_{f}^{-1}\Psi_{g_y}   \Psi_{h_z}^{-1} \times \chi^{-1}  \}(\epsilon_{\cyc}))  \lra K.
 %\\  \nonumber  \log_p &:& H^1(\Q_p, K_x\{ \psi_{f}^{-1} \Psi_{\hg}^{-1} \Psi_{\hh}  \times \chi \}(\epsilon_{\cyc}))\lra K_x .
    \end{eqnarray} 
We refer to \eqref{id-B} in \S \ref{subsec:PR} below for a more detailed discussion of this fact in a more general setting and a discussion of the periods involved in the above isomorphisms.



The goal of this section is to give an explicit formula for the Bloch-Kato $p$-adic logarithms of the classes $\kappa_p^f(\testf^*,\testg_y,\testh_z)$ and $\kappa_p^g(\testf,\testg_y^*,\testh_z)$ in terms of the image of the twisted diagonal cycles introduced in the previous chapter under the $p$-adic syntomic Abel-Jacobi map \eqref{AJpsyn}. 
 





Recall the de Rham cohomology classes of \eqref{eqn:define-eta-f-star} and \eqref{eqn:define-eta-phistar}, and let 
\begin{align*}
  & \omega_{\testf}^\circ := (w w_s)^{-1} \omega_{\testf}, \qquad
\omega_{\testg_y}^{\circ} := (w w_{s})^{-1} \omega_{\testg_y}, \qquad \omega_{\testh_z}^{\circ} = (w w_s)^{-1} \eta_{\testh_z},\\
  & \eta_{\testf^*}^\circ := (w w_s)^{-1} \eta_{\testf^*}, \qquad
\eta_{\testg_y^*}^{\circ} := (w w_{s})^{-1} \eta_{\testg_y^*}, \qquad \eta_{\testh_z^*}^{\circ} = (w w_s)^{-1} \eta_{\testh_z^*},
\end{align*}
viewed as elements of  the anti-ordinary part of $H^1_{\dR}(X_0(Np)_{/\Q_p})$ and $H^1_{\dR}(X_{s /\Q_p(\zeta_s)})$, respectively.
The triple tensor classes
  $$  
  \eta_{\testf^*}^{\circ} \otimes \omega_{\testg_y}^\circ \otimes \omega_{\testh_z}^\circ, 
  \qquad
    \omega_{\testf}^{\circ} \otimes \eta_{\testg_y^*}^\circ \otimes \omega_{\testh_z}^\circ, 
    \qquad
  \omega_{\testf}^{\circ} \otimes \omega_{\testg_y}^\circ \otimes \eta_{\testh_z^*}^\circ, 
  $$
 are thus elements in the anti-ordinary part of 
$H^3_{\dR}(W^\dag_{{s,s}/\Q_p(\zeta_s)}) =
   H^3_{\dR}(W_{{s,s}/\Q_p(\zeta_s)})$. Since the $N$-part (resp.\,the $p$-part) of the nebentype characters of $g_y$ and $h_z$ are inverse one 
   of another (resp.\,equal), these classes are invariant under the action of the group
$D_s$ of diamond operators acting {\em diagonally at $N$} and
 {\em anti-diagonally at $p$} on $W_{s,s}$, hence they descend to 
 classes in $H^3_{\dR}(W_{s/\Q_p(\zeta_s)})$, denoted 
$$
\eta_{\testf^*}^\circ\otimes \omega_{\testg_y}^\circ \omega_{\testh_z}^\circ, \quad \omega_{\testf}^\circ\otimes \eta_{\testg^*_y}^\circ \omega_{\testh_z}^\circ, \quad \omega_{\testf}^\circ\otimes \omega_{\testg_y}^\circ  \eta_{\testh^*_z}^\circ \in H^3_{\dR}(W_{s/\Q_p(\zeta_s)}),
$$  
which are characterized by the relation
\begin{equation}
\label{eqn:class-on-quotient}
 \pr_s^*(\eta_{\testf^*}^\circ\otimes \omega_{\testg_y}^\circ \omega_{\testh_z}^\circ) = \eta_{\testf^*}^\circ\otimes \omega_{\testg_y}^\circ \otimes \omega_{\testh_z}^\circ,
 \end{equation}
and likewise for the other two. Faltings' comparison isomorphism \eqref{eqn:faltings-comparison} together with K\"unneth's formula \eqref{kun} allows to regard them as elements of $\Fil^0(D_{f g_y h_z}(N))$ and it thus makes sense to evaluate the functional $ \AJ_p(\Delta_{s})$ at these classes. 
\begin{proposition}
\label{prop:from-log-to-AJ}
For all arithmetic points $x=(y,z)$ of weight $(2,\epsilon)$ as above, the specialisations
$$ \kappa_p^f(\testf^*, \testg_y, \testh_z) :=   \kappa_p^f(\testf^*, \underline{\testg \testh})(y,z), 
\qquad 
 \kappa_p^g(\testf, \testg_y^*, \testh_z) :=   \kappa_p^g(\testf, \underline{\testg^* \testh})(y,z) $$
 satisfy the formulae
\begin{eqnarray*}
\log_p(\kappa_p^f(\testf^*,\testg_y,\testh_z)) &=&  \alpha_g^{-s}   \AJ_p(\Delta_{s})(\eta^{\circ}_{\testf^*} \otimes \omega_{\testg_y}^{\circ}   
  \omega_{\testh_z}^{\circ}), \\
\log_p(\kappa_p^g(\testf,\testg_y^*,\testh_z)) &=&       \AJ_p(\Delta_{s})(\omega_{\testf}^\circ \otimes \eta_{\testg_y^*}^{\circ}  
  \omega_{\testh_z}^{\circ}).
\end{eqnarray*}
\end{proposition}
\begin{proof}
Let $\langle \ , \ \rangle_{W_s}$ denote the Poincar\'e pairing (in both its \'etale and de Rham incarnations) on the $(1,1,1)$-K\"unneth component of the middle cohomology of $W_s$, which is related to the Poincar\'e duality on the curves $X_0(Np)$ and  $X_s $ by the rules
$$ \langle \eta_1\otimes \eta_2  \eta_3, \omega_1\otimes\omega_2  \omega_3\rangle_{W_s} = 
\langle \eta_1,\omega_1\rangle_{X_0(Np)} \langle \eta_2,\omega_2\rangle_s \langle \eta_3,\omega_3\rangle_s.$$
Let $ \langle \ , \ \rangle_{W_s,\Gamma_s}$ denote the group-ring valued pairing constructed
from the above by setting
$$ \langle \eta_1\otimes \eta_2  \eta_3, \omega_1\otimes\omega_2  \omega_3\rangle_{W_s, \Gamma_s} = 
\langle \eta_1,\omega_1\rangle_{X_0(Np)} \langle \eta_2,\omega_2\rangle_{\Gamma_s} \langle \eta_3,\omega_3\rangle_{\Gamma_s}.$$
The formulae for $\underline{c}_g^+(\testhg)$ and $\underline{c}_h^+(\testhh)$  described in equation
\eqref{eqn:explicit-for-c+g}  lead to the following equality in the local cohomology group
 $H^1(\Q_p, K_x(\Upsilon_f^+(y,z)))$, where $\kappa_p(f, \underline{gh})_s$ denotes the natural projection of the ordinary $\Lambda$-adic class $\kappa_p(f, \underline{gh})$ to the $s$-th component in the inverse limit:
\begin{equation}
\label{eqn:formula-for-kappa-f}
 \kappa_p^f(\testf^*, \testg_y,\testh_z) := \nu_{(y,z)} (\kappa_p^f(\testf^*, \testhg,\testhh)) = 
\nu_{(y,z)} (\langle \kappa_p(f, \underline{gh})_s, \eta_{f^*}^\circ \otimes \omega_{\testg_s}^\circ \omega_{\testh_s}^\circ \rangle_{W_s, \Gamma_s}).
\end{equation}
It then follows from Lemma \ref{Gammas} that
\begin{equation}
\label{eqn:formula-for-kappa-f-bis}
 \kappa_p^f(\testf^*, \testg_y,\testh_z) =   
\langle \kappa_p(f, g_y, h_z)_s, \eta_{f^*}^\circ \otimes \omega_{\testg_y}^\circ \omega_{\testh_z}^\circ \rangle_{W_s}.
\end{equation}
Proposition \ref{prop:specialisation-wt-2} thus yields $\kappa_p^f(\testf^*,\testg_y,\testh_z) = 
\alpha_{g_y}^{-s}
\langle  \AJ_{\et}(\Delta_s), \eta_{\testf^*}^\circ \otimes \omega_{\testg_y}^\circ \omega_{\testh_z}^\circ \rangle_{W_s}$
and the first assertion of the proposition follows by appying the Bloch-Kato logarithm to both sides.
The second is proved by the same calculation, but replacing 
\eqref{eqn:formula-for-kappa-f} by the analogous formula arising from
both \eqref{eqn:explicit-for-c+g} and \eqref{eqn:explicit-for-c-g}, namely
%\begin{equation}
%\label{eqn:formula-for-kappa-g}
$\kappa_p^g(\testf, \testg_y^*,\testh_z) =  
\alpha_{g_y}^s  \nu_{(y,z)}(\langle \kappa(f,\underline{gh}), \eta_{f}^\circ \otimes \eta_{\testg_s^*}^\circ \omega_{\testh_s}^\circ \rangle_{W_s, \Gamma_s})$.
%\end{equation}
\end{proof}

For the later calculations, it will be convenient to dispose of a simpler formula for 
$\log_p(\kappa_p^f(\testf^*,\testg_y,\testh_z))$  and 
 $\log_p(\kappa_p^g(\testf,\testg_y^*,\testh_z))$ involving the product variety $W_{s,s}$ rather than its quotient $W_{s}$, and the classes 
 $\eta_{\testf^*} \otimes \omega_{\testg_y}   
  \omega_{\testh_z}$ and 
  $\omega_{\testf} \otimes \eta_{\testg_y^*}   
  \omega_{\testh_z}$  instead of their anti-ordinary counterparts. To do this, 
%let $((ww_1)^{-1},  (w w_s)^{-1}, (w w_s)^{-1})$ denote the automorphism of the three-folds 
%  $W_{s,s}$ and $W_s$ associated to this triple of automorphisms of $X_0(Np)$ and $X_s$, and let 
%$\Delta_{s,s}^\circ$    
define the cycle
\begin{align*}
 \Delta_{s,s}^\circ & := ( (w w_1)^{-1}, (w w_s)^{-1}, (w w_s)^{-1})_* \Delta_{s,s}\in \CH^2(W_{s,s})_0(\Q_p(\zeta_N \zeta_s)).
  \end{align*}
Equations \eqref{eqn:def-Deltaf} and 
\eqref{def-Deltas} show that $\Delta_{s,s}^{\circ}$ 
is given by the simple equation 
\begin{align}\label{eqn:def-Deltaf-circ}
 \Delta_{s,s}^\circ  &  =   ( (w w_1)^{-1}, (w w_s)^{-1}, (w w_s)^{-1})_*  
 \varepsilon_{s,s*}
(j_1 \circ \varpi_{2}^{s-1} \circ w_{s}, \id ,w_{s})_* \delta_* (X_s)  \\ \nonumber
& = \varepsilon_{s,s*} (j_1 \circ \varpi_1^{s-1},  w_s^{-1}, \Id)_* (w^{-1}, w^{-1},w^{-1})_*  \delta_* (X_s) = \varepsilon_{s,s*} (j_1 \circ \varpi_1^{s-1},  w_s^{-1}, \Id)_*  \delta_* (X_s).
 \end{align}
 \begin{corollary}
\label{cor:from-log-to-AJ}
For all arithmetic points $x=(y,z)$ of weight $(2,\epsilon)$,  
\begin{eqnarray*}
\log_p(\kappa_p^f(\testf^*,\testg_y,\testh_z)) &=&  \alpha_g^{-s}   d_s^{-1} \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*} \otimes \omega_{\testg_y}  
  \omega_{\testh_z}), \\
\log_p(\kappa_p^g(\testf,\testg_y^*,\testh_z)) &=&    d_s^{-1}   \AJ_p(\Delta_{s,s}^{\circ})(\omega_{\testf} \otimes \eta_{\testg_y^*} 
  \omega_{\testh_z}).
\end{eqnarray*}
\end{corollary}
\begin{proof} This follows directly from Proposition \ref{prop:from-log-to-AJ}
in light of the functorial 
properties of Abel-Jacobi maps with respect to automorphisms.
\end{proof}


Corollary \ref{cor:from-log-to-AJ}
  motivates the study of  the image of the twisted diagonal cycle $\Delta_{s,s}^\circ$ under the $p$-adic syntomic Abel-Jacobi map,  which is taken up in the next two chapters.


 
\section{The syntomic Abel-Jacobi map on products of semistable curves}

\subsection{The cohomology of semistable curves}
\label{sec:cohomology-semistable}
 This section recalls the description (following 
%\cite{coleman-bu}, 
\cite{CI-Duke}, and \cite{CI-hidden}) of  the de Rham cohomology of a semistable
curve  and  the attendant structures with which it is equipped. Let $F$ be  a finite extension of $\Q_p$, let $\cO_F$ denote its
 ring of integers, fix an uniformizer $\pi_F$ and let $k_F = \cO_F/\pi_F$ denote the residue field.  Let $F_0$ denote the maximal unramified subfield of $F$, which is
 isomorphic to the fraction field of the ring of Witt vectors of $k_F$. Write $\sigma_0\in \Gal(F_0/\Q_p)$ for the Frobenius element generating the Galois group of $F_0/\Q_p$. Let $X$ be a smooth, connected, proper curve over $F$ and assume it 
possesses a regular semistable model $\cX$ over $\cO_F$, meaning that locally at every point, $\cX$ is either smooth over $\mathrm{Spec}(\cO_F)$ or \'etale over $\mathrm{Spec}(\cO_F[X,Y]/(XY-\pi_F))$.

Let $\tilde{\cX}$ denote the 
special fiber of $\cX$. It is  assumed that the irreducible components of $\tilde{\cX}$ are 
  smooth, geometrically irreducible, and defined over $k_F$, 
  that there are at least two of them, and that distinct components
  intersect only in ordinary double points defined over $k_F$.
  Let $\cG$ denote the dual graph of the special fiber, whose set $\cV(\cG)$ of
   vertices is 
  in bijection with the irreducible components  of $\tilde{\cX}$ and whose  set $\cE(\cG)$ of oriented edges
  is in bijection with the singular points of the special fiber, together with an ordering of the two components which intersect at that point.
  Given $v\in \cV(\cG)$, let $\tilde{\cX}_{v}$ denote the associated component, and let 
  $\tilde{\cX}_v^{\mathrm{sm}}$ denote the smooth locus of $\tilde{\cX}_v$, that is to say, the complement in $\tilde{\cX}_v$ of the  singular points of $\tilde{\cX}$. 
  If $e\in \cE(\cG)$ is an edge
  of $\cG$, write $s(e)$ and $t(e)$ for its source and target respectively, and $x_e$ for the associated 
  ordinary double point, satisfying
  $\tilde{\cX}_{s(e)} \cap \tilde{\cX}_{t(e)} = \{ x_e\}$. Write also $\bar e$ for the reversed edge, satisfying $s(\bar e)=t(e)$ and $t(\bar e) = s(e)$; note that $x_{\bar e} = x_e$.
  
  
  
 
Let 
$$\red: X(\C_p) \lra \tilde{\cX}(\bar \F_p) $$
denote the reduction map and, for each $v\in \cV(\cG)$,  let 
$$\cW_v:= \red^{-1}(\tilde{\cX}_v(\bar \F_p)), \qquad  \cA_v:= \red^{-1}(\tilde{\cX}_v^{\mathrm{sm}}(\bar \F_p)) \subset \cW_v$$ 
denote the  wide open subspace  and underlying affinoid  
associated to the component $\tilde{\cX}_v$. 
If $e$ is an edge of $\cG$, the wide opens
$\cW_{s(e)}$ and $\cW_{t(e)}$ intersect in the wide open  annulus 
$$\cW_e := \cW_{s(e)} \cap \cW_{t(e)} = \red^{-1}(\{ x_e\}).$$
  
  The collection $\{\cW_v\}_{v\in \cV(\cG)}$ gives rise to an {\em admissible  cover}
of $X(\C_p)$ by wide open subsets defined over $F$, whose nerve  is encoded in the 
 graph $\cG$. The de Rham cohomology group $H^1_{\dR}(X)$ can be described rigid analytically  as the set of 
 hypercocycles 
 $$\left\{ (\{\omega_v\}_v, \{f_e\}_e) \in \!\!\! \prod_{v\in\cV(\cG)} \!\!\!\Omega^1(\cW_v) \times \!\!\!\prod_{e\in\cE(\cG)}\!\!\! \cO_{\cW_e} \, \mbox{ such that } df_e = \omega_{t(e)} - \omega_{s(e)},  f_{\bar e} = -f_e
 \mbox{ for all } e\in \cE(\cG) \right\},$$
  taken modulo the hyper-coboundaries, which are hypercocycles
 of the form $(df_v, f_{t(e)}-f_{s(e)})$ attached to a collection $\{f_v\}_v$   of elements 
 of $\cO_{\cW_v}$ indexed by     $v\in \cV(\cG)$. 
 
The non-trivial rigid cohomology groups of $\cW = \cW_v$ or $\cW_e$ are $$H^0_{\mathrm{rig}}(\cW) = F \quad \mbox{and} \quad H^1_{\mathrm{rig}}(\cW) = \Omega^1(\cW)/d\cO_{\cW}.$$ Following \cite[(1.1)]{CI-Duke}, 
the Mayer-Vietoris sequence attached to this admissible covering  places $H^1_{\dR}(X)$
in the middle of a short 
exact sequence  
\begin{equation}
\label{eqn:coleman-iovita}
 0 \lra \frac{\oplus_{e} H^0_{\mathrm{rig}}(\cW_e)^-}{\delta(\oplus_v H^0_{\mathrm{rig}}(\cW_v))} 
\stackrel{i}{\lra}
  H^1_{\dR}(X) \stackrel{r}{\lra} \ker\left(\oplus_v H^1_{\mathrm{rig}}(\cW_v)
   \stackrel{\delta}{\rightarrow} \oplus_{e} H^1_{\mathrm{rig}}(\cW_e)^-\right) \lra 0,
  \end{equation}
 where 
 \begin{itemize}
\item $\oplus_{e} H^i_{\mathrm{rig}}(\cW_e)^-$ is the $(-1)$-eigen-subspace of $\oplus_{e} H^i_{\mathrm{rig}}(\cW_e)$ which respect to the involution $e\mapsto \bar e$; 
 \item the indices $e$ and $v$ run over $\cE(\cG)$ and $\cV(\cG)$ respectively;
  \item the map $\delta$ sends $\{\kappa_v\}_v$ to  the class of $\{\kappa_{t(e)}- \kappa_{s(e)}\}_e$;
 \item the map $i$ sends  a collection $\{\lambda_e\}_e$ to the class of the hypercocycle
 $(\{0\}_v, \{\lambda_e\}_e)$;
 \item the map $r$ is the natural restriction map sending  the class of
 $(\{\omega_v\}_v, \{f_e\}_e)$ to $\{\omega_v\}_v$.
 \end{itemize}
 The cohomology group $H^1_{\dR}(X)$ is an $F$-vector space equipped with the following further structures:
 \begin{enumerate}
\item[(i)] A decreasing Hodge filtration of $F$-vector spaces given by $\Fil^i = H^1_{\dR}(X)$ for $i\leq 0$, $\Fil^1 = \Omega^1(X)$ and $\Fil^i=\{ 0 \}$ for $i>1$.

 \item[(ii)] A canonical $F_0$-structure $H^1_{\mathrm{log-cris}}(\tilde{\cX})$ afforded by the log-crystalline cohomology of $\tilde{\cX}$ with coefficients in $F_0$. This is a $F_0$-vector space for which there is a canonical comparison isomorphism 
\begin{equation}\label{comp-iso-0}
H^1_{\mathrm{log-cris}}(\tilde{\cX})\otimes_{F_0} F \simeq H^1_{\dR}(X).
\end{equation}

\item[(iii)]  A $\sigma_0$-semilinear Frobenius operator $\varphi$  on $H^1_{\mathrm{log-cris}}(\tilde{\cX})$ arising from the Frobenius
 on $k_F$. 

 \item[(iv)] A $F_0$-linear {\em monodromy map} $N: H^1_{\mathrm{log-cris}}(\tilde{\cX}) \lra H^1_{\mathrm{log-cris}}(\tilde{\cX})$ which via \eqref{comp-iso-0} extends to a $F$-linear map
\begin{equation}
\label{eqn:display-N}
N: H^1_{\dR}(X) \lra H^1_{\dR}(X),
\end{equation}
denoted with the same letter. It sends the class of $(\{\omega_v\}_v, \{f_e\}_e)$ to the class of $(\{0\}_v, \{\lambda_e\}_e) $ with
 $\lambda_e = \partial_{\cW_e}\omega_{s(e)}$. Here, $\partial_{\cW_e}:H^1_{\mathrm{rig}}(\cW_e) \lra H_{\rig}^0(\cW_e)$ 
 denotes the $p$-adic annular residue on $\cW_e$,
 taken relative to an orientation on $\cW_e$ which is determined by $e$.
 \item[(v)] A section
\begin{equation}
\label{eqn:display-s}
 s: H^1_{\dR}(X) \lra \oplus_{e} H^0_{\mathrm{rig}}(\cW_e)^-/\delta\left(\oplus_v H^0_{\mathrm{rig}}(\cW_v)\right)
 \end{equation}
 of the map $i$ in the exact sequence \eqref{eqn:coleman-iovita}, 
 defined 
%via Coleman's theory of $p$-adic integration, 
by the rule
 $$ s(\{\omega_v\}_v, \{f_e\}_e) := \{\lambda_e\}_e, \qquad \lambda_e := f_e + F_{\omega_{s(e)}} - F_{\omega_{t(e)}},$$
 where $F_{\omega_v}$ denotes the {\em Coleman primitive} of the differential 
 $\omega_v$. This primitive is a canonically defined up to addition of a single constant of integration. It is analytic on each residue disk in $\cA_v$ and
its restriction to the annulus
 $\cW_{e}$ (for $e$ any edge having $v$ as either source or target) is  generated by a single logarithmic function over the ring of
 rigid analytic functions on $\cW_e$.

 
 \end{enumerate}

Set $d=[F_0: \Q_p]$. Since $\Phi := \varphi^d$ is $F_0$-linear, it extends via the comparison isomorphism \eqref{comp-iso-0} to a $F$-linear operator on $H^1_{\dR}(X)$ that is still written as $\Phi$.


The action of $\Phi$ on $H^1_{\dR}(X)$ admits a direct description in terms 
 of the exact sequence
 \eqref{eqn:coleman-iovita}, in which  the action
  of $\Phi$ on the left and right-hand terms is  realized concretely by choosing a system $\Phi_v$ of characteristic
 $0$ lifts of Frobenius to a system $\cA_v\subset\cW_v[\epsilon)\subset \cW_v$
 of Frobenius neighbourhoods of the underlying affinoids $\cA_v$
 in $\cW_v$, indexed by a real parameter $0<\epsilon<1$. 
 While $\Phi_v$ is only defined on  $\cW_v[\epsilon)$ for suitable $\epsilon>0$ and does not preserve this domain,
 the map $\omega_v \mapsto \Phi^* \omega_v$ still induces a well-defined transformation
 \begin{equation}\label{defPhi-W}
 \xymatrixcolsep{5pc}
 \xymatrix{ 
 \Phi: \frac{\Omega^1(\cW_v)}{d\cO_{\cW_v}} \ar[r]^{\res_{\cW_v[\epsilon)}} &
 \frac{\Omega^1(\cW_v[\epsilon))}{d\cO_{\cW_v[\epsilon)}} \ar[r]^{\Phi^*} &
 \frac{\Omega^1(\cW_v[\epsilon'))}{d\cO_{\cW_v[\epsilon')}}  
 \ar[r]^{\res_{\cW_v[\epsilon')}^{-1}}  & \frac{\Omega^1(\cW_v)}{d\cO_{\cW_v}} }
\end{equation}
 where $\res_{\cW_v[\epsilon)}$ is the isomorphism
  induced   by restriction to the wide open $\cW_v[\epsilon)$.  This action of Frobenius 
 on the cohomology does not depend  on the choice of characteristic zero liftings $\{\Phi_v\}$. 
Following the definitions in  \cite{CI-Duke}, the action of $\Phi$  then  extends to the full
 $H^1_{\dR}(X)$ via the requirement that
 $\Phi$  be compatible with the section $s$ of \eqref{eqn:display-s}. 


 \begin{definition}
 \label{def:pure}
 A class in $H^1_{\dR}(X)$ is said to be {\em pure} if it is in the kernel of both $N$ and $s$. 
 \end{definition}
 The set of
 pure classes, denoted $H^1_{\dR}(X)_{\pure}$, is a subgroup of $H^1_{\dR}(X)$ which is preserved  by the action of
  Frobenius and is
 identified via the map $r$ of \eqref{eqn:coleman-iovita} with
 \begin{equation}
 \label{eqn:H1-stable}
 H^1_{\dR}(X)_{\pure} = \bigoplus_v \frac{\Omega^1_{\sk}(\cW_v)}{d\cO_{\cW_v}},
 \end{equation}
 where $\Omega^1_{\sk}(\cW_v) \subset \Omega^1(\cW_v)$ denotes the space of rigid differentials on $\cW_v$ {\em of the second kind}, i.e., have vanishing annular residues.
  This identification is compatible with the action of
 $\Phi$ on both groups. 
  \begin{proposition}
 \label{prop:pure-of-weight-1}
 The eigenvalues of   $\Phi$ on $H^1_{\dR}(X)_{\pure}$ have complex absolute value $\sqrt{p}$.
 \end{proposition}
 \begin{proof}
 Equation \eqref{eqn:H1-stable} shows that $H^1_{\dR}(X)_{\pure}$ is isomorphic as a Frobenius module to the crystalline cohomology of the disjoint union of the components $\tilde{\cX}_v$ of the special fiber $\tilde{\cX}$. More precisely, there is an exact sequence
\begin{equation}\label{ex-se}
0 \ra H^1_{\cris}(\tilde{\cX}_v) \ra H^1_{\cris}(\tilde{\cX}^{\mathrm{sm}}_v) \ra \oplus_{e} F(-1) \stackrel{\Sigma}{\ra} F(-1) \ra 0
\end{equation}
where the comparison theorems allow us to identify $$H^1_{\cris}(\tilde{\cX}_v)\otimes_{F_0} F \simeq \frac{\Omega^1_{\sk}(\cW_v)}{d\cO_{\cW_v}} \quad \mbox{ and } \quad H^1_{\cris}(\tilde{\cX}^{\mathrm{sm}}_v)\otimes_{F_0} F \simeq \frac{\Omega^1(\cW_v)}{d\cO_{\cW_v}} = H^1_{\mathrm{rig}}(\cW_v).$$ Since the components $\tilde{\cX}_v$ are smooth and 
 projective, the theorem follows from the Riemann hypothesis for curves.
 \end{proof}
 In light of Proposition \ref{prop:pure-of-weight-1}, it would have been more appropriate to use the expression 
 {\em pure of weight $1$}
  to designate the classes
 in $H^1_{\dR}(X)_{\pure}$. The vaguer but more concise terminology adopted in
 Definition \ref{def:pure}
  was preferred because the qualifier
 {\em weight $1$} is also used in  the setting of modular forms  with a  different meaning. It follows from Proposition \ref{prop:pure-of-weight-1} that the exact sequence \eqref{ex-se} has a natural Frobenius-equivariant  splitting and hence the  injective map
 $r: H^1_{\dR}(X)_{\pure} \lra \oplus_{v} H^1_{\mathrm{rig}}(\cW_v)$
  admits a natural Frobenius-equivariant section
 \begin{equation}
 \label{eqn:def-r-prime}
 r^{\iota}: \oplus_v H^1_{\mathrm{rig}}(\cW_v) \lra H^1_{\dR}(X)_{\pure}.
 \end{equation}
 
 
%If $X$ and $X'$ are two semistable curves over $F$ as above, a map $\pi: X\lra X'$ is said to be a {\em map of semistable curves}
%if it extends to the regular semistable models of $X$ and $X'$, denoted $\cX$ and $\cX'$ respectively.
%This means that $\pi$ maps the 
%smooth locus of the special fiber of $\cX$ to the corresponding 
%smooth locus for $\cX'$, and  maps singular points
%  to singular points. 
%  In particular, it    induces functorial  morphisms
%  $$ \pi_{*}: \cG_X \lra \cG_{X'}, \qquad \pi^*: H^1_{\dR}(X')_{\pure}\lra H^1_{\dR}(X)_{\pure}.$$
%The map $\pi^*$  is simply 
%  induced by the natural pull-back morphism
%$$ \pi^*: \oplus_{v\in \cV(\cG_{X'})} \Omega_{\sk}^1(\cW_v) \lra \oplus_{w\in \cV(\cG_X)} \Omega_{\sk}^1(\cW_w), \quad  \pi^*\left(\{\omega_v\}_{v\in \cV(\cG_{X'})}\right)
 %:=   \{ \pi^*\omega_{\pi(w)} \}_{w\in \cV(\cG_X)}.$$
 
 The Poincar\'e duality on $H^1_{\dR}(X)$ descends to a perfect pairing on $H^1_{\dR}(X)_{\pure}$,
 which is described by  the formula
 \begin{equation}
\label{eqn:pd-sums}
 \left\langle \{\omega_v\}, \{ \eta_v\} \right\rangle_X =
 \sum_{e \in \cE(\cG)}  \res_{\cW_e}(F_{e} \eta_{s(e)}),
 \end{equation}
 where the sum is taken over all the oriented edges of $\cG$ and  $F_e$ is an
  analytic primitive of the restriction to the annulus $\cW_e$ of $\omega_{s(e)}$,
 which exists because $\omega_v$ has vanishing annular residues for all $v\in \cV(\cG)$, and
 is well-defined up to a constant. (The choice of constant in the  local primitive $F_e$
 does not affect the term on the right
 in \eqref{eqn:pd-sums}, since $\eta_v$ also has vanishing residues for all $v$.)
  
  
 
 \subsection{An analytic recipe for the syntomic Abel-Jacobi map}
 \label{sec:recipe}
 The goal of this section is to give  an analytic expression for the  
 syntomic Abel-Jacobi map on the Chow group of codimension two cycles on the 
triple product of semistable curves,
 in terms of Coleman's theory of $p$-adic integration of rigid differential forms.
It will be convenient to place oneself in the 
   abstract setting given by conditions (a)-(d) below. While ostensibly somewhat special,
this setting  will nonetheless   suffice  for the applications in this paper.
 \begin{itemize}
 \item[(a)] The ambient three-fold is  a product 
 $W = X_1 \times X_2 \times X_3$
 of three semistable curves over $F$. 
 \end{itemize}
  For $i=1,2,3$, write
  $\cG_i$ for the dual graph of the special fiber of $\cX_i$,  and
 let $\Phi_i=\{\Phi_{i,v}\}_{v\in \cV(\cG_i)} $ denote a system of liftings of Frobenius to each of the basic
 wide open spaces in the admissible covering of $X_i(\C_p)$ described in the previous section.
 Denote by $\Phi_{23} := \Phi_2\otimes\Phi_3$ the corresponding Frobenius endomorphism on 
 $H^1_{\dR}(X_2)_{\pure}\otimes H^1_{\dR}(X_3)_{\pure}\subset H^2_{\dR}(X_2\times X_3)$. 
 Our second assumption, concerning the cycle $\Delta$ itself, is:
 \begin{itemize}
  \item[(b)]  The   cycle $\Delta\in \CH^2(W)$ is obtained 
 from the data of a semistable curve $X$, of three semistable morphisms
 $ \pi_i: X \lra X_i$ and of three correspondences
 $\varepsilon_i\in \mathrm{Corr}(X_i)$,
 by setting
 $$ \Delta:=  (\varepsilon_1,\varepsilon_2,\varepsilon_3)_*   (\pi_1,\pi_2,\pi_3)_*(X).$$
%where 
% $$ \pi:= (\pi_1,\pi_2,\pi_3): X \lra W, \qquad
% \varepsilon:= (\varepsilon_1,\varepsilon_2,\varepsilon_3): W \lra W$$
% are defined in the natural way. 
  \item[(c)]
 The correspondences $\varepsilon_i$ annihilate the $H^0$ and $H^2$ terms in the cohomology of
 $X_i$. By the same reasoning as in the proof of Proposition \ref{prop:null-homologous}, this implies that
 $\Delta$ is null-homologous, i.e., that it belongs to $\CH^2(W)_0$.
 \end{itemize}
Let
\begin{equation}\label{AJp}
\AJ_p: \CH^2(W)_0 \lra (\Fil^2 H^3_{\dR}(W))^\vee
\end{equation}
denote the $p$-adic syntomic Abel-Jacobi map, as discussed e.g. in \cite[Introduction and \S 5]{Bes-inv}, \cite[\S 3]{besser-loeffler-zerbes} and \cite[(3.7)]{Ne1}. It follows from \cite[Theorem B]{NN} that \eqref{AJp} is canonically identified with the composition of the \'etale Abel-Jacobi map with Bloch-Kato's logarithm, as    in \eqref{AJpsyn}.


Even under   assumptions (a),  (b), and (c), we  fall short of giving a formula for the full image 
 $\AJ_p(\Delta)$ but only for its values on cohomology  classes of the form
 $$ \eta_1\otimes\omega_2\otimes\omega_3 \in H^1_{\dR}(X_1)\otimes \Omega^1(X_2)\otimes\Omega^1(X_3) \subset
 \Fil^2(H^3_{\dR}(W)),$$ 
assuming that:
 \begin{itemize}
 \item[(d)] $\eta_1$, $\omega_2$ and $\omega_3$  belong  to the pure subspaces of 
 $H^1_{\dR}(X_1)$, $H^1_{\dR}(X_2)$, and
  $H^1_{\dR}(X_3)$ respectively.
 \end{itemize}
Our recipe for describing $\AJ_p(\Delta)(\eta_1\otimes \omega_2\otimes\omega_3)$ 
rests on the following lemma. 
 \begin{lemma}
 \label{lemma:choose-P}
 There exists a polynomial $P\in \Q[X]$ satisfying:
 \begin{enumerate}
 \item $P(\Phi_{23})$ annihilates the class of $\omega_2\otimes \omega_3$ in 
 $H^2_{\dR}(X_2\times X_3)$;
 % \item $P(x)$ is divisible by $x-p$, so that $P(\Phi_X)$ sends rigid $1$-forms on  $\cW_{v}$   to differential forms of the second kind on this wide open,   for all $v\in \cV(\cG_X)$. 
 \item $P(\Phi_{X})$ acts invertibly on $\ker  N\subset H^1_{\dR}(X)$. 
  \end{enumerate}
  \end{lemma}
  \begin{proof}
By  Proposition \ref{prop:pure-of-weight-1}, there is a polynomial $P$ satisfying the first  condition,
all of whose roots have complex absolute value $p$.  But any such $P$ automatically satisfies
the second condition, since Frobenius acts on the kernel of $N$ with eigenvalues of complex absolute value
either $1$ or $\sqrt{p}$. 
\end{proof}
Fix a polynomial $P$ as in the above lemma. Since $P(\Phi_{23})$ annihilates the class of $\omega_2\otimes \omega_3$ in
$H^2_{\dR}(X_2\times X_3)$, there is a system of rigid $1$-forms  
\begin{equation}
\label{eqn:choose-rho}
\rho:=  \rho_P^{v,w}(\omega_2,\omega_3)  \in \Omega^1(\cW_v\times\cW_w), \qquad v\in \cV(\cG_2), \quad
w\in \cV(\cG_3)
\end{equation}
satisfying
\begin{equation*}
  d \rho_P^{v,w}(\omega_2,\omega_3) = P(\Phi_{23}) \left((\omega_2\wedge\omega_3)_{|\cW_v\times\cW_w}\right).
  \end{equation*}

The system $\rho_P^{v,w}(\omega_2,\omega_3)$ 
of rigid $1$-forms is well-defined, up to a system of closed $1$-forms on each
product of basic wide opens in $X_2\times X_3$.   
% The system  $\rho_P^{v,w}(\omega_2,\omega_3)$ is therefore well-defined, up to a system of closed $1$-forms on  $\cW_v\times \cW_w$ with trivial residues, since $\Phi_{23}$ acts as multiplication by $p$ on the target of the residue map.
 One next defines a system $\{\tilde\xi_P^{v}(\omega_2,\omega_3)\}_{v\in \cV(\cG_X)}$
  of  rigid $1$-forms   on each basic wide open of  $X$ by pulling back the 
differentials $\rho_P^{v,w}(\omega_2,\omega_3)$ via the maps $(\pi_2,\pi_3):X\lra X_2\times X_3$ and
$(\varepsilon_2,\varepsilon_3) X_2\times X_3 \lra X_2\times X_3$:
  $$ \{\tilde\xi_P^v(\omega_2,\omega_3)\}_{v\in\cV(\cG_X)} = 
  (\pi_2,\pi_3)^*(\varepsilon_2,\varepsilon_3)^*
  \left(\{\rho_P^{v,w}(\omega_2,\omega_3)\}\right).
  $$ 
 Condition (c) combined with the K\"unneth formula for
 $H^1_{\mathrm{rig}}(X_2\times X_3)$ 
   implies that for each $v\in \cV(\cG_X)$, the   class of the differential 
 $ \tilde\xi_P^{v}(\omega_2,\omega_3)$  in $H^1_{\mathrm{rig}}(\cW_v)$  does not depend on the choice of
 system $\rho_P^{v,w}(\omega_2,\omega_3)$ 
 satisfying 
 \eqref{eqn:choose-rho}. 
Hence
 the class of  
 $\tilde \xi_P(\omega_2,\omega_3)\in \oplus_{v\in \cV(\cG_X)} H^1_{\mathrm{rig}}(\cW_v)$
  is  independent of this choice.
  Let
  $$ \xi_P(\omega_2,\omega_3) := r^{\iota}(\tilde\xi_P(\omega_2,\omega_3)) \in H^1_{\dR}(X)_{\pure},$$
  where $r^{\iota}$ is the Frobenius-equivariant section introduced in 
  \eqref{eqn:def-r-prime}. 

  
 Applying   the second condition in Lemma \ref{lemma:choose-P} shows  that there is a unique class
 $$\xi(\omega_2,\omega_3)  \in H^1_{\dR}(X)_{\pure} \quad \mbox{ satisfying } \quad  P(\Phi_C) \xi(\omega_2,\omega_3) = \xi_P(\omega_2,\omega_3).$$ 
 %satisfying
% $$ P(\Phi_X) \xi(\omega_2,\omega_3) = \xi_P(\omega_2,\omega_3).$$

%The statement below follows (by computations similar to the ones performed in \cite[\S 3]{DR1}) from %the work \cite[\S 4]{besser-loeffler-zerbes} of Besser, Loeffler and Zerbes on syntomic regulators and %Abel-Jacobi maps for semistable curves:

 \begin{theorem}
 \label{thm:recipe}
 For all $\Delta\in \CH^2(W)_0$ and classes 
 $\eta_1$, $\omega_2$ and $\omega_3$ satisfying conditions $(a)-(d)$,
  $$ \AJ_p(\Delta)(\eta_1\otimes\omega_2\otimes\omega_3) = \langle \pi_1^*\varepsilon_1^*(\eta_1), \xi(\omega_2,\omega_3)\rangle_X.$$
% where the pairing is the natural Poincar\'e duality pairing on $H^1_{\dR}(X)_{\pure}$. 
 \end{theorem}
 
  
  \begin{proof} The proof of this formula closely follows 
   the one given in \cite[\S 3.3]{DR1} for the Gross-Kudla-Schoen diagonal 
   cycle on the cube of a modular curve with good reduction at $p$, 
   which exploits Besser's finite polynomial cohomology for smooth 
   varietes over $\Z_p$ introduced in \cite{Bes-inv}. This theory has
    been extended recently to the setting of semistable varieties by 
    Besser, Loeffler and Zerbes in \cite{besser-loeffler-zerbes}, building
  crucially on prior work of Nekovar and Niziol \cite{NN}. Namely, \cite[Proposition 3.3]{besser-loeffler-zerbes} asserts in this case that
$$
\AJ_p(\Delta)(\eta_1\otimes\omega_2\otimes\omega_3) = \tr_{\cX}\big((\pi_1,\pi_2,\pi_3)^* (\varepsilon_1,\varepsilon_2,\varepsilon_3)^*(\tilde \eta_1\otimes \tilde\omega_2\otimes\tilde\omega_3)\big) \qquad \mbox{ where}
$$
\begin{itemize}
\item $\tilde \eta_1 \in H^1_{\fp}(\cX_1,0)$, $\tilde{\omega}_2\in H^1_{\fp}(\cX_2,1)$, $\tilde{\omega}_3\in H^1_{\fp}(\cX_3,1)$ are respectively lifts of $\eta_1$, $\omega_2$, $\omega_3$ to $P$-syntomic cohomology in the sense of \cite[\S 2.2]{besser-loeffler-zerbes}. In order to lighten the notations,   the polynomials implicit in the construction shall be omitted.

\item $(\pi_1,\pi_2,\pi_3)^* (\varepsilon_1,\varepsilon_2,\varepsilon_3)^*(\tilde \eta_1\otimes \tilde\omega_2\otimes\tilde\omega_3) \in H^3_{\fp}(\cX,2)$ stands for the pull-back to $\cX$ of the tensor product of these classes in syntomic cohomology.

\item $\tr_{\cX}: H^3_{\fp}(\cX,2)\overset{\sim}{\lra} F$ is the trace isomorphism of \cite[Definition 3.1]{besser-loeffler-zerbes}. 

\end{itemize}

Let $\langle \, , \rangle_{\fp}: H^1_{\fp}(\cX,0)\times H^2_{\fp}(\cX,2) \lra F$ denote the cup-product described in \cite[\S 1.3, \S 2.4]{besser-loeffler-zerbes}.
By the compatibility between the trace and cup-product in syntomic cohomology afforded by \cite[Theorem 2.20]{besser-loeffler-zerbes}, the above quantity may be recast as
$$
 \tr_{\cX}\big((\pi_1,\pi_2,\pi_3)^* (\varepsilon_1,\varepsilon_2,\varepsilon_3)^*(\tilde \eta_1\otimes \tilde\omega_2\otimes\tilde\omega_3)\big) = \langle \pi_1^*\varepsilon_1^*(\tilde\eta_1),(\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*(\tilde\omega_2\otimes\tilde\omega_3)\rangle_{\fp}.
$$

%By the functoriality of \cite[(8)]{Bes-inv}, there is a commutative
% diagram with exact rows
%\begin{equation}
%\label{Besser51b}
%\xymatrix {
%0 \ar[r]  & H^1_{\dR}(X^2) \ar[r]^{\mathrm{i}} \ar[d]_{ (\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*}   & H^2_{\fp}(\cX^2,2)  \ar[r]^{\mathrm{p}\quad }
%\ar[d]_{ (\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*}   & \Fil^2 H^2_{\dR}(X^2) \ar[r] \ar[d]_{  (\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*} & 0 \\
%  0  \ar[r]  & H^1_{\dR}(X) \ar[r]^{\mathrm{i}}  & H_{\fp}^2(\cX,2) \ar[r]^{\mathrm{p}\qquad}  & \Fil^2 H^2_{\dR}(X)=0.    & }
%\end{equation} 
%In light of \cite[Def.\,2.2 and eq.\,(6)]{Bes-inv}, a pair 
%$$(\alpha,\beta)\in \Omega^1_{\rig}(\cW_\epsilon^2)\oplus \Fil^2 H^2_{\dR}(X^2)=\Omega^1_{\rig}(\cW_\epsilon^2)\oplus \Omega^2(X^2)$$
% satisfying  $d\alpha=P(\Phi)(\beta)$ gives rise to a class in $H^2_{\fp}(\cX^2,2)$. Since the map $\mathrm{p}$ of the upper row of \eqref{Besser51b} is the %projection $[\alpha,\beta]\mapsto [\beta]$ to the second factor, 

%Since $\Fil^2 H^2_{\dR}(X) = 0$, the natural injection $H^1_{\dR}(X)\hookrightarrow H^2_{\fp}(\cX,2)$ is an isomorphism. 
According to the explicit description of finite polynomial cohomology provided in \cite[Definition 2.4]{besser-loeffler-zerbes}, the class $\tilde{\omega}_2\otimes \tilde{\omega}_3$ in $H^2_{\fp}(\cX^2,2) $ may be represented by the pair $(\rho,\omega_2\otimes \omega_3)$
%\in \Omega^1_{\rig}(\cW_\epsilon^2)\oplus \Omega^2(X^2),$$
 where $\rho$ is as in \eqref{eqn:choose-rho}, and hence
%Since $\Fil^2 H^2_{\dR}(X)=0$, the class $(\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*(\tilde{\omega}_2\otimes \tilde{\omega}_3)$ has a preimage in %$H^1_{\dR}(X)$ under the map $\mathrm{i}$ of the lower row of \eqref{Besser51b}. Such preimage is 
 the class $(\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*(\tilde{\omega}_2\otimes \tilde{\omega}_3)$  in $H^2_{\fp}(\cX,2)$ is represented by the image of $\xi=\xi(\omega_2,\omega_3)$ under the natural morphism $\mathrm{i}: H^1_{\dR}(X)\lra H^2_{\fp}(\cX,2)$, $\mathrm{i}(\xi) = P(\Phi)(\xi)$.
%\begin{equation}\label{sss}
%i(\xi):=[P(\Phi)\xi,0] = [(\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*\rho,0] =(\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*(\tilde{\omega}_2\otimes %\tilde{\omega}_3)\in H^2_{\fp}(\cX,2).
%\end{equation}
It follows that
$$ \langle \pi_1^*\varepsilon_1^*(\tilde \eta_1), (\pi_2,\pi_3)^* (\varepsilon_2,\varepsilon_3)^*(\tilde\omega_2\otimes\tilde\omega_3) \rangle_{\fp} = \langle \pi_1^*\varepsilon_1^*(\tilde \eta_1),\mathrm{i}(\xi(\omega_2,\omega_3)) \rangle_{\fp} = \langle \pi_1^*\varepsilon_1^*(\eta_1),\xi(\omega_2,\omega_3)\rangle_X,
$$
where the second equality follows as in \cite[(14)]{Bes-inv}. 
 \end{proof}
  


 




 

    
 \section{Bloch-Kato logarithms of weight two specializations}
\label{sec:bloch-kato-AJ}

As was explained at the end of \S \ref{sec:Ohta},  the 
 aim of this chapter is to show that the one-variable $p$-adic $L$-functions 
%\begin{eqnarray*}
%\Lp^{f_\alpha}(\testf^*, \underline{\testg \testh}) &:=& \Lp^{f_\alpha}(\testhf^*, \testhg, \testhh)_{|\Omega_{f \underline{g h}}}, \\
%\Lp^{g_\alpha}(\testf, \underline{\testg^* \testh})  &:=& \Lp^{g_\alpha}(\testhf, \testhg^*, \testhh)_{|\Omega_{f \underline{g h}}}, \\
%\Lp^{h_\alpha}(\testf,\underline{\testg \testh^*})  &:=&   \Lp^{h_\alpha}(\testhf, \testhg, \testhh^*)_{|\Omega_{f \underline{g h}}}
%\end{eqnarray*}
\begin{equation}\label{one-var}
\Lp^{f_\alpha}(\testf^*, \underline{\testg \testh}) := \Lp^{f_\alpha}(\testhf^*, \testhg, \testhh)_{|\Omega_{f \underline{g h}}}, \quad 
\Lp^{g_\alpha}(\testf, \underline{\testg^* \testh})  := \Lp^{g_\alpha}(\testhf, \testhg^*, \testhh)_{|\Omega_{f \underline{g h}}}
\end{equation}
arising as the restriction to $\Omega_{f \underline{g h}}$ of the Garrett-Hida $p$-adic $L$-functions discussed in the introduction can be recovered as the image of the $\Lambda$-adic classes $\kappa_p^f(\testf^*,\underline{\testg\testh})$ and $ \kappa_p^g(\testf, \underline{\testg^*\testh})$ and  under Perrin-Riou's $\Lambda$-adic logarithm map. We shall do so by comparing their values at classical points $(y,z)$ of weight $2$.  
Corollary \ref{cor:from-log-to-AJ} asserts that the Bloch-Kato logarithm of the specialisation of these classes at such points may be computed as the value of the functional $\AJ_p(\Delta_{s,s}^\circ)$ at  the classes
$
\eta_{\testf^*} \otimes \omega_{\testg_y}  \otimes \omega_{\testh_z}$ and $\omega_{\testf} \otimes \eta_{\testg_y^*} \otimes \omega_{\testh_z}$, respectively. 

Armed with 
Theorem \ref{thm:recipe}, we can now proceed with this program, starting with 
an analysis of  the above forms in the rigid analytic cohomology 
of the modular curves $X_0(Np)$ and $X_s$.



\subsection{The rigid geometry of modular curves
}\label{sec:rig}


Let  $\cX_1(Np^s)$ be the proper, flat, regular model of $X_s$ over $\Z_p[\zeta_s]$ considered in \cite[\S 12]{KaMa}. Its special fiber is the union of a
   finite number of reduced Igusa curves $\mathrm{Ig}_j$ over $\F_p$, meeting (not always transversally) at their supersingular points. Among these 
   irreducible components there are two which are isomorphic to the
    Igusa curve $\mathrm{Ig}(Np^s)$ representing the moduli problem $(\Gamma_1(N),\mathrm{Ig}(p^s))$ over $\F_p$ specified in \cite[\S 12]{KaMa}. These are called the   {\em good components},
   and are  labelled $\mathrm{Ig}_\infty$ and $\mathrm{Ig}_0$. More precisely, $\mathrm{Ig}_\infty$ classifies 
   ordinary elliptic curves with $\Gamma_1(N)$-level structure and a chosen section of order 
    $p^s$ in its canonical subgroup of order $p^s$. 

Let $F/\Q_p(\zeta_s)$ be a finite extension over which $X_s$ acquires semistable reduction, with valuation ring $\cO_F$ and residue field $k_F$. 
Let $\pi: \cX_s \ra \cX_1(Np^s) \times \cO_F$
be a semistable model of $X_s$ over $\cO_F$, obtained by applying sufficiently many blow-ups to the base change of $\cX_1(Np^s)$ to $\cO_F$. The map $\pi$ is birational and induces an isomorphism between the generic fibers and also between two of the components of the closed fibre of $\cX_s$ with $\mathrm{Ig}_\infty \times k_F$ and $\mathrm{Ig}_0\times k_F$, respectively.  By an abuse of notation, continue to denote $\mathrm{Ig}_\infty$ and $\mathrm{Ig}_0$  the two components of the closed fibre of $\cX_s$ corresponding to the two latter. Let 
 $$ \red: \cX_s(F)   \lra  \tilde{\cX}_s(k_F)$$
 denote the ``mod $p$" reduction map. The inverse images of the irreducible components of the special fiber
 of $\cX_s$  give a finite collection of wide open subsets 
 of $\cX_s(F)$ which form an admissible
 cover of this rigid analytic space, as in the discussion in \S \ref{sec:cohomology-semistable}.
 Let $WO(\cX_s)$ denote the collection  of wide open spaces in this admissible cover.



Denote by   
 $\cW_\infty (p^s)$ and $\cW_0(p^s)$  the   wide open spaces in $WO(\cX_s)$
  arising as the inverse image of $\mathrm{Ig}_\infty$ and $\mathrm{Ig}_0$, respectively, under the reduction map. The inverse image of the smooth locus of 
$\mathrm{Ig}_\infty$, denoted $\cA_\infty(p^s)$,    is 
the  underlying affinoid  of   $\cW_\infty(p^s)$ and an analogous definition is made for  
$\cA_0(p^s)$. 


The $F$-valued  points of the rigid analytic space $\cA_\infty(p^s)$ are in bijection with
triples $(A, i_N, i_p)$ where
$A$ is an ordinary (generalised) elliptic curve over $\cO_F$, 
$i_N:\mu_N \lra A[N]$ is an injective map of group schemes over $\cO_F$, and
$i_p: \mu_{p^s} \lra A\{p^s\}$ is an isomorphism between $\mu_{p^s}$ and the canonical
subgroup $A\{p^s\}$ 
of $A$ of order $p^s$. As explained in \cite{coleman-bordeaux} and \cite[Lemme 4.4.1]{BrEm}, the $F$-vector 
spaces 
$$H^1_{\mathrm{rig}}(\cW_\infty(p^s))  = \frac{\Omega^1_{\rig}(\cW_\infty(p^s))}{d\cO_{\cW_\infty(p^s)}},
\qquad
H^1_{\mathrm{rig}}(\cW_0(p^s))  = \frac{\Omega^1_{\rig}(\cW_0(p^s))}{d\cO_{\cW_0(p^s)}}
$$
are equipped with a natural action of Hecke operators $T_\ell$ for $\ell\nmid Np$. As already explained in \eqref{defPhi-W}, they are also equipped with natural $F$-linear Frobenius endomorphisms, which are defined by choosing characteristic zero lifts $\Phi_\infty$ and $\Phi_0$ of the Frobenius endomorphism in characteristic $p$ to a system of wide open neighbourhoods of the affinoids
$\cA_\infty(p^s)$ and $\cA_0(p^s)$ in $\cW_\infty(p^s)$ and $\cW_0(p^s)$, respectively.
In the setting at hand, a natural choice of $\Phi_\infty$ is given by the Deligne-Tate mapping
defined on triples $(A,i_N,i_p)$ as above by the rule
$$ \Phi_{\infty}(A,i_N,i_p) = (A/A\{p\}, \pi_{\{p\}} \circ i_N, \tilde i_p),$$
where
\begin{enumerate}
\item $A\{p\}$ is the canonical subgroup of $A$ of order $p$;
\item  $\pi_{\{p\}} $ is the natural isogeny $A \lra A/A\{p\}$;
\item $\tilde i_p: \mu_{p^s}\lra A$ sends $\zeta_s$ to 
$\pi_{\{p\}}(P_{s+1})$ where $P_{s+1}\in A\{p^{s+1}\}$ satisfies
$p P_{s+1} = i_p(\zeta_s)$. 
\end{enumerate}

Let $(\Gm/q^{\Z}, i_N, i_p)$ denote the Tate curve, where $i_N$ and $i_p$ are induced by the natural embeddings $\mu_N, \mu_{p^s}\hookrightarrow \Gm$. Then $\Phi_{\infty}$ acts on it by the rule
$ \Phi_\infty(\Gm/q^{\Z}, i_N, i_p) = (\Gm/q^{p\Z}, i_N^p, i_p),$
and therefore 
\begin{equation}
\label{eqn:Phi-infty-V}
 \Phi_\infty = p \langle p; 1\rangle V
 \end{equation}
where  $V$ is the operator on $p$-adic modular forms whose
 action
is given by
$V\left(\sum_{n=1}^\infty a_n q^n\right) = \sum_{n} a_{n} q^{np}$  on $q$-expansions.
To define $\Phi_0$,  recall the Atkin-Lehner automorphism $w_{s}$ 
 attached to the choice of primitive $p^s$-th root of unity $\zeta_s$.
It   interchanges the 
wide opens $\cW_\infty(p^s)$ and $\cW_0(p^s)$,  and  one defines
\begin{equation}
\label{eqn:def-Phi0}
 \Phi_0 := w_{s}^{-1}   \Phi_\infty w_{s} = \langle p^{-1}; 1 \rangle w_{s} \Phi_\infty
 w_{s}.  
 \end{equation}
 

For each $\cW\in WO(\cX_s)$,  recall that 
$$\res_{\cW}: H^1_{\dR}(X_1(Np^s)) \lra
 H^1_{\mathrm{rig}}(\cW) $$ is the  the map induced by restriction
 to $\cW$, and write $\res_0$ and $\res_\infty$ for $\res_{\cW_0(p^s)}$ and $\res_{\cW_\infty(p^s)}$.
 
 % The maps $\res_{\cW_\infty(p^s)}$ and $\res_{\cW_0(p^s)}$, which  will be abbreviated to $\res_\infty$ and $\res_0$ respectively,  are part of a commutative diagram $$  \xymatrix{  H^1_{\dR}(X_1(Np^s)\times F) \ar[r]^{\quad \res_0} \ar[d]^{w_{s}} & H^1_{\mathrm{rig}}(\cW_0(p^s)) \ar[d]^{w_{s}} \\ H^1_{\dR}(X_1(Np^s)\times F) \ar[r]^{\quad\res_\infty}  & H^1_{\mathrm{rig}}(\cW_\infty(p^s)). } $$

Let $H^1_{\dR}(X_s/F)^{\prim}$ denote the subspace of the de Rham cohomology of 
$X_s$ associated to the   {\em primitive subspace} 
$S_2(Np^s)_F^{\prim}$  in the sense of \cite[\S 3]{mazur-wiles-mc}.
\begin{theorem}
\label{thCol}
\cite{coleman-bordeaux}, \cite{BrEm} 
The restriction maps $\res_\infty$ and $\res_0$ induce  an isomorphism  
$$
H^1_{\dR}(X_{s/F})^{\prim} \simeq H^1_{\mathrm{rig}}(\cW_\infty(p^s))^{\pure} \oplus  H^1_{\mathrm{rig}}(\cW_0(p^s))^{\pure},
$$
where $H^1_{\mathrm{rig}}(\cW_\infty(p^s))^{\pure}$ denotes the subspace of the de Rham cohomology which is pure of weight one,
i.e., the space generated by classes of rigid differentials with vanishing annular residues.
The restriction morphism 
is equivariant with respect to 
\begin{itemize}
\item the Hecke operators $T_\ell$ for $\ell\nmid Np$ acting on both sides;
\item the Frobenius endomorphism $\Phi$ acting
 on $H^1_{\dR}(X_{s/F})$;
 \item 
  the Frobenius
  endomorphism $(\Phi_\infty,\Phi_0)$ acting on $H^1_{\mathrm{rig}}(\cW_\infty(p^s)) \oplus 
  H^1_{\mathrm{rig}}(\cW_0(p^s))$.
\end{itemize}
\end{theorem}

A similar but  simpler analysis also holds for the modular curve $X_0(Np)$, since it admits
 a proper regular model over $\Z_p$, whose special fiber is the union of two irreducible 
 components, each isomorphic to the special fiber of the smooth integral model of
  $X_0(N)$. Write  $\{\cW_\infty,\cW_0\}$ for the  standard admissible covering of $X_0(Np)$
   by wide open neighbourhoods of the Hasse domain of $X_0(N)$. As above, the 
   group $H^1_{\dR}(X_0(Np))$ is again endowed with the action of a Frobenius map 
   $\Phi$ and the Hecke operator $U_p$, which mutually commute (by
    functoriality of log-crystalline cohomology with respect to correspondences).


The {\em ordinary unit root subspace} 
$$H^1_{\dR}(X_0(Np)_{/\Q_p})^{\ord,\ur} \subset e_{\ord} H^1_{\dR}(X_0(Np)_{/\Q_p})$$
is defined to be the subspace  spanned by the eigenvectors of $\Phi$   whose eigenvalue is a $p$-adic unit. 

%, as described in \cite{coleman-bu} for example.
\begin{lemma}
\label{restrict-to-X0N}
The map $H^1_{\dR}(X_0(Np)_{/F})^{\ord,\ur}\lra H^1_{\rig}(\cW_\infty)$ induced by restriction is the zero map, and for any eigenform $\phi$ of weight $2$ and level dividing $N$ the map
\begin{equation}
\label{eqn:restrict-to-X0N}
 H^1_{\dR}(X_0(Np)_{/F})^{\ord,\ur}[\phi] \lra H^1_{\rig}(\cW_0)^{\ur}[f] \simeq H^1_{\dR}(X_0(N)_{/F})^{\ur}[\phi]
 \end{equation}
is an isomorphism.
\end{lemma}
\begin{proof} This follows from noting that the action of $\Phi=\Phi_\infty$ and $U$ on $H^1_{\rig}(\cW_\infty)$ 
are related by the rule
$\Phi_\infty = \langle p \rangle p V = \langle p \rangle p U^{-1}.$
\end{proof}
 


\subsection{Test vectors and their associated de Rham cohomology classes} As 
was explained in the introduction and recalled above, the $\Lambda$-adic 
cohomology classes and the Garrett-Hida $p$-adic $L$-functions considered
 in this work depend on a choice of test vectors $\testf$, $\htg$, $\hth$ and
  dual test vectors $\testf^*$, $\htg^*$, $\hth^*$ that were fixed  
  in \eqref{eqn:test-vectors2} and \eqref{eqn:dual-test-vectors2}. 
%In this section we pin down more precisely these choices and analyze the classes in the de Rham cohomology of modular curves associated to their weight two specialisations.
%and $\testf^*_0\in S_2(N)^\vee[f]$ 
We focus first on the former, and analyze the regular differential forms associated to $\testf$ and the weight two specialisations of $\htg$ and $\hth$.

Assume that  $\testf\in S_2(Np)[f]$  arises as the ordinary $p$-stabilisation 
of some test vector in $S_2(N)[f]$, and  recall that
 $\omega_{\testf}\in \Omega^1(X_0(Np))$ denotes the regular differential 
 form asociated to $\testf$. 
Recall also the maps
 $\xymatrix{X_s  \ar[r]^-{\varpi^{s-1}_1}  &  X_1 \ar[r]^-{j_1} & X_0(Np) }$
 that arise in the equation 
  \eqref{eqn:def-Deltaf-circ}
  describing the cycle $\Delta_{s,s}^\circ$,
and define
\begin{equation}
\label{f,s,omega}
\omega_{\testf,s}:= (\varpi_1^{s-1})^*\circ j_1^* (\omega_{\testf}),
\end{equation}
to be the pull-back  of $\omega_{\testf}$ to $X_s$ via these   projections. 


 Let $(y,z) \in \Omega_g \times_\Omega \Omega_h$ be a classical point of weight $2$ and character $\epsilon \ne \omega$ of primitive conductor $p^s$. In order to lighten notations, set $g=g_y$, $h=h_z$ and
$$
\testg=\testg_y \in S_2(Np^s,\chi \epsilon \omega^{-1}),  \qquad  \testh=\testh_z \in S_2(Np^s,\chi^{-1} \epsilon \omega^{-1}).
$$ 
These notations are in force only in  Chapter   \ref{sec:bloch-kato-AJ}, and 
 $g$ and $h$ will revert to denoting  modular forms of weight $1$ in subsequent chapters, as 
 was done
  in the introduction. 
   Let  $h^\iota\in S_2(N_h p^s, \chi^{-1}\omega  \epsilon^{-1})$ be the newform
 whose $q$-expansion coefficients are determined by
 $$   a_{m}(h^\iota) =  
 \left\{ \begin{array}{ll} 
 \omega \epsilon^{-1}(m) \cdot a_{m}(h) & \mbox{ if } p \nmid m; \\
     \chi^{-1}(p)\overline{a}_{p}(h)^r & \mbox{ if } m = p^r.  
          \end{array} \right.$$
If $\testh = \sum_{d|N/N_h} \lambda_d \cdot h(q^d)$, 
set also 
$$\testh^\iota := \sum_{d|N/N_h} \lambda_d h^\iota(q^d) 
\in S_2(N p^s, \chi^{-1}\omega  \epsilon^{-1}),$$
so that for all $n$ with $p\nmid n$, the fourier coefficient $a_n(\testh^\iota)$ is given by 
$a_n(\testh^\iota) := (\omega \epsilon^{-1})(n) \cdot a_{n}(\testhh).$ 

For any Dirichlet character $\alpha$, let $\gauss(\alpha)\in \bar\Q_p$ denote the
 Gauss sum attached to $\alpha$, on which $G_{\Q_p}$ acts through this character.

 \begin{proposition} 
 \label{prop:atkin-li} The identity $w_{s}(\testh) = \gauss(\omega^{-1}\epsilon)  \alpha_h^{-s} \testh^\iota$ holds in $S_2(N_h p^s, \chi^{-1}\omega  \epsilon^{-1})$.
 \end{proposition}
 \begin{proof}
This follows from \cite[Theorem 2.1]{atkin-li} (with $k=2$ and $Q = p^s$), taking into account that $w_{s}$ commutes with the level-raising operators
 at the divisors of $N/N_h$, which are prime to $p$. 
 \end{proof} 

For any regular differential form $\omega\in \Omega^1(X_{s/F})$, let $\omega_\infty$ and $\omega_0$ denote its restrictions to $\cW_\infty(p^s)$ and $\cW_0(p^s)$ respectively, and write $[\omega_\infty]$, $[\omega_0]$ for their classes in cohomology. 

\begin{proposition}
 \label{prop:frobenius-on-primitives}  For $\omega= \omega_{\testf,s}$, $\omega_{\testg}$ and $\omega_{\testh}$,  
 \begin{equation}
\label{eqn:eigen-restrict}
 \Phi_\infty[\omega_{\infty}] = \beta \cdot [\omega_{\infty}], 
 \qquad 
   \Phi_0[\omega_{0}] = \alpha\cdot [\omega_{0}],
   \end{equation}
where $(\alpha,\beta)=(\alpha_f,\beta_f)$, $(a_p(g),\chi(p)\overline{a_p(g)})$ and $(a_p(h),\chi^{-1}(p)\overline{a_p(h)})$ respectively. 
   \end{proposition}
  \begin{proof} Let us prove the claim for $\omega=\omega_{\testg}$, as the other cases follow similarly.
Together with $V$, let $U=U_p$ be the usual operator on $p$-adic modular forms defined on $q$-expansions as in \eqref{eqn:Up-on-differentials}.
A direct calculation shows that 
$$ U(\testg) = a_p(g) \testg, \qquad  UV=VU=1 \mbox{ on } H^1_{\mathrm{rig}}(\cW_\infty(p^s)),
$$
hence $V([\omega_{\testg,\infty}]) = a_p(g)^{-1} \cdot [\omega_{\testg,\infty}]$.
The first equality  in \eqref{eqn:eigen-restrict}
now follows from the fact that the canonical lift $\Phi_\infty$ of
Frobenius agrees with $p\langle p;1\rangle V$, as stated in
\eqref{eqn:Phi-infty-V},
 and that $a_p(g)$ is a Weil number of weight one, i.e.,
a complex number of absolute value $\sqrt{p}$. 
To study the action of $\Phi_0$ on $\omega_{\testg,0}$,  note that
$$
 \Phi_0 (\omega_{\testg,0}) =  w_{s}^{-1} \Phi_\infty w_{s} (\omega_{\testg,0})
=   w_{s}^{-1} \beta_{g^\iota} w_{s} \omega_{\testg,0},$$
where the first equality follows from \eqref{eqn:def-Phi0} and the second from  
Proposition \ref{prop:atkin-li}. The second equality  in \eqref{eqn:eigen-restrict} now follows 
from the fact that $\beta_{g^\iota} = a_p(g) = \alpha_g$. 
\end{proof}
 If $\phi(q)$ is any overconvergent $p$-adic modular form, its {\em $p$-depletion}  is defined to be
 the overconvergent modular form whose $q$-expansion is given by 
 $$\phi^{[p]}(q) := (1-VU)\phi(q)  = \sum_{p\nmid n} a_n(\phi) q^n,$$
 and $\omega^{[p]}_{\phi}$ denotes the associated rigid differential on a wide open
 neighbourhood of $\cA_\infty(p^s)$ in $\cW_\infty(p^s)$. 
 
 
 \begin{corollary}
 \label{cor:deplete-via-Phi}
 \begin{enumerate}
\item The class of the rigid differential
  $\omega_{\testf,\infty}^{[p]} = (1-\beta_f^{-1}\Phi_\infty)(\omega_{\testf,\infty})$  is trivial in $H^1_{\mathrm{rig}}(\cW_\infty)$, i.e.,  $\omega_{\testf,\infty}^{[p]} = d\testF$ for some $\testF \in \cO_{\cW_\infty}$.
   The $q$-expansion of $\testF$ is 
   $$\testF (q) =  \sum_{p\nmid n} \frac{a_n(\testf)}{n}q^n.$$
 \item
 The class of the rigid differential
  $\omega_{\testg,\infty}^{[p]} = (1-\beta_g^{-1}\Phi_\infty)(\omega_{\testg,\infty})$  is trivial in $H^1_{\mathrm{rig}}(\cW_\infty(p^s))$, i.e., $\omega_{\testg,\infty}^{[p]}  = d \testG$
 for some  $\testG_\infty \in \cO_{\cW_\infty(p^s)}$.
The  $q$-expansion of $G$ is given by
 $$\testG(q) = \sum_{p\nmid n} \frac{a_n(\testg)}{n} q^n.$$
  \item
   The class of the rigid differential $\omega_{\testh,0}^{[p]} = (1-\alpha_h^{-1}\Phi_0)(\omega_{\testh,0})$  is trivial in $H^1_{\mathrm{rig}}(\cW_0(p^s))$, i.e.,  $\omega_{\testh,0}^{[p]} = d\testH^\circ$ for some  $\testH\circ \in \cO_{\cW_0(p^s)}$.  
The
 $q$-expansion of $w_s \testH^\circ$ is  given by
 $$ (w_s \testH^\circ)(q) =  \gauss(\omega^{-1}\epsilon)   \alpha_h^{-s} H^\iota(q), 
 \qquad H^\iota(q) =  \sum_{p\nmid n}  \frac{a_n(\testh^\iota)}{n} q^n, \qquad  a_n(\testh^\iota) = (\omega\epsilon^{-1})(n) a_n(\testh).$$
 \end{enumerate}
 \end{corollary}
\begin{proof}
 The triviality statements follow from Prop.\,\ref{prop:frobenius-on-primitives}
and the $q$-expansion of the primitives in  (1) and (2) are obtained by a direct calculation.
A similar reasoning, using Prop.\,\ref{prop:atkin-li}, implies (3).
\end{proof}






%Having discussed the classes $\omega_{\testg}$ and $\omega_{\testh}$ attached to the primitive cusp forms
%$\testg$ and $\testh$,  it is time to
%elucidate some issues concerning the classes $\omega_{\testf}$ and $\eta_{\testf^*}$ in $H^1_{\dR}(X_0(Np))$.

We turn now to analyze the dual test vectors and their associated de Rham cohomology classes.
Given a dual test vector $\testf^* \in S_2^{\ord}(\Gamma_0(Np))^\vee[f]$, 
recall that  $\eta_{\testf^*} \in   H^1_{\dR}(X_0(Np))^{\ord,\ur}[f]$ 
denotes the unique class satisfying 
\begin{equation}
\label{eqn:testf-star-bis}
\Phi(\eta_{\testf^*}) = \alpha_f\cdot \eta_{\testf^*}, \quad \mbox{ and } \quad 
 \langle \eta_{\testf^*}, w_1 \omega \rangle_{X_0(Np)} = \testf^*(\omega) \quad
 \forall \omega\in S_2^{\ord}(\Gamma_0(Np)). 
 \end{equation}
 As in equation \eqref{f,s,omega}, define
\begin{equation}
\label{eqn:eta-fs-pullback}
 \eta_{\testf^*,s}:= (\varpi_1^{s-1})^*\circ j_1^* (\eta_{\testf^*}).
\end{equation}
Likewise, set 
$$\testg^* = \testg^*_y\in S_2(Np^s,\chi^{-1} \epsilon \omega^{-1})^\vee, \qquad
\testh^* = \testh^*_z\in S_2(Np^s,\chi \epsilon \omega^{-1})^\vee,$$ 
 and let $\eta_{\testg^*}$ and $\eta_{\testh^*}\in H^1_{\dR}(X_{s/F})$ be the de Rham cohomology classes associated to them as in \eqref{eqn:define-eta-phistar}. Note that 
\begin{equation}
\label{frobf}
\Phi(\eta) = \alpha \cdot \eta \, \mbox{ and } \,  U_p (\eta) = \alpha\cdot \eta
\end{equation}
for $\eta = \eta_{\testf^*,s}$, $\eta_{\testg^*}$ and $\eta_{\testh^*}$, where $\alpha = \alpha_f$, $\alpha_g$ and $\alpha_h$, respectively.
 \begin{lemma}
 \label{lemma:etaf-support}
  The classes  $\eta_{\testg^*}$ and $\eta_{\testh^*}$ are supported on $\cW_0(p^s)$. 
 \end{lemma}
\begin{proof}
  Arguing as in Lemma \ref{restrict-to-X0N} and taking \eqref{frobf} into account, it  follows that the restriction of $\eta_{\testg^*}$ and $\eta_{\testh^*}$ to 
  $\cW_\infty(p^s)$ vanish, and hence they are  supported on $\cW_0(p^s)$ 
  by Theorem \ref{thCol}.
\end{proof}
\begin{remark}
Lemma \ref{lemma:etaf-support} is consistent with the fact that the value of 
$\langle \eta_{\testg^*}, w_s \omega\rangle_s$ depends only on the restriction of the ordinary class 
$\omega$ to
$\cW_\infty(p^s)$, i.e., on the  overconvergent  modular form attached to $\omega$.
\end{remark} 



\subsection{Twisted diagonal cycles under the syntomic Abel-Jacobi map}
\label{sec:AJDelta}

We are now in a position to evaluate the image of the twisted diagonal 
cycles under the syntomic Abel-Jacobi map. 
% We start with the following simple observation, which places our computation  in the setting of the previous chapter:  \begin{lemma}  \label{lemma:reduce-to-triple} Let $d_s := \frac{1}{2} \varphi(N)(p-1)p^{s-1}$. Then $$\AJ_p(\Delta_s)(\eta_{\testf^*} \otimes \omega_{\testg}\omega_{\testh}) =d_s^{-1} \times  \AJ_p(\Delta_{s,s})(\eta_{\testf^*}\otimes \omega_{\testg} \otimes \omega_{\testh})$$ and likewise for $\AJ_p(\Delta_s)(\omega_{\testf} \otimes \eta_{\testg^*}\omega_{\testh})$ and $\AJ_p(\Delta_s)(\omega_{\testf} \otimes \omega_{\testg}\eta_{\testh^*})$. \end{lemma} \begin{proof} Note that, over $F$, the varieties $W_s^\dag$ and $W_s$ are canonically isomorphic, and that the cycles  $\Delta_s$ on both varieties are equal. The lemma  is then a direct consequence of the functoriality properties of the Abel-Jacobi map in light of the first equality in \eqref{Up-comp} and \eqref{eqn:class-on-quotient}.  \end{proof}

 
\subsubsection{Evaluation of $\AJ_p(\Delta_{s,s}^\circ)$ at $\eta_{\testf^*} \otimes \omega_{\testg}  \otimes \omega_{\testh}$.} 

\begin{lemma}
\label{lemma:verify-conditions}
The data $(\Delta, \eta_1,\omega_2,\omega_3) = (\Delta_{s,s}^\circ, \eta_{\testf^*}, \omega_{\testg}, \omega_{\testh})$
%, $ (\Delta_{s,s}, \omega_{\testf}, \eta_{\testg^*}, \omega_{\testh})$ and $ (\Delta_{s,s}, \omega_{\testf}, \omega_{\testg}, \eta_{\testh^*})$ 
satisfies conditions (a)-(d) of \S \ref{sec:recipe}.
\end{lemma}
\begin{proof}
Condition (a) is clear, since $W_{s,s}$ is a triple product of curves with semistable reduction over $F$. 
The cycle $\Delta_{s,s}^\circ$ is described as in condition (b) of Section \ref{sec:recipe}
by setting 
$$ X=X_s,  \qquad (\pi_1, \pi_2, \pi_3) = ( j_1 \circ \varpi_1^{s-1}, w_s, \Id),  \qquad 
(\varepsilon_1,\varepsilon_2,\varepsilon_3) = (\varepsilon, \varepsilon_s,\varepsilon_s).
$$
Condition (c) is built into the assumptions, while 
Condition (d) follows from the fact that the form $\eta_{\testf^*}$ belongs to the $f$-isotypic subspace attached to an  eigenform of prime-to-$p$ level, and $\testg$ and $\testh$ are
primitive at $p$.
% and hence their associated cohomology classes lie in the pure subspace of the de Rham cohomology by Theorem \ref{thCol}.
\end{proof}

Let  $\xi(\omega_{\testg},\omega_{\testh}) \in H^1_{\dR}(X_{s/F})$ be the cohomology class attached to the data $(\Delta_{s,s}^\circ, \omega_{\testg}, \omega_{\testh})$ following 
 the recipe of \S \ref{sec:recipe}.  Then Theorem \ref{thm:recipe} applies and asserts that 
\begin{equation}
\label{eqn:first-AJ}
 \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})  
 = \langle \eta_{\testf^*,s}, \xi(\omega_{\testg},\omega_{\testh}) \rangle_{s}.
 \end{equation}



 \begin{lemma} 
 \label{le}
 The class $\xi(\omega_{\testg},\omega_{\testh})$ arises
 as the pull-back of a class in $H^1_{\dR}(X_0(Np^s)_{/F})$  via $j_s$.
 \end{lemma}
 \begin{proof}
% Observe first that for each $1\leq j \leq 4$, the map $\pi:=(\pi_2^{(j)}, \pi_3^{(j)}): X_s \lra X_s \times X_s$ satisfies  \begin{equation} \label{eqq} \pi \circ \langle a; b\rangle = (\langle a; b\rangle,\langle a; b^{-1}\rangle) \circ \pi \quad \mbox{ for all } \quad (a,b)\in (\Z/N\Z)^\times\times (\Z/p^s\Z)^\times.  \end{equation}
Let  
$$
\rho_P^{v,w}(\omega_{\testg},\omega_{\testh})  \in \Omega^1(\cW_v\times\cW_w), \qquad v,w\in \cV(\cG(\tilde\cX_s)), 
$$
be a system of rigid $1$-forms satisfying \eqref{eqn:choose-rho}. The operators $(\langle a; b\rangle,\langle a; b^{-1}\rangle)$ in $D_s$ leave the system $\{ \rho_P^{v,w} \}$ invariant {\em up to systems of closed $1$-forms in $\Omega^1(\cW_v\times\cW_w)$}; this follows because $D_s$ fixes the class $\omega_{\testg}\otimes \omega_{\testh}\in H^2_{\dR}(X_s \times X_s)$. It follows  that the class 
\begin{equation}
\label{xiPP}
\xi_P(\omega_{\testg},\omega_{\testh})\in H^1_{\dR}(X_{s/F})
\end{equation}
introduced in \S \ref{sec:recipe}  is well-defined, and fixed by the diamond operators.
 Since these operators  commute with Frobenius,
  the same holds for  $\xi(\omega_{\testg},\omega_{\testh})$, 
  and  the lemma follows.
 \end{proof}
 
   The degeneracy maps $ \pi_1, \pi_2: X_s^\flat \lra X_s$ that were introduced 
in \eqref{eqn:factoring-the-maps}
 induce similar maps from $X_0(Np^{s+1}) \lra X_0(Np^s)$ by passing to the quotient
by the action of the group $(\Z/Np^s)^\times$ of diamond operators of level $Np^s$ on both sides.
Let us continue to denote these maps as $\pi_1$ and $\pi_2$, by a slight abuse of notation.
The class $\eta_{\testf^*,s}$ likewise arises as the pull-back via $j_s$
of a class on $H^1_{\dR}(X_0(Np^s)_{/F})$, which shall be denoted by the same symbol.  
(Any ambiguity arising from this double use of notation  shall be avoided by 
consistently specifying   the modular  curve on which the Poincar\'e pairing being computed. Furthermore, the object $\eta_{\testf^*,s}$ will play only a provisional role, disappearing after the proof of 
Lemma \ref{lemma:AJfourth}.)
With  these new  notations,    
\begin{equation}
\label{eqn:eta-fs-pullback-bis}
 \eta_{\testf^*,s}  = (\pi_1^{s-1})^* (\eta_{\testf^*}),
\end{equation}
as follows directly from \eqref{eqn:eta-fs-pullback}. Since $d_s=\deg(j_s)$, the functoriality of Poincar\'e duality relative to pull-backs now allows us to rewrite \eqref{eqn:first-AJ}  in terms of the Poincar\'e pairing on $X_0(Np^s)$, namely
as
\begin{equation}
\label{eqn:AJ-second}
 \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})  
 =  d_s \cdot \langle \eta_{\testf^*,s}, \xi(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)}.
 \end{equation}
 
 

 
 Recall the ordinary   projector  $e_{\ord}$ of \eqref{eqn:def-eord-eordstar} 
 that was already exploited in the  discussion of Hida theory
 in \S \ref{sec-Hida}
as well as its anti-ordinary counterpart
 $e_{\ord}^*$. Both 
 will now be considered as   endomorphisms of 
 $H^1_{\dR}(X_0(Np^s)_{/F})$. Just like $U_p$ and $U_p^*$, 
 these endomorphisms are adjoint to each other relative to 
 Poincar\'e duality.
 \begin{lemma}
 \label{anti-ord}
 For all $s\ge 1$,
  $$  \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})  
 =  d_s \langle \eta_{\testf^*,s},  e_{\ord}^* \xi(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)}.$$
 \end{lemma}
 \begin{proof} 
 This follows from  \eqref{eqn:AJ-second} in light of 
  the fact that the class $\eta_{\testf^*,s}$ is ordinary
  and hence equal to its image under the ordinary projector, so that
  $$ \langle \eta_{\testf^*,s},    \xi(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)} = 
 \langle e_{\ord}\eta_{\testf^*,s},    \xi(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)}
  =  \langle \eta_{\testf^*,s},    e_{\ord}^* \xi(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)}.$$\end{proof}
  

  \begin{lemma}
  \label{lemma:ord-pullback-coho}
The pulback $(\pi_1^{s-1})^*$ 
  induces an isomorphism
  $$(\pi_1^{s-1})^*: e_{\ord} H^1_{\dR}(X_0(Np)) \lra e_{\ord} H^1_{\dR}(X_0(Np^s)).$$
 \end{lemma}
 \begin{proof} 
The 
 second displayed equation in the proof of Thm.~4.1. of \cite{Emerton} (after replacing $s$ 
 by $1$ and $r$ by $s$)  asserts that the   
 homomorphism 
 $$  e_{\ord} ((\Gamma_1(Np) \cap \Gamma_0(p^s))^{\rm ab} \otimes \Z_p) 
 \lra e_{\ord}( \Gamma_1(Np)^{\rm ab}\otimes \Z_p)$$
 induced from the natural inclusion $  \Gamma_1(Np)\cap \Gamma_0(p^s) \subset \Gamma_1(Np)$
 is an isomorphism. The abelianisation  of a congruence subgroup (tensored with
 $\Z_p$)  is naturally identified with  the homology
 with $\Z_p$-coefficients of the associated open
 modular curve. Taking a further quotient by the subgroup generated by the parabolic elements and passing to $\Z_p$-duals, 
 one concludes  that the natural map
 $$ e_{\ord} H^1(X_1(Np),\Z_p)  \lra e_{\ord} H^1(X_1(Np) \times_{X_0(Np)} X_0(Np^s),\Z_p)$$
 is an isomorphism. This map agrees with $(\pi_1^{s-1})^*$ (viewed as a map on singular cohomology).
  Hence 
Lemma \ref{lemma:ord-pullback-coho}
 follows after taking invariants under the action of the group $(\Z/Np\Z)^\times$
 of diamond operators acting on both sides, and invoking the comparison isomorphism between 
 singular and de Rham cohomology.
 \end{proof}
 
 
 
 
 
 Applying the ordinary projector to $ \xi^\iota(\omega_{\testg},\omega_{\testh}) := w_s \xi(\omega_{\testg},\omega_\testh)$, it  follows from Lemma \ref{lemma:ord-pullback-coho} that
\begin{equation}
\label{eqn:bring-in-p-adic-forms}
  e_{\ord}\xi^\iota(\omega_{\testg},\omega_{\testh})  = (\pi_1^{s-1})^*(\xi^\iota(\omega_{\testg},\omega_{\testh})_{1}), 
  \end{equation} 
 for a suitable class $ \xi^\iota(\omega_{\testg},\omega_{\testh})_{1} \in e_{\ord} H^1_{\dR}(X_0(Np)_{/F})$. 
 Let $\xi^\iota(\omega_{\testg},\omega_{\testh})_{\cW_\infty}$ denote the restriction of the
  class $\xi^\iota(\omega_{\testg},\omega_{\testh})_{1}$
   to the Hasse domain $\cW_\infty$ viewed as a  $p$-adic modular form of weight two. 
  \begin{lemma}
  \label{lemma:AJfourth}
   For all $s\ge 1$,
 $$ \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})   
 =  d_s  \alpha_f^{s-1} \testf^*(\xi^\iota(\omega_{\testg},\omega_{\testh})_{\cW_\infty}).$$
 \end{lemma}
 \begin{proof}  Since   $w_s =w_s^{-1}$   on $X_0(Np^s)$,  one has
  $e_{\ord}^* = w_s e_{\ord} w_s$.
 Lemma \ref{anti-ord} implies that 
$$
\AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh}) 
=  d_s \langle \eta_{\testf^*,s}, w_s   e_{\ord}  \xi^\iota(\omega_{\testg},\omega_{\testh}) \rangle_{X_0(Np^s)}  .$$
It folllows from  \eqref{eqn:bring-in-p-adic-forms}
that
%$$
%\AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})  
% =  d_s \langle \eta_{\testf^*,s},  w_s  (\pi_1^{s-1})^*\xi^\iota(\omega_{\testg},\omega_{\testh})_{1}
% \rangle_{X_0(Np^s)}  =  d_s \langle   \pi_{2*}^{s-1}\eta_{\testf^*,s}, 
%  w_1^* \xi^\iota(\omega_{\testg},\omega_{\testh})_{1} \rangle_{X_0(Np)},
%$$
\begin{align*}
\AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})  
& =  d_s \langle \eta_{\testf^*,s},  w_s  (\pi_1^{s-1})^*\xi^\iota(\omega_{\testg},\omega_{\testh})_{1}
 \rangle_{X_0(Np^s)} \\ 
 & =  d_s \langle   \pi_{2*}^{s-1}\eta_{\testf^*,s}, 
  w_1^* \xi^\iota(\omega_{\testg},\omega_{\testh})_{1} \rangle_{X_0(Np)},
  \end{align*}
where  the facts that $w_s^*(\pi_1^{s-1})^* = (\pi_2^{s-1})^* w_1^*$ 
and that  the adjoint of $\pi_2^*$ relative to Poincar\'e duality is 
$\pi_{2*}$ have been used to derive the last equation.
By  \eqref{eqn:eta-fs-pullback-bis}
    and \eqref{eqn:def-Up},
$$ \pi_{2*}^{s-1}\eta_{\testf^*,s}= \pi_{2*}^{s-1} (\pi_{1}^{s-1})^* \eta_{\testf^*} = U_p^{s-1} \eta_{\testf^*} 
= \alpha_f^{s-1} \eta_{\testf^*},$$
and therefore 
\begin{align*}
\AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh}) 
 &= d_s \alpha_f^{s-1}  \langle   \eta_{\testf^*}, 
  w_1^* \xi^\iota(\omega_{\testg},\omega_{\testh})_{1} \rangle_{X_0(Np)} \\ 
  &= d_s \alpha_f^{s-1} \testf^*(\xi^\iota(\omega_{\testg},\omega_{\testh})_{\cW_\infty}), 
  \end{align*}
  where the last equality follows from equation \eqref{eqn:testf-star-bis} defining $\eta_{\testf^*}$. 
  \end{proof}
  Since the pull-back $\pi_1^*$ induces the identity on $q$-expansions,  Lemma
   \ref{lemma:AJfourth} implies that
\begin{equation}
\label{eqn:AJfourth}
 \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*}\otimes \omega_{\testg}\otimes \omega_{\testh})   
 =  d_s  \alpha_f^{s-1} \testf^*(e_{\ord}\xi^\iota(\omega_{\testg},\omega_{\testh})),
 \end{equation}
in light of \eqref{eqn:bring-in-p-adic-forms}. We now turn to the calculation of the ordinary  $p$-adic modular form 
 $e_{\ord}\xi^\iota(\omega_{\testg},\omega_{\testh})$.
   Let $P\in \C_p[t]$ be a polynomial satisfying the conditions of 
      Lemma \ref{lemma:choose-P}, i.e.\,such that 
\begin{enumerate}
\item[(i)] $P(\Phi\times \Phi)$ annihilates  the class of $\omega_{\testg,\infty}\otimes \omega_{\testh,0}$ in $H^2_{\rig}(\cW_\infty(p^s) \times \cW_0(p^s))$.
%\item[(ii)]$P(\Phi)$ annihilates $\C_p(-1)$, that is to say,. 
\item[(ii)] None of the roots of $P(t)$ are of complex absolute value $\sqrt{p}$.
\end{enumerate}
In light of Proposition \ref{prop:frobenius-on-primitives}, one may take $P(x) :=  2(1- \beta_{g}^{-1} \alpha_h^{-1} x) = 2 (1- p^{-1} \chi^{-1}(p) \alpha_g \alpha_h^{-1} x)$.
The polynomial identity
$$P(xy) = (1-\beta_g^{-1}x)(1+\alpha_h^{-1} y) + (1+\beta_g^{-1} x)(1-\alpha_h^{-1}y)$$
combined with Corollary \ref{cor:deplete-via-Phi}
implies that
$$   P(\Phi_2\times\Phi_3)(\omega_{\testg,\infty}\otimes\omega_{\testh,0})
= (d\testG)  \otimes (1+\alpha_h^{-1} \Phi_0)\omega_{\testh,0}
 + (1+\beta_g^{-1}\Phi_\infty)\omega_{\testg,\infty}     \otimes  
 (d\testH^\circ).$$
It follows that the  rigid analytic differential $1$-form
 $\varrho_{P}(\omega_{\testg},\omega_{\testh})\in \Omega^1(\cW_\infty(p^s) \times \cW_0(p^s))$  
 satisfying
$$
d\varrho_{P}(\omega_{\testg},\omega_{\testh}) 
= P(\Phi_2\times\Phi_3)(\omega_{\testg,\infty} \otimes \omega_{\testh,0})
$$
can be chosen to be
\begin{equation}
\label{eqn:formula-rho}
\varrho_P(\omega_{\testg},\omega_{\testh}) = 
 \testG \otimes (1+\alpha_h^{-1} \Phi_0) \omega_{\testh,0} 
 - (1+\beta_g^{-1}\Phi_\infty)\omega_{\testg,\infty} \otimes  \testH^\circ,
 \end{equation}
where  $\testG  \in \cO_{\cW_\infty(p^s)}$ and $\testH^\circ \in\cO_{\cW_0(p^s)}$
are the primitives  described in Corollary \ref{cor:deplete-via-Phi}.

After setting $\xi_P^\iota(\omega_{\testg},\omega_{\testh}) := w_s \xi_P(\omega_{\testg},\omega_{\testh})$ and observing that the correspondences $\varepsilon_s$ fix the differentials $\omega_{\testg}$,
$\omega_{\testh}$ as well as their primitives $\testG$ and $\testH^\circ$, 
 it follows  that
\begin{align}
\label{eqn:formula-for-xiP}
  \xi_P(\omega_{\testg},\omega_{\testh})  & =       
 \left( (w_s^{-1}\testG) \times (1+\chi^{-1}(p)\alpha_h^{-1} \Phi_\infty) \omega_{\testh,\infty} 
- (1+\beta_g^{-1} \Phi_\infty) ( w_s^{-1}\omega_{\testg,\infty}) \times \testH^\circ \right), \\
\xi_P^\iota(\omega_{\testg},\omega_{\testh})  & =       
 \left( \testG  \times (1+\chi^{-1}(p)\alpha_h^{-1} \Phi_\infty) (w_s\omega_{\testh,\infty}) 
- (1+\beta_g^{-1} \Phi_\infty) \omega_{\testg,\infty} \times (w_s\testH^\circ)\right).
\end{align}
By Corollary \ref{cor:deplete-via-Phi},    the $q$-expansion of this $p$-adic modular form is given by 
\begin{equation}
\label{eqn:formula-for-xiP-bis}
  \xi_P^\iota(\omega_{\testg},\omega_{\testh})(q) =      \gauss(\omega^{-1}\epsilon) \alpha_h^{-s} 
 \left(\testG  \times    (1+\chi^{-1}(p)\alpha_h^{-1} \Phi_\infty) {\testh^\iota } 
- (1+\beta_g^{-1} \Phi_\infty) \testg_{\infty} \times \testH^\iota \right)(q) \frac{dq}{q}.
\end{equation}
 This is a rigid differential on a wide open neighbourhood of $\cA_\infty(p^s)$ in
 $\cW_\infty(p^s)$, and hence corresponds to an 
 overconvergent modular form of weight $2$ on $\cW_\infty(p^s)$. 
 Since   the nebentypus character  of ${\testg}$ and $\testG$
 is $\chi\omega^{-1}\epsilon$, and that of ${\testh^\iota}$ and
$\testH^\iota$ is $\chi^{-1}\omega \epsilon^{-1}$, it follows 
that $\xi_P^\iota(\omega_{\testg},\omega_{\testh})$
arises as a pullback  under the projection $j_s$ of a rigid differential on $\cW_\infty$. 
 \begin{proposition}
 \label{prop:eval-e-ord-xi-P}
The class of $e_{\ord}(\xi_P^\iota(\omega_{\testg},\omega_{\testh}))$ in $H^1_{\rig}(\cW_\infty)$ is equal
to the class of the weight two $p$-adic modular form
$$ -2    \gauss(\omega^{-1}\epsilon) \times \alpha_h^{-s} 
      \times \testg \testH^\iota \in \Omega^1(\cW_\infty).$$
\end{proposition}
\begin{proof} 
If $f_1$ and $f_2$ are overconvergent modular forms,
then $f_1^{[p]} \times (Vf_2)$ lies in the kernel of $U$ and hence a fortiori in the kernel
of $e_{\ord}$. Since $\Phi_\infty = \langle p \rangle p V$, after applying $e_{\ord}$ to
\eqref{eqn:formula-for-xiP-bis} one obtains
\begin{eqnarray}
\nonumber
  e_{\ord}(\xi_P^\iota(\omega_{\testg},\omega_{\testh})) &=&    
   \gauss(\omega^{-1}\epsilon) \times \alpha_h^{-s} \times
 e_{\ord}(\testG  \times   \testh^\iota_{\infty}
-   \testg_{\infty} \times \testH^\iota) \\
\label{eqn:eliminate-messy-terms}
&=&  
  \gauss(\omega^{-1}\epsilon) \times \alpha_h^{-s} \times
 e_{\ord}(\testG  \times   \testh_{\infty}^{\iota[p]}
-   \testg_{\infty}^{[p]} \times \testH^{\iota}).
\end{eqnarray}
Applying $e_{\ord}$ to the identity 
$d(\testG  \testH^\iota) =  \testG \times   \testh^{\iota[p]}_{\infty} 
+ \testg_{\infty}^{[p]} \times \testH^\iota,$
and using the fact that the image of $d$ on overconvergent forms is in the kernel of
$e_{\ord}$,
it follows that
\begin{equation}
\label{eqn:product-rule}
 e_{\ord}( \testG\times   \testh^{\iota[p]}_\infty) = - e_{\ord}(\testg_{\infty}^{[p]} \times \testH^\iota),
 \end{equation}
 and the result follows from \eqref{eqn:product-rule} combined with 
 \eqref{eqn:eliminate-messy-terms}.
\end{proof}
We can now state the main formula concerning
 the value of 
  $\AJ_p(\Delta_{s,s}^\circ)$ at the vector 
  $\eta_{\testf^*} \otimes \omega_{\testg}   \otimes \omega_{\testh}$: 
 \begin{theorem}
 \label{thm:AJ-p-Delta} 
 Setting $\cE_{f}(f, g,h) := -2 (1-\chi^{-1}(p) \alpha_f^{-1} \alpha_g \alpha_h^{-1})^{-1} $, 
one has
\begin{equation*}
\AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*} \otimes \omega_{\testg} \otimes  \omega_{\testh}) =
\cE_{f}(f,g,h)   \times   \gauss(\omega^{-1}\epsilon) \times \alpha_f^{s-1} \alpha_h^{-s} \times   d_s\times
\testf^*(\testg \testH^{\iota}). 
 \end{equation*}
 \end{theorem}
  \begin{proof}
 By combining \eqref{eqn:AJfourth} with Proposition  \ref{prop:eval-e-ord-xi-P},
\begin{equation*}
 \AJ_p(\Delta_{s,s}^\circ)(\eta_{\testf^*} \otimes \omega_{\testg}  \otimes \omega_{\testh})  =  
 -2    \times   \gauss(\omega^{-1}\epsilon) \times  \alpha_f^{s-1} \alpha_h^{-s}  \times d_s 
  \times {\testf^*}(P(\Phi_\infty)^{-1} ( \testg\testH^\iota)).
 \end{equation*}
But for all classes $\xi\in H^1_{\mathrm{rig}}(\cW_\infty)$,  
${\testf^*}(\Phi \xi ) =  p\alpha_f^{-1} {\testf^*}(\xi) 
 = \beta_f \testf^*(\xi)$.
  More generally, $\testf^*(Q(\Phi)\xi)  = Q(\beta_f) \testf^*(\xi)$, 
   for all rational functions $Q$ with no pole at $\beta_f$.
 The  result follows. 
  \end{proof}
  
  
  
  

\subsubsection{Evaluation of $\AJ_p(\Delta_{s,s}^\circ)$ at $\omega_{\testf} \otimes \eta_{\testg^*}  \otimes\omega_{\testh}$.}  
Thanks to Lemma \ref{lemma:verify-conditions}, the data $(\Delta_{s,s}, \omega_{\testf}, \eta_{\testg^*}, \omega_{\testh})$ satisfies conditions (a)-(d) of \S \ref{sec:recipe}. 
% However, in this case it is more convenient to work with the projections described below (which still give rise to the same cycle, as argued in the proof of Proposition \ref{Up-compf}): $$ \begin{array}{|c|r|l|}  \hline  j & m_j & ( \pi_{1},\pi_2^{(j)}, \pi_3^{(j)})  \\ \hline  1 & 1 & (j_1\circ \varpi_2^{s-1},w^{-1}_{\zeta_s},\id) \\  2 & -1 & (j_1\circ \varpi_2^{s-1},\langle a\rangle \circ w^{-1}_{\zeta_s},\id) \\  3 & -1 & (j_1\circ \varpi_2^{s-1},w^{-1}_{\zeta_s}, \langle a \rangle)  \\  4 & 1 & (j_1\circ \varpi_2^{s-1}, \langle a \rangle \circ w^{-1}_{\zeta_s}, \langle a \rangle) \\ \hline      \end{array}     $$
The class $w_{s}^{-1} \eta_{\testg^*}$ is anti-ordinary and supported on $\cW_\infty(p^s)$ by Lemma \ref{lemma:etaf-support}.
Theorem \ref{thm:recipe} therefore applies and asserts   that
\begin{equation}\label{asserts}
\AJ_p(\Delta_{s,s})(\omega_{\testf} \otimes \eta_{\testg^*} \otimes \omega_{\testh}) 
= \langle w_{s}^{-1} \eta_{\testg^*},\xi(\omega_{\testf}, \omega_{\testh}) \rangle_s 
= \langle   \eta_{\testg^*}, w_s e_{\ord}(\xi(\omega_{\testf}, \omega_{\testh})) \rangle_s
\end{equation} 
where $\xi(\omega_{\testf}, \omega_{\testh})\in H^1_{\dR}(X_{s/F})$ is the class associated to the triple $(\Delta_{s,s},\omega_{\testf},\omega_{\testh})$ as in \S \ref{sec:recipe} and the second equality follows as in Lemma \ref{anti-ord}. 

In order to compute the class $\xi(\omega_{\testf}, \omega_{\testh})$ explicitly, consider the polynomial $P(x) :=  2(1- \beta_{f}^{-1} \beta_h^{-1} x)$. It satisfies the conditions 
of Lemma \ref{lemma:choose-P}, since
 the endomorphism $P(\Phi_1\times \Phi_3)$ annihilates 
 the class of $\omega_{\testf,\infty} \otimes \omega_{\testh,\infty}$ in $H^2_{\rig}(\cW_\infty\times \cW_\infty(p^s))$ by Proposition \ref{prop:frobenius-on-primitives} and \eqref{frobf}. The
 polynomial  identity 
 $$P(x y) = (1-\beta_f^{-1}x)(1+\beta_h^{-1}y)+(1+\beta_f^{-1}x)(1-\beta_h^{-1}y)$$
 combined with  Corollary \ref{cor:deplete-via-Phi} 
implies that
$$P(\Phi_1\times\Phi_3)(\omega_{\testf,\infty}\otimes\omega_{\testh,\infty})
= \omega_{\testf,\infty}^{[p]} \otimes (1+\beta_h^{-1} \Phi_\infty)\omega_{\testh,\infty}
 + (1+\beta_f^{-1}\Phi_\infty)\omega_{\testf,\infty}     \otimes  
  \omega_{\testh,\infty}^{[p]}.$$
It follows that the form
 $\varrho_{P}(\omega_{\testf},\omega_{\testh})\in \Omega^1(\cW_\infty\times \cW_\infty(p^s))$  
satisfying \eqref{eqn:choose-rho}
%$$
%d\varrho_{P}(\omega_{\testf},\omega_{\testh}) 
%= P(\Phi_1\times\Phi_3)(\omega_{\testf,\infty} \otimes \omega_{\testh,\infty})
%$$
can be chosen to be
\begin{equation}
\label{eqn:formula-rho-bis}
\varrho_P(\omega_{\testf},\omega_{\testh}) = 
 \testF \otimes (1+\beta_h^{-1} \Phi_\infty) \omega_{\testh,\infty} 
 - (1+\beta_f^{-1}\Phi_\infty)\omega_{\testf,\infty} \otimes  \testH,
 \end{equation}
 where $\testH\in\cO_{\cW_\infty(p^s)}$ is the  rigid analytic primitive of  $\omega_{\testh,\infty}^{[p]}$
 following the definitions of Corollary
  \ref{cor:deplete-via-Phi} (with $g$ and $G$ replaced by $h$ and $H$). By the analogous calculation that led to  \eqref{eqn:formula-for-xiP}, the
 pull-back $\xi_P(\omega_{\testf},\omega_{\testh})$ of $\varrho_P(\omega_{\testf},\omega_{\testh})$ under the correspondence
$$ (\varepsilon,\varepsilon_s) ( j_1 \circ \varpi_1^{s-1}, \Id): X_0(Np)\times X_s \lra X_0(Np)\times X_s$$
arising in the definition of the cycle $\Delta_{s,s}^\circ$ 
 is given by
\begin{equation}\label{eqn:formula-for-xiP-ter}
  \xi_P(\omega_{\testf},\omega_{\testh}) =  \testF  \times   (1+\chi(p)\alpha_h \Phi_\infty)  \omega_{\testh,\infty}
- (1+\beta_f^{-1} \Phi_\infty) \omega_{\testf,s,\infty} \times \testH,
\end{equation}
where  the fact that 
 $\varpi_1^*$ and $j_1^*$ induce the identity on $q$-expansions has been used to show that 
  $\testF$ pulls back  to the same overconvergent modular form (viewed this time as a rigid differential on $\cO_{\cW_\infty(p^s)}$). 
In this calculation, copious use has been made of 
 the fact that the correspondence $\varepsilon$ was chosen so as to leave 
$\omega_{\testf}$  and $\testF$ invariant, and likewise that the correspondence $\varepsilon_s$ fixes 
$\omega_{\testh}$ and $\testH$.

 Arguing exactly as in the proof of Proposition \ref{prop:eval-e-ord-xi-P}, it follows that 
\begin{equation}
\label{prop:eval-e-ord-xi-P-bis}
e_{\ord}\xi_P(\omega_{\testf},\omega_{\testh})_{\cW_\infty} =
  2\testF  \cdot \omega_{\testh,\infty}  \, \mbox{ and hence } \,  e_{\ord}\xi(\omega_{\testf},\omega_{\testh})_{\cW_\infty} =  2 P(\Phi_\infty)^{-1}\big(\testF  \cdot \omega_{\testh,\infty}\big).
\end{equation}
\begin{theorem}
 \label{thm:AJ-p-Delta-g} 
Let  $\cE_{g}(f,g,h) : = 2(1- \chi^{-1}(p)\beta_f^{-1} \beta_g \beta_h^{-1})^{-1} = 2(1-\chi(p) p^{-1} \alpha_f \alpha_g^{-1} \alpha_h)^{-1}$. Then 
\begin{equation*}
\AJ_p(\Delta_{s,s}^\circ)(\omega_{\testf} \otimes \eta_{\testg^*}  \omega_{\testh}) = \cE_g(f,g,h)   \times  
 {\testg^*}(\testF  \times \testh).
\end{equation*}
\end{theorem}
\begin{proof} Combining \eqref{asserts} and \eqref{prop:eval-e-ord-xi-P-bis},
 and taking into account that the adjoint of $w_{s}^{-1}$ with respect to
  $\langle \, , \rangle_s$ is $w_{s}$, 
  it follows that 
$$
\AJ_p(\Delta_{s,s}^\circ)(\omega_{\testf} \otimes \eta_{\testg^*}  \omega_{\testh})   = 2 \langle\eta_{\testg^*},  w_s P(\Phi_\infty)^{-1} \big(\testF  \cdot   \omega_{\testh,\infty}\big)\rangle_s 
  =  \cE_g(f,g,h) \langle \eta_{\testg^*},  w_s  \big(\testF \cdot   \omega_{\testh,\infty}\big)\rangle_s.
$$
   %\begin{align}
%\AJ_p(\Delta_{s,s}^\circ)(\omega_{\testf} \otimes \eta_{\testg^*}  \omega_{\testh})  & = 2 \langle\eta_{\testg^*},  w_s P(\Phi_\infty)^{-1} \big(\testF  \cdot   \omega_{\testh,\infty}\big)\rangle_s \\
% & =  \cE_g(f,g,h) \langle \eta_{\testg^*},  w_s  \big(\testF \cdot   \omega_{\testh,\infty}\big)\rangle_s \\ 
% & = \testg^*(\testF \times \testh),
% \end{align}
The theorem is then a consequence of the definition of $\eta_{\testg^*}$.
   \end{proof}
 

  

  
\subsection{Garrett-Hida $p$-adic $L$-functions} 
\label{sec:garrett-rankin}
We now come to the main results of this chapter, expressing the Bloch-Kato logarithms of the specialisation of the classes
$\kappa_p^f(\testf^*,\underline{\testg\testh})$ and 
$\kappa_p^g(\testf,\underline{\testg^*\testh})$  at weight $2$ points in terms of the values of
the one-variable Garret-Hida $p$-adic $L$-functions introduced in \eqref{one-var}.
\begin{theorem} 
\label{prop:bloch-kato-log-in-families}
 For all arithmetic points $x=(y,z)\in \Omega_g\times_\Omega \Omega_h$ of weight $(2,\epsilon)$,
  $$   \log_p(\kappa_p^f(\testf^*,\testg_y,\testh_z))  = \cE_{f}(f, g_y,h_z) \times \gauss(\omega^{-1}\epsilon) \times \alpha_f^{s-1} \alpha_{g_y}^{-s} \alpha_{h_z}^{-s} \times \Lp^{f_\alpha}(\testf^*,\underline{\testg\testh})(y,z).$$
%  where   $ \cE_{f}(f, g_y,h_z)$ is the factor defined in  Theorem \eqref{thm:AJ-p-Delta}.
  \end{theorem}
  \begin{proof}
  Combining    Corollary  \ref{cor:from-log-to-AJ} with Theorem  \ref{thm:AJ-p-Delta} 
  leads to the equality
 \begin{equation}
 \label{eqn:starting-point-now}
\log_p(\kappa_p^{f_\alpha}(\testf^*,\testg_y,\testh_z)) =  \cE_{f}(f,g_y,h_z) \times \gauss(\omega^{-1}\epsilon) \times \alpha_f^{s-1} \alpha_{g_y}^{-s} \alpha_{h_z}^{-s} \times \testf^*(\testg_y  \times d^{-1} \testh_z^{\iota}).
\end{equation}
The  formal $q$-expansion with coefficients in $\Lambda_h$ which is given by
$\testhH^\iota(q) := \sum_{p\nmid n} \langle n^{-1}\rangle a_n(\testh) q^n,$ 
where $\langle n \rangle\in \Lambda$ denotes the group-like element associted to $n\in \Z_p^\times$, 
satisfies the following properties:
\begin{enumerate}
\item[(i)] For any arithmetic point $z\in \Omega_h$ of weight-character $(2,\epsilon)$ as above, $\testhH^{\iota}_z =  d^{-1} \testh_z^{\iota}.$
 %\begin{equation}
% \label{eqn:H-iota-2}
% \testhH^{\iota}_z =  d^{-1} \testh_z^{\iota}.
% \end{equation}
 \item[(ii)]
For any point $z\in \Omega_{\hh}$  of weight $k\ge 2$ with $k\equiv 1$ (mod $p-1$) and trivial nebentypus character at $p$, $\testhH^\iota_z = d^{1-k} \testh_z$.
%\begin{equation}
%\label{eqn:H-iota-k}
%\testhH^\iota_z = d^{1-k} \testh_z .
%\end{equation}
In particular, it is an overconvergent modular form of weight $2-k$.
\end{enumerate}
  If $\testhg$ is any $\Lambda$-adic modular form
 of level $N$ and character $\chi$, then the product $\testhg\times \testhH^\iota$ 
therefore  specialises, at all  points $(y,z)$ with common weights  $(k,\epsilon)$,
 to a  $p$-adic overconvergent modular form  of weight $2$
on $\Gamma_0(Np)$. Since ordinary overconvergent modular forms of weight $2$ are classical, it follows that
 $e_{\ord}(\testhg \times \testhH^\iota)$ belongs to the space $S_2(\Gamma_0(Np))\otimes \Lambda_{gh}$ of classical modular forms tensored with the 
 ring  $\Lambda_{gh} =   \Lambda_g \otimes_{\Lambda} \Lambda_h$. 
Property (ii) together with the interpolation property   
\eqref{eqn:garrett-hida-f} implies  that $\testf^*(\testhg    \testhH^{\iota})= \Lp^{f_\alpha}(\testf^*,\underline{\testg \testh})$,
since these two elements of $\Lambda_{f\underline{gh}}$ coincide on a dense set of points
in $\Omega_g\times_\Omega \Omega_h$.  In particular, $\testf^*(\testg_y  \times d^{-1} \testh_z^{\iota}) = \Lp^{f_\alpha}(\testf^*,\underline{\testg \testh})(y,z)$
and the theorem follows from \eqref{eqn:starting-point-now}.
\end{proof}



  
     
   \begin{theorem} 
\label{prop:bloch-kato-log-in-families-g}
 For all arithmetic points $x=(y,z)\in \Omega_g\times_\Omega \Omega_h$ of weight $(2,\epsilon)$,
  $$   \log_p(\kappa_p^g(\testf^*,\testg_y,\testh_z))  = \cE_{g_y}(f,g_y,h_z) \times   \Lp^{g_\alpha}(\testf,\underline{\testg^* \testh})(y,z).$$
 % where   $ \cE_{g_y}(f,g_y,h_z)$ is described in the statement of Theorem \ref{thm:AJ-p-Delta-g}.
  \end{theorem}
\begin{proof}
Combining the second statement in Corollary \ref{cor:from-log-to-AJ} with 
Theorem \ref{thm:AJ-p-Delta-g}, 
\begin{equation}
\label{eqn:victor1}
\log_p(\kappa_p^g(\testf,\testg_y^*,\testh_z)) = d_s^{-1} \times  \cE_{g_y}(f,g_y,h_z)   \times  
 {\testg_y^*}(\testF \times \testh_{z}). 
 \end{equation} 
 By definition,
\begin{equation}
\label{eqn:victor2}
  \testg_y^*(\testF \times \testh_z)  = \langle \eta_{\testg_y^*}, w_s (\testF \times  \testh_z) \rangle_s   
       = d_s \times  (y,z)(\langle \eta_{\testg_s^*}, w_s(\testF \times \testh_s)\rangle_{\Gamma_s}) 
       = d_s  \times (y,z)(\testhg^*(\testF\times \testhh)_s),
\end{equation}
   where
   the penultimate equality 
   follows from   Lemma \ref{Gammas}, 
   and   the expression  
   $\testhg^*(\testF\times \testhh)_s \in \Lambda_{f\underline{gh}}\otimes_{\Lambda} \Lambda_s$ 
  occuring  in the last term
   denotes the natural  projection ``to level $s$" of an element of $\Lambda_{f\underline{gh}}$.
The  interpolation property   \eqref{eqn:garrett-hida-f}  of the 
 $p$-adic $L$-function $\Lp^{g_\alpha}(\testf,\testhg^*, \testhh)$
    at points of weight $(2,\ell,\ell)$ and trivial nebentypus character shows that
    $$  \testhg^*(\testF \times \testhh) =  \Lp^{g_\alpha}(\testf,\testhg^*,\testhh), $$ and hence,
     since $ \Lp^{g_\alpha}(\testf,\underline{\testg^* \testh})$ is just the restriction of $\Lp^{g_\alpha}(\testf,\testhg^*, \testhh)$ to the one-dimensional rigid analytic space $\Omega_{f \underline{g h}}$,
     that
\begin{equation}
\label{eqn:victor3}
(y,z)(\testhg^*(\testF\times \testhh)_s)  =  \Lp^{g_\alpha}(\testf, \underline{\testg^* \testh})(y,z).
\end{equation}
The theorem follows by combining \eqref{eqn:victor1}, \eqref{eqn:victor2} and \eqref{eqn:victor3}.
\end{proof}
 
  


 
\section{The explicit reciprocity law}
\label{sec:erl}

\subsection{Perrin-Riou's $\Lambda$-adic regulator}\label{subsec:PR}



Let $\tLambda$ be a finite flat extension of $\Lambda$ and let $\Psi: G_{\Q_p} \lra \tLambda^\times$
be an unramified $\tLambda$-valued character. Let 
\begin{equation}\label{ukappa}
\ukappa^g \in H^1(\Q_p,\tLambda\{\Psi\}(1)),  \qquad \ukappa^f \in H^1(\Q_p,\tLambda\{\Psi\}(\uepsilon))
\end{equation}
be $\tLambda$-adic classes with values in the $\tLambda[G_{\Q_p}]$-modules $\tLambda\{\Psi\}(1) = \tLambda\{\Psi\}(\epsilon_{\cyc})$ and $\tLambda\{\Psi\}(\uepsilon))$, 
respectively. The notations used to 
designate these classes are motivated  by 
 \eqref{eqn:ohta-classes}, which asserts  that $\ukappa^g = \kappa^g_p(\testf,\underline{\testg^*\testh})$ and $\ukappa^f = \kappa^f_p(\testf^*,\underline{\testg\testh})$ yield particular instances of \eqref{ukappa}, after taking
$$
\Psi = \psi_{f}^{-1}\Psi_{\hg}   \Psi_{\hh}^{-1} \times \chi^{-1} \qquad \mbox{and} \quad \Psi = \psi_{f}    \Psi_{\hg}^{-1} \Psi_{\hh}^{-1}
$$
for $\ukappa^g$ and $\ukappa^f$ respectively. 

In general, the $\Lambda$-adic classes $\ukappa^g$ and $\ukappa^f$ gives rise to a collection of classical specialisations 
$$
\kappa_x^g \in H^1(\Q_p,   K_x\{\Psi_x\}(1)), \qquad \kappa^f_x \in H^1(\Q_p,   K_x\{\Psi_x\}(\epsilon_x)) \quad \mbox{where } \epsilon_x =  \epsilon_{\cyc}^{\ell-1} \omega^{1-\ell} \epsilon,
$$
as $x\in \tilde\Omega_{\cl}$ ranges over the set of classical points of $\tilde\Omega = \mathrm{Spf}(\tLambda)$ satisfying $\mathrm{w}(x)=\nu_{\ell,\epsilon}$ for some $\ell\geq 1$ and a Dirichlet character $\epsilon$ of conductor $p^s$, $s\geq 1$. 

%Let $t$ denote Fontaine's $p$-adic analogue of $2\pi i$ on which $G_{\Q_p}$ acts as multiplication by
%$\epsilon_{\cyc}$ and set $t_x :=   t^{1 - \ell} \in B_{\dR}$. 

One has
  canonical isomorphisms of Dieudonn\'e modules
 for any $x\in \tilde\Omega_{\cl}$,
\begin{align}
\label{id-B}
\D_{\dR}(K_x\{\Psi_x\}(1))  &:=  \D_{\dR}(K_x\{\Psi_x\}) \otimes \D_{\dR}(K_x(1))  = 
  \D_{\dR}(K_x(1)) \stackrel{\cdot t}{\simeq}  K_x \\ \nonumber
\D_{\dR}(K_x\{\Psi_x\}(\epsilon_x))  & :=   \D_{\dR}(K_x\{\Psi_x\}) \otimes \D_{\dR}(K_x(\epsilon_x)) 
=    \D_{\dR}(K_x(\epsilon_x)) \stackrel{\cdot t^{\ell-1}}{\simeq}     K_x,
\end{align}
where
\eqref{eqn:the-whole-point-of-introducing-this-notation-in-the-first-place} 
has been invoked
 in deriving  the penultimate equalities,
 and the last isomorphism is given by multiplication by the corresponding power of Fontaine's period $t\in B_{\dR}$, the $p$-adic analogue  of $2\pi i$  on which $G_{\Q_p}$ acts as multiplication by
$\epsilon_{\cyc}$. (Note that the field $K_x$ is assumed to contain $\zeta_s$, and hence,
that it contains the periods of the finite order character $\omega^{1-\ell} \epsilon$.) 

Assume that for any $x\in \tilde\Omega_{\cl}$ the character $\Psi_x$ is not trivial. Arguing as in \cite[Lemma 3.8]{BDR2}, it then follows that the Bloch-Kato logarithm and dual exponential map induce isomorphisms 
%$$
%H^1(\Q_p,   K_x\{\Psi_x\}(\epsilon_{\cyc}^j)) = H^1_{\exp}(\Q_p,   K_x\{\Psi_x\}(\epsilon_{\cyc}^j)) \quad \mbox{ for any }  x\in \tilde\Omega_{\cl}
%$$
%and 
%$$
%H^1(\Q_p, K_x\{\Psi_x\}(\epsilon_x)) = H^1_{\exp}(\Q_p, K_x\{\Psi_x\}(\epsilon_x)) \quad \mbox{ for any }  x\in \tilde\Omega_{\cl} \mbox{ such that } \ell \geq 2.
%$$ 
\begin{eqnarray}\label{log-exp}
\log_p: & H^1(\Q_p, K_x\{\Psi_x\}(1))  & \lra   K_x \quad \mbox{ for any }  x\in \tilde\Omega_{\cl}, \\ \nonumber
\log_p: & H^1(\Q_p, K_x\{\Psi_x\}(\epsilon_x))  & \lra  K_x  \quad \mbox{ for any }  x\in \tilde\Omega_{\cl} \mbox{ such that } \ell \geq 2, \\ \nonumber
 \exp^*_p: &  H^1(\Q_p, K_x\{\Psi_x\}(\epsilon_x)) & \lra K_x\quad \mbox{ for any }  x\in \tilde\Omega_{\cl} \mbox{ such that } \ell = 1.
\end{eqnarray}

Note that the logarithm maps quoted in \eqref{logs} are particular cases of the above. The two propositions below describe how  the images of the classes $\kappa_x^g$ and $\kappa_x^f$ under the maps in \eqref{log-exp} can be interpolated $p$-adically. 
The $p$-adic interpolation of the logarithms of the  classes $\kappa_x^g$ is  treated first in Proposition \ref{LZ-Lambda-expdual0} below, 
inspired by the elegant treatment of the $p$-adic interpolation of unramified periods given
in  \cite{LZ}. 
The corresponding interpolation problem for the classes $\kappa_x^f$, which is taken up  in
Proposition
\ref{LZ-Lambda-expdual},
 lies considerably deeper and
relies crucially on the ``explicit reciprocity law" of 
Perrin-Riou (as refined in   \cite{LZ}).
% which in turn builds on the ideas and methods of   Coleman, Yager and Perrin-Riou.

\begin{proposition}
\label{LZ-Lambda-expdual0}
There exists an element $\cL(\ukappa^g)\in \tLambda$ such that $\cL(\ukappa^g)(x) =  \log_p(\kappa_x^g)$ for all $x\in \tilde\Omega_{\cl}.$
 \end{proposition}
\begin{proof}
Let 
$$ G^{\ur} = \Gal(\Q_p^{\ur}/\Q_p) = \hat \Z$$
be the Galois group of the maximal unramified extension of $\Q_p$, which is canonically isomorphic to 
${\hat \Z}$ as a profinite group via the canonical topological generator given
by the  Frobenius automorphism $\sigma_p
\in G^{\ur}$.   
Let 
$\Z_p[[ G^{\ur}]]$ be  the completed group ring for  $G^{\ur}$, and 
write $\Z_p^\circ[[G^{\ur}]]$ for the same completed group ring viewed as a rank one module  over itself and
equipped with the ``tautological"  unramified action of 
$G_{\Q_p}$ in which  the Frobenius automorphism   acts  as  multiplication
 by the group-like element 
$\sigma_p$.
If $R$ is a  $p$-adic ring, and $\alpha\in R^\times$ is any invertible element of $R$, 
the 
 assignment
$\sigma_p \mapsto \alpha$ extends uniquely to  a continuous group
homomorphism $\Psi: G^{\ur} \lra R^\times$ since $R^\times$ is a profinite group,
and this homomorphism extends by $\Z_p$-linearity to a 
continuous ring homomorphism $$\Psi: \Z_p[[G^{\ur}]] \lra R. $$
Let  $R\{\Psi\}$ denote the free $R$-module of rank one equipped with an
action of $G_{\Q_p}$ in which $G_{\Q_p}$ acts on $R$ via the unramified character
$\Psi$. 
With these notations, the     representation $R(\Psi)(1)$ of $G_{\Q_p}$  can be written as 
$$ R(\Psi)(1)  = \Z_p^{\circ}[[G^{\ur}]](1)\hat\otimes_{\Psi} R.$$
 It follows, after invoking Shapiro's Lemma for the second equality,  that
$$
H^1(\Q_p,R(\Psi)(1)) =
 H^1(\Q_p,\Z_p^{\circ}[[ G^{\ur}]](1))  \hat\otimes_{\Psi} R = 
 (\invlim_n H^1(\Q_{p^n}, \Z_p(1))) \hat\otimes_{\Psi} R, 
$$
where the inverse limit is taken with respect the corestriction maps and
 $\Q_{p^n}$ denotes the unramified extension of $\Q_p$ of degree $n$ for each $n\geq 1$.
Hilbert's theorem 90 implies that 
$$
H^1(\Q_p,R(\Psi)(1)) =
   (\invlim_n (\Q_{p^n}^\times \otimes \Z_p))   \hat\otimes_{\Psi} R =  (\invlim_n (\cO_{n}^\times \otimes \Z_p))  \hat\otimes_{\Psi} R,$$
   where the inverse limit is taken with respect to the norm maps, and $\cO_n$ denotes the ring of integers of
   $\Q_{p^n}$. 
The standard $p$-adic logarithm $\log_p: \cO_{n}^\times  \ra \cO_{n}$ 
 therefore gives rise to a natural homomorphism
\begin{equation}
\label{cLL}
\log_p: H^1(\Q_p,R(\Psi)(1)) \lra (\invlim_n \cO_{n}) \hat\otimes_{\Psi} R,
\end{equation}
where the inverse limit is taken relative to the trace maps.
The elements of $\invlim_n \cO_{n}$ can be interpreted as $\hat\Z_p^{\ur}$-valued 
measures on $G^{\ur}$  by
associating to an element $ a= \{a_n\}$  of this inverse limit  the measure $\mu_a$  defined
by 
$$ \mu_a( \sigma  \cdot \Gal(\Q_p^{\ur}/\Q_{p^n})) := \sigma a_n, \qquad \forall n\ge 1, \quad \sigma\in \Gal(\Q_{p^n}/\Q_p).$$
The assignment
$$ 
a\otimes 1 \mapsto \int_{G^{\ur}} \Psi(t) d\mu_a(t)$$
gives rise to a natural   homomorphism
\begin{equation}
\label{eqn:measures}
(\invlim_n \cO_{n}) \hat\otimes_{\Psi} R \lra 
(R  \hat\otimes\hat\Z_p^{\ur})^{\sigma_p = \alpha^{-1}} = (R(\Psi)\hat\otimes\hat\Z_p^{\ur})^{G_{\Q_p}}.
\end{equation}
Combining \eqref{cLL} and \eqref{eqn:measures} 
allows one to
 parlay the logarithm map
into a canonical homomorphism
\begin{equation}
\label{cLL-bis-prelim}
 H^1(\Q_p,R(\Psi)(1)) \lra   (R(\Psi)\hat\otimes\hat\Z_p^{\ur})^{G_{\Q_p}}.
\end{equation}
Equivalently, after 
 invoking
\eqref{eqn:the-whole-point-of-introducing-this-notation-in-the-first-place}, 
one obtains a canonical logarithm map
\begin{equation}
\label{cLL-bis}
\cL: H^1(\Q_p,R\{\Psi\}(1)) \lra   R.
\end{equation}
The homomorphism $\cL$  is functorial in the sense that 
for all $G^{\ur}$-equivariant homomorphisms $\varphi: R\{\Psi\} \lra R'\{\Psi'\}$,
the diagram
$$ \xymatrix{
H^1(\Q_p,R\{\Psi\}(1)) \ar[r]^-{\cL}  \ar[d]_{\varphi} &   R \ar[d]_{\varphi}  \\
H^1(\Q_p, R'\{\Psi'\}(1)) \ar[r]^-{\cL}  &   R' }
$$ 
commutes.
In particular, after setting $R:= \tLambda\{\Psi\}$ and letting $\varphi:\tLambda\{\Psi\} \lra K_x\{\Psi_x\}$
be the homomorphism induced by the evaluation map at $x$,  
 the element $\cL(\ukappa^g)\in \tLambda$ satisfies the interpolation property 
claimed in Proposition  \ref{LZ-Lambda-expdual0}, in light of the fact
 that the map $\cL: H^1(\Q_p,K_x\{\Psi_x\}(1)) \lra K_x$ 
agrees with the  $p$-adic logarithm map $\log_p$.
\end{proof}
In order to treat the analogous question for the $\Lambda$-adic cohomology class $\ukappa^f$, define 
\begin{equation}
\label{eqn:def-period-Jy}
\mho_x := (-1)^{1 - \ell} \cdot \Gamma^*(2-\ell) \cdot \gauss(\epsilon^{-1}\omega^{\ell-1}) \cdot  \Psi_x(\mathrm{Frob}_p)^{-s}   \cdot \mathcal{E}_x \cdot  
 p^{s(2-\ell)}  \in K_x,
\end{equation}
where 
$$
\Gamma^*(2-\ell) = \frac{1}{(\ell-2)!} \, \mbox{ if } \,  \ell\ge 2; \qquad \Gamma^*(2-\ell) = (1-\ell)! \,  \mbox{ if } \ell \le 1,
$$
and 
$$
\mathcal{E}_x = (1-p^{1-\ell} \Psi_x(\mathrm{Frob}_p)^{-1})(1-p^{\ell-2} \Psi_x(\mathrm{Frob}_p))^{-1}\, \mbox{ if } \, \epsilon = \omega^{\ell-1}; \qquad \mathcal{E}_x = 1 \mbox{ otherwise.}
$$
%Note that $\mathcal{E}_x \ne 0$ when $\ell=1$ and $\Psi_x(\mathrm{Frob}_p)\ne 1$. 


\begin{proposition}
\label{LZ-Lambda-expdual}
There exists an element $\cL(\ukappa^f)\in \tLambda$ such that for all  $x\in \tilde\Omega_{\cl}$ with $\mathrm{w}(x)=\nu_{\ell,\epsilon}$,
$$ 
\cL(\ukappa^f)(x) =  \left\{ \begin{array}{ll}
\mho_x   \cdot \log_p(\kappa^f_x), & \mbox{ if } \ell \ge 2, \\
\mho_x   \cdot \exp_p^*(\kappa^f_x), & \mbox{ if } \ell = 1.
 \end{array}\right.
$$
 \end{proposition}

\begin{proof} Let $K_\infty$ denote the abelian extension of $\Q_p$ through which the 
$\tLambda$-adic character $\Psi\uepsilon$ factors, let 
$G=\Gal(K_\infty/\Q_p)$, and let 
$\Lambda_{G} := \Z_p[[G]]$ be the completed group ring attached to $G$,
equipped with its tautological action of $G_{\Q_p}$.  The decomposition
$K_\infty = K_\infty^{\rm ur} \Q_p(\mu_{p^\infty})$,
where $K_\infty^{\ur}$ is the maximal unramified subfield of $K_\infty$,   
determines a canonical decomposition  $G  = U\times \Z_p^\times$. 
The unramified  character $\Psi:U \lra \tLambda^\times$ extends by linearity to
a natural $G_{\Q_p}$-equivariant projection 
\begin{equation}\label{thetapsi}
\theta_{\Psi}: \Lambda_{G} \lra \tLambda\{\Psi\}(\uepsilon).
\end{equation}
The    $p$-adic regulator 
 of \cite{LZ}, specialised to the  case where $V$ is
 the trivial one-dimensional representation of $G_{\Q_p}$, yields 
  a ``two-variable regulator map"
$$\cL^G: H^1(\Q_p, \Lambda_G) \lra \Lambda_G \,\hat\otimes \, \hat\Z_p^{\ur}.$$

%Tensoring $\cL^G$  with $\tLambda$ via the map $\theta_{\Psi}$ gives rise to 
%the ``one variable regulator map" attached to $\Psi$:
%$$ \cL^{\Psi}: H^1(\Q_p,\tLambda(\Psi \uepsilon)) \lra \tLambda \,\hat\otimes \,\hat\Z_p^{\ur}.$$

Let $\tilde\theta_\Psi: H^1(\Q_p,\Lambda_{G}) \lra H^1(\Q_p,\tLambda\{\Psi\}(\uepsilon))$
 denote the map induced by \eqref{thetapsi} in cohomology. 
 Note that $\tilde\theta_\Psi$ is surjective because the cohomological dimension 
 of $\Q_p$ is $1$.
Let $\tilde\ukappa^f\in H^1(\Q_p,\Lambda_{G}) $ be any lift of 
the class $\ukappa^f$ under $\tilde\theta_\Psi$,  and set 
\begin{equation}
\label{eqn:def-L-ukappa}
 \cL(\ukappa^f) :=      
% \cL^{\Psi}(\ukappa)  = 
  \theta_{\Psi}(\cL^G(\tilde\ukappa^f)) \in \tLambda\{\Psi\}  \,\hat\otimes \,\hat\Z_p^{\ur}.
 \end{equation}
It is not hard to see (and follows, for instance, from Prop.~4.9. of \cite{LZ}) that
$\cL(\ukappa^f)$ is fixed under the action of $\Frob_p$ and hence $\cL(\ukappa^f)$ belongs to $ \hD(\tilde\Lambda\{\Psi\})$, which is canonically isomorphic to $\tLambda$ by
\eqref{eqn:the-whole-point-of-introducing-this-notation-in-the-first-place}. 
Now \cite[Theorem 4.15]{LZ}, with $j=1-\ell$ and $\Phi^n = \Psi_x(\mathrm{Frob}_p)^{-s}$ applied 
to the character $\omega= \epsilon_{\cyc}^{\ell-1}$ in the notation of loc.cit.~implies that the element $\cL(\ukappa^f)$ 
has the required  interpolation 
property.
 \end{proof}


\subsection{The triple product $p$-adic L-functions via $\Lambda$-adic cohomology classes}
\label{sec:Lp}

We are now in position to prove the following result, which  lends support to Perrin-Riou's vision according to which $p$-adic $L$-functions 
 ought to arise as the images of $p$-adic families of distinguished global elements
 (referred to loosely as the Iwasawa theoretic incarnations of  ``Euler systems") 
 under suitable $p$-adic regulator maps. 
Realising a $p$-adic $L$-function in this way has strong arithmetic
 consequences, some of which shall be explored in the remainder of this article.

 \begin{theorem}
\label{pL-PR} 
The following equalities hold:
\begin{align*}
\Lp^{f_\alpha}(\testf^*,\underline{\testg \testh}) &=
 \alpha_f/2 \cdot (1-\chi^{-1}(p) \alpha_f^{-1} a_p(\hg) a_p(\hh)^{-1})\times  \cL(\kappa^f_p(\testf^*,\underline{\testg\testh})), \\
\Lp^{g_\alpha}(\testf,\underline{\testg^*\testh}) & = 1/2 \cdot  (1-\chi(p) p^{-1} \alpha_f a_p^{-1}(\hg) a_p(\hh)) \times \cL(\kappa^g_p(\testf,\underline{\testg^*\testh})).
\end{align*}
\end{theorem}
\begin{proof}
By specialising Propositions \ref{LZ-Lambda-expdual0}  and  \ref{LZ-Lambda-expdual} 
 to  $\ukappa^g := \kappa^g_p(\testf,\underline{\testg^*\testh})$ and  
 $\ukappa^f := \kappa^f_p(\testf^*,\underline{\testg\testh})$ respectively, 
 one obtains from \eqref{eqn:specialised-ohta-classes} that
\begin{align*}
\log_p(\kappa_p^f(\testf^*,\testg_y,\testh_z))  &= 
 \alpha_f^s a_p(g_y)^{-s} a_p(h_z)^{-s} \times 
\cL(\kappa^f_p(\testf^*,\underline{\testg\testh}))(y,z), \\
\log_p(\kappa_p^g(\testf,\testg^*_y,\testh_z))   & = \cL(\kappa^f_p(\testf^*,\underline{\testg\testh}))(y,z),
\end{align*}
at  all classical points $x=(y,z) \in \Omega_g\times_\Omega \Omega_h$ 
of weight-character $(2,\epsilon)$ with $\epsilon\omega^{-1}$ of
 conductor $p^s$ with $s\ge 1$.
Comparing this identity with Theorems \ref{prop:bloch-kato-log-in-families}  and \ref{prop:bloch-kato-log-in-families-g} respectively
shows that 
\begin{align*}
\Lp^{f_\alpha}(\testf^*,\testhg, \testhh)(y,z) & =  \alpha_f/2\cdot (1-\chi^{-1}(p) \alpha_f^{-1} \alpha_{g_y} \alpha_{h_z}^{-1}) \times \cL(\kappa^f_p(\testf^*,\underline{\testg\testh}))(y,z), \\
\Lp^{g_\alpha}(\testf,\testhg^*, \testhh)(y,z) & = 
1/2 \cdot  (1-\chi(p) p^{-1} \alpha_f \alpha^{-1}_{g_y} \alpha_{h_z}) \times 
\cL(\kappa^g_p(\testf,\underline{\testg^*\testh}))(y,z),
\end{align*}
for infinitely many points $(y,z)$ of weight-character of the form $(2,\epsilon)$.
But 
  a non-zero element of $\Lambda_{f \underline{g h}}$ can only vanish at
  finitely many points of $\Omega_g\times_\Omega \Omega_h$. 
  The theorem follows, since $a_p(g_y)=\alpha_{g_y}$ and $a_p(h_z)=\alpha_{h_z}$.
\end{proof}





\section{Application to the Birch and Swinnerton-Dyer conjecture}
\label{sec:BSD}

\def\fp{{\mathfrak{p}}}
As in the introduction, let $$f\in S_2(N_f), \qquad g\in S_1(N_g,\chi), \qquad h\in S_1(N_h,\chi^{-1})$$ be three newforms and
assume that  the level $N_f$ of $f$ 
is relatively prime to $N_gN_h$.
Assume also  that the weight two modular 
form $f$ has rational fourier coefficients, and hence is associated to an
elliptic curve $E$. (This last assumption is only made for notational simplicity
and the arguments of this chapter would  extend to the setting where $E$ is replaced
by any simple abelian variety quotient of $J_0(N)$, at the cost of slight technical complications.)
The  goal of this chapter is to prove 
 Theorems A,  B, C and D
  and their corollaries stated in the introduction,
  concerning the arithmetic of the twist of $E$ by
 the 
four-dimensional self-dual Artin representation  attached  to
$$  \varrho_{gh}:= \varrho_g\otimes\varrho_h: G_\Q\lra \GL_4(L),$$
where $\varrho_g$ and $\varrho_h$ 
are the odd, irreducible, two-dimensional Artin representations 
attached to $g$ and $h$. 
 
 
\subsection{Bounding Mordell-Weil groups}
For this section, let $W$ be a general $d$-dimensional 
 self-dual Artin representation, with coefficients in a finite
extension $L$ of $\Q$, and factoring through the Galois group of a finite
extension $H$ of $\Q$.
Our goal is to present general results, based on local Tate duality and the 
global Poitou-Tate exact sequence, for bounding
the   $W$-isotypic part of $E(H)$,
 defined to be 
 the $L$-vector space
$$ E(H)_L^W := \Hom_{\Gal(H/\Q)}(W, E(H)\otimes L).$$
 

As in the introduction, fix a  rational prime $p$ and an embedding $L\subset L_p$ of $L$ into
 a finite extension of $\Q_p$.
Associated  to $E$ and $W$  are
 the continuous $p$-adic representations 
$$ V_p(E) := H^1_{\et}(E_{\bar\Q},\Q_p)(1) \otimes_{\Q_p} L_p, \qquad W_p := W\otimes_L L_p, 
\qquad V_p(E) \otimes_{L_p} W_p$$
of $G_\Q$,
which are $L_p$-vector spaces of dimensions $2$, $d$ and $2d$ 
respectively. 
 Restriction to the absolute Galois group
$G_{H}$ induces an isomorphism
\begin{eqnarray}
\label{eqn:coho-induced}
 H^1(\Q, V_p(E)\otimes W_p) &\simeq&  (H^1(H, V_p(E))\otimes W_p)^{\Gal(H/\Q)} \\
 \nonumber
  &=&  \Hom_{\Gal(H/\Q)}(W_{p}, H^1(H,V_p(E))),
  \end{eqnarray}
  where the self-duality of $W_p$ is used to obtain the second equality. 
  Thanks to this identification, the  connecting homomorphism $$\delta: E(H) \otimes L   \lra H^1(H,V_p(E))$$ 
of Kummer theory gives rise to a homomorphism
  \begin{equation}
  \label{eqn:def-delta-global}
\delta: E(H)_{L}^{W_p} \lra
    H^1(\Q,V_p(E) \otimes W_p). 
\end{equation}
    For each rational prime $\ell$, the 
    maps \eqref{eqn:coho-induced} and \eqref{eqn:def-delta-global}
    admit local counterparts
    \begin{eqnarray*}
    H^1(\Q_\ell, V_p(E)\otimes W_p) &\simeq&    \Hom_{\Gal(H/\Q)}(W_{p}, \oplus_{\lambda|\ell} H^1(H_\lambda,V_p(E))), \\
    \delta_\ell: \left(\oplus_{\lambda|\ell} E(H_\lambda)\right)_{L_p}^{W_p}  &\lra&
    H^1(\Q_\ell,V_p(E) \otimes W_p),
    \end{eqnarray*}
   for which the following diagram commutes:
\begin{equation}
   \label{eqn:comm-diag-resp}
   \xymatrix{  
    E(H)_{L}^{W} \ar[r]^\delta \ar[d]^{\res_\ell}  & H^1(\Q, V_p(E)\otimes W_p)  \ar[d]^{\res_\ell} \\
    \left(\oplus_{\lambda|\ell} E(H_\lambda)\right)_{L_p}^{W_p} \ar[r]^{\delta_\ell} &
    H^1(\Q_\ell, V_p(E)\otimes W_p).  }
\end{equation}
The Bloch-Kato submodule $H^1_{\fin}(\Q_\ell,V_p(E)\otimes W_p)$ of the local cohomology group 
$H^1(\Q_\ell, V_p(E)\otimes W_p)$ coincides with the image of the local connecting homomorphism $\delta_\ell$, 
and the 
{\em singular quotient} is defined to be
$$H^1_{\sing}(\Q_\ell,V_p(E) \otimes W_p) := \frac{H^1(\Q_\ell,V_p(E) \otimes W_p)} {H^1_{\fin}(\Q_\ell,V_p(E) \otimes W_p)}. $$
When $\ell=p$ is a prime of good reduction for $E$ at which $W$ is unramified, 
the subspace 
$ H^1_{\fin}(\Q_p,V_p(E)\otimes W_p)$ 
consists of classes of cristalline extensions of Galois representations. For each rational prime $\ell$, let
$$ \res_\ell: H^1(\Q, V_p(E) \otimes W_p) \lra H^1(\Q_\ell, V_p(E)\otimes W_p)$$ 
denote the restriction map
from the global to the local cohomology at $\ell$.
The composition
$$ \partial_\ell: H^1(\Q, V_p(E)\otimes W_p) \lra H^1_{\sing}(\Q_\ell,V_p(E)\otimes W_p)$$
 of $\res_\ell$ with the natural projection to the singular quotient is called the {\em residue map at $\ell$}.
 
The Selmer group $\Sel_p(E, W)$ is defined in terms of these maps as 
\begin{equation}\label{defSelmer}
\Sel_p(E,W) = H^1_{\fin}(\Q,V_p(E)\otimes W_p) := \bigcap_\ell \ker\big(H^1(\Q,V_p(E)\otimes W_p) \stackrel{\partial_\ell}{\lra} H^1_{\sing}(\Q_\ell,V_p(E)\otimes W_p)\big).
\end{equation}



 
 


\begin{lemma}
The local cohomology group $H^1(\Q_p, V_p(E) \otimes W_p)$ is a $2d$-dimensional
$L_p$-vector space. 
The finite subspace 
$ H^1_{\fin}(\Q_p,V_p(E)\otimes W_p)$
 and the 
singular quotient
$H^1_{\sing}(\Q_p,V_p(E) \otimes W_p)$
are each 
 $d$-dimensional and in perfect duality under the local Tate pairing.
 \end{lemma}
 \begin{proof}
 This follows by  choosing a $G_{\Q_p}$-stable lattice
 in $V_p(E) \otimes W_p$,  applying (for instance)
   Theorem 2.17 of \cite{ddt} to its finite 
   $G_{\Q_p}$-stable quotients, and passing to the limit.
 \end{proof}



 
 \begin{proposition}
 \label{prop:es-argument}
 If the map $\partial_p$ is surjective, then the map 
 \begin{equation}
 \label{eqn:resp-h1-fin}
 \res_p: \Sel_p(E, W) \lra H^1_{\fin}(\Q_p,V_p(E)\otimes W_p)
 \end{equation}
 is the zero map.
 \end{proposition}
\begin{proof}
The standard calculations arising from the global Poitou-Tate exact sequence
imply that the image of the restriction map
$$ \res_p: \Sel_p(E, W) \lra H^1(\Q_p, V_p(E) \otimes W_p)$$ 
is $d$-dimensional. 
(This can be shown, for example, by choosing a $G_\Q$-stable
 lattice in $V_{p}(E)\otimes W_p$, applying 
Theorem 2.18 of \cite{ddt} to its finite quotients, and taking the inverse limit.)
It follows that the restriction of $\res_p$ to $H^1_{\fin}(\Q_p, V_p(E)\otimes W_p)$ is the zero map.
\end{proof}
\begin{proposition}
\label{prop:bounding-mw}
Assume that  the residue  map $\partial_p$ attached to the representation $V_p(E)\otimes W_p^\sigma$ is a
surjective map of $L_p$-vector spaces, for all $\sigma\in\Gal(L/\Q)$. 
Then $E(H)_L^W=0$. 
\end{proposition}
\begin{proof}
   By Proposition \ref{prop:es-argument},  the map $\res_p$ of \eqref{eqn:resp-h1-fin} is the zero map, and therefore
   the commutativity of the diagram 
   \eqref{eqn:comm-diag-resp}
implies that the 
   natural map
   $$ \res_p:E(H)_L^{W^\sigma} \lra ((\oplus_{\fp|p} E(H_{\fp}))\otimes L_p)^{W^\sigma}$$ 
   is the zero map, for each $\sigma\in \Gal(L/\Q)$.
   Since the local connecting homomorphism $\delta_\ell$ is injective, this implies that the  vector space 
   $$ \oplus_{\sigma\in \Gal(L/\Q)} E(H)_L^{W^\sigma} = E(H)_L^{\tilde W}, \qquad \mbox{ where }
   \tilde W = \oplus_{\sigma\in\Gal(L/\Q)} W^\sigma,$$
   has trivial image in the group $\oplus_{\fp|p} E(H_{\fp}) \otimes L$ of local points.
   But $\tilde W$ is a rational representation and hence
    admits an $L$-basis consisting of elements of $E(H)$. 
    Since the natural map $E(H) \ra \oplus_{\fp|p} E(H_{\fp})$ is injective modulo torsion,
   it follows that 
   $\dim_{\Q} E(H)_{\Q}^{\tilde W}= 0$, and therefore the same is true of $\dim_L E(H)_L^W$.
  The proposition follows.
  \end{proof}
  
  \subsection{Proof of Theorem C}
\label{sec:reclaw1}



The most interesting application of Theorem \ref{pL-PR} arises without a doubt when 
$\ell=1$. A simplified version of the following statement 
was stated in  Theorem C of the introduction. As in loc.\,cit.\,let $g\in S_1(N_g,\chi)$ and $h\in S_1(N_h,\chi^{-1})$ be classical newforms of weight $1$ and let $g_\alpha\in S_1(N_g p,\chi)$ and $h_\alpha\in S_1(N_h p,\chi^{-1})$ be ordinary $p$-stabilisations. 

Let $\hg$ and $\hh$ be Hida families specialising to $g_y=g_\alpha$ and $h_z=h_\alpha$ at suitable weight one
points $y$ and $z$ of the parameter spaces $\Omega_{\hg}$ and $\Omega_{\hh}$ respectively, and
fix test vectors $\testf^*$, $\htg$ and $\hth$ as in \eqref{eqn:test-vectors2} and \eqref{eqn:dual-test-vectors2}. Let $$\kappa(f,g_\alpha,h_\alpha) \in H^1(\Q,V_{fgh}(N))$$ be the global cohomology class introduced in \eqref{eqn:classes-wt1}.

\begin{theorem}
\label{main2}
The central critical value $L(f,g,h,1)$ is non-zero if and only if
 the global cohomology class 
  $\kappa(f,g_\alpha,h_\alpha)$ is {\em not} crystalline at $p$.
\end{theorem}
\begin{proof} 
Since the triplet $(2,1, 1)$ is unbalanced, 
with $2$ as its dominant weight, the point $(y,z)$ lies in
 the region of interpolation defining the $p$-adic L-function $\Lp^{f_\alpha}(\testf^*,\underline{\testg \testh})$. 
By \cite[Theorems 4.2 and 4.7]{DR1}, $L(f, g, h,1) \ne 0$ if and only if there exists a choice $(\testf^*, \testg_\alpha, \testh_\alpha)$ 
  of test vectors  such that $\Lp^{f_\alpha}(\testf^*,\underline{\testg \testh})(y,z)\ne 0$. 
  By Theorem \ref{pL-PR}, this in turn is equivalent to the non-vanishing of the value of $\cL(\kappa_p^f(\testf^*,\underline{\testg \testh}))$ at the point $(y,z)$, which Proposition \ref{LZ-Lambda-expdual} recasts as a non-zero multiple of 
  $\exp_p^*(\kappa_p^f(\testf^*,\testg_\alpha,\testh_\alpha))$. 
  Because the dual exponential map appearing in \eqref{log-exp} is an isomorphism,  one concludes that
  $L(f,g,h,1)\ne 0$ if and only if there is a triple $(\testf^*,\testg_\alpha,\testh_\alpha)$ of test vectors
  for which the local class $\kappa_p^f(\testf^*,\testg_\alpha,\testh_\alpha)$ is non-trivial.
  
  As $G_{\Q_p}$-modules, $V_{f}(Np)^-\otimes V_{g h}^{\beta\beta}(N)$ is isomorphic to the direct sum of a finite number of  copies of $K_x\{ \psi_{f}    \Psi_{g_\alpha}^{-1} \Psi_{h_\alpha}^{-1} \}$, and the local class $\kappa_p^f(f,g_\alpha,h_\alpha)$ is determined by the collection of  its projections to the cohomology groups $H^1(\Q_p, K_x\{ \psi_{f}    \Psi_{g_\alpha}^{-1} \Psi_{h_\alpha}^{-1} \})$ indexed by the  set of all possible triples $(\testf^*, \testg_\alpha, \testh_\alpha)$  of test vectors. The non-vanishing
  of $\kappa_p^f(f, g_\alpha,h_\alpha)$ is therefore equivalent to the non-vanishing
  of  the local class $\kappa_p^f(\testf^*,\testg_y,\testh_z)\in H^1(\Q_p, K_x\{ \psi_{f}    \Psi_{g_\alpha}^{-1} \Psi_{h_\alpha}^{-1} \})$ introduced in \eqref{eqn:specialised-ohta-classes} for some such triple. 
  Theorem \ref{main2}  now follows from  Proposition \ref{prop:later}.
\end{proof}
 


  
\subsection{Proof of Theorem A and its corollaries}
We now specialise the results of the previous section 
to the setting where $W=V_{gh}$ in order to prove Theorem A of the introduction and its corollaries. Henceforth, the Artin representation $V_{gh}$ is
viewed as having coefficients in a finite extension $L$ of $\Q$, just as the 
representation $W$ of the previous section. In the notation of \S \ref{sec:lambda-adic-class-hida} we thus have $L_p=K_{f g h}$.


\begin{proposition}
\label{cor:main-prelim}
Assume $L(f,g,h,1)\ne 0$. 
For all pairs of eigenvalues 
$$\lambda  \in \{ (\alpha_g, \alpha_h), (\alpha_g, \beta_h), (\beta_g, \alpha_h), (\beta_g, \beta_h)\},$$  
there exists a global class $\kappa_\lambda\in H^1(\Q,V_{fgh})$ 
whose natural  image in $H^1_{\sing}(\Q_p, V_{fgh})$
is non-zero and belongs to  $H^1_{\sing}(\Q_p, V_f \otimes V_{gh}^{\lambda})$.
\end{proposition}
\begin{proof} Up to re-ordering the eigenvalues
 of $\mathrm{Frob}_p$ acting on $V_g$ and $V_h$, it may be
 assumed without loss of generality that $\lambda=(\alpha_g, \alpha_h)$. 
  By Theorem \ref{main2}, the associated
 cohomology class $  \kappa(f,g_\alpha,h_\alpha)$ is non-cristalline, and therefore there exists 
 a Galois-equivariant surjection $j: V_{fgh}(N) \lra V_{fgh}$ for which 
 $\kappa_\lambda := j(\kappa(f,g_\alpha,h_\alpha))$ is also  non-cristalline.
Since the natural  image of $\kappa_\lambda$  in $H^1_{\sing}(\Q_p, V_{fgh})$
 belongs to  $H^1_{\sing}(\Q_p, V_f \otimes V_{gh}^{\lambda})$
 by  Proposition \ref{prop:later},   the 
 proposition follows.
\end{proof}

 
\begin{theorem}
\label{thm:main-prelim-surj}
If $L(E,V_{gh},1)\ne 0$, there
 exists a rational prime $p$ for which the natural map
$$\partial_p: H^1(\Q, V_f \otimes V_{gh}) \lra H^1_{\sing}(\Q_p, V_f\otimes V_{gh})$$
is surjective.
\end{theorem}
\begin{proof}
 The  Chebotarev
 density theorem  shows the existence of a prime $p\geq 5$
whose associated  Frobenius element acts on both $V_g$ and $V_h$ with distinct eigenvalues,
and such that $a_p(f)\ne 0$ (so that $f$ is ordinary at $p$).
 By Proposition \ref{cor:main-prelim},
the non-vanishing of $L(E,V_{gh},1)$ implies the existence of global classes 
\begin{equation}
\label{eqn:classes-bis}
\kappa_{\alpha_g,\alpha_h}, \quad
\kappa_{\alpha_g,\beta_h}, \quad
\kappa_{\beta_g,\alpha_h}, \quad
\kappa_{\beta_g,\beta_h} \in H^1(\Q,V_{f g h}) 
\end{equation}
whose image under $\partial_p$ is non-zero in $H^1_{\sing}(\Q_p,V_{fgh})$. Since $\partial_p(\kappa_\lambda)$ belongs to $H^1_{\sing}(\Q_p,V_{f}\otimes V^\lambda_{gh})$ for each pair $\lambda  \in \{ (\alpha_g, \alpha_h), (\alpha_g, \beta_h), (\beta_g, \alpha_h), (\beta_g, \beta_h)\}$, the four classes are linearly independent and generate this singular quotient. 
  \end{proof}

We can  now prove Theorem A of the introduction:
\begin{theorem}
\label{theoremA}
Assume that $\varrho_{gh}$ is regular.
If $L(E,\varrho_{gh},1)\ne 0$, then  $E(H)_L^{\varrho_{gh}}=0$.
\end{theorem}
\begin{proof}
 For all $\sigma\in\Gal(L/\Q)$, the non-vanishing of $L(E,\varrho_{gh},1)$ at the central point
 implies that the same is true for $L(E,V_{gh}^\sigma,1)$, for all
 $\sigma\in\Gal(L/\Q)$. Furthermore, there exists a prime $p$ for which $V_{gh}^\sigma$ is
   regular, for all such $\sigma$. 
 Theorem \ref{thm:main-prelim-surj} implies that  the map
 $$ \partial_p: H^1(\Q,V_p(E) \otimes V_{gh}^\sigma) 
 \lra H^1_{\sing}(\Q_p,V_p(E)\otimes V_{gh}^\sigma)$$
 is surjective for 
 each
 $\sigma\in \Gal(L/\Q)$.
 The theorem now follows from Proposition \ref{prop:bounding-mw}.
\end{proof}


Turning  to  Corollary A1 of the introduction, 
let  $K$ be a quadratic field of discriminant prime to the conductor  $N_E$ 
of $E$.
If $\psi$ is a finite order ray class character of $K$, let $\psi'$ denote the character obtained from it by composing with the automorphism of $K/\Q$. A ring class 
character is a character of $K$ with trivial central character,
which therefore  satisfies $\psi' = \psi^{-1}$.
Let $\Pi_{K}$ denote the set of all ring class characters 
 of conductor
prime to $N_E$.
When $K$ is a real quadratic field, the set $\Pi_K$ can be partitioned
into two subsets $\Pi_{K}^+$ and $\Pi_{K}^-$, consisting of totally even and
totally odd ring class characters, respectively.
Let $\Pi_{E/K}$, $\Pi_{E/K}^+$ and $\Pi_{E/K}^-$ denote the subsets of
$\Pi_K$, $\Pi_K^+$ and $\Pi_K^-$ respectively consisting of the characters for which
$L(E/K,\psi,1)\ne 0$.

 To the pair $(E,K)$ one can associate the sign
  ${\rm sgn}(E,K)\in \{\pm 1\}$ of the functional equation
 for the $L$-function  $L(E/K,s)$. 
 It turns out that this is the very same sign that occurs
  in the functional equation of $L(E/K,\psi,s)$ for any
  $\psi\in \Pi_K$. In particular, $\Pi_{E/K}$ is empty 
   when ${\rm sgn}(E,K)=-1$.
   
   When ${\rm sgn}(E,K)=1$ and $K$ is imaginary quadratic,
  a non-vanishing theorem of 
    Cornut and Vatsal shows that $\Pi_{E/K}$ is infinite. 
    If $K$ is real quadratic, 
      a simple argument involving congruences for $L$-values
   shows that  $\Pi_{E/K}^+$ is either empty or infinite, and likewise
   for $\Pi_{E/K}^-$. The scenario where $\Pi_{E/K}^\pm=\emptyset$ is highly
   unlikely, since it would provide us with a systematic supply of 
   ring class characters for which the (primitive) $L$-function 
   $L(E/K,\psi,s)$ admits at least a double zero   at the center.
 \begin{definition}
 \label{def:non-vanishing} 
 The  pair $(E,K)$  is said to satisfy the {\em non-vanishing hypothesis } 
  if $\Pi_{E/K}^+$ and $\Pi_{E/K}^-$ are both non-empty (and hence, infinite).
 \end{definition}
 When $K$ is real, a proof of this non-vanishing hypothesis does not seem out of reach of
 current techniques in analytic number theory, although the unavailability of
 an ``anti-cyclotomic $\Z_p$-extension" of $K$ prevents a straightforward application of the 
 methods of Cornut and Vatsal.
 \begin{lemma}
 \label{lemma:psi0}
 For all $\psi\in \Pi_K$, there exists a ray class character 
 $\psi_0$ of $K$ of conductor prime to   $N_E$  such that $\psi = \psi_0/\psi'_0$.
 \end{lemma}
 \begin{proof}
Let $M/K$ be the cyclic 
 extension of $K$ (of degree $n$, say) which is cut out by the character
 $\psi$. This field is Galois over $\Q$, and $\Gal(M/\Q)$ is isomorphic to the dihedral group
 $D_n$ of order $2n$, furnished  with a natural embedding  into the semi-direct
 product  $(\C^\times\rtimes\Z/2\Z)$ via the character $\psi$.
  Embed $D_n$ into  $\PGL_2(\C)$
  by sending   $\lambda\in \C^\times$ to the class of any diagonal matrix of the form $\smallmat {\lambda_1}00{\lambda_2}$, where $\lambda_1/\lambda_2 = \lambda$ modulo the center, 
  and sending some  reflection in
  $D_n$ to the matrix $\smallmat 0110$. 
 This yields
 a projective representation
 $\bar r: G_\Q\ra \PGL_2(\C)$.
  By a classical result of Tate on the vanishing of $H^2(\Q,\Q/\Z(1))$, the projective representation
   $\bar r$
  may be lifted to a linear representation $r: G_\Q \ra \GL_2(\C)$.
  By construction, the representation $r$ has dihedral projective image, and hence
  is induced  from some character $\psi_0$ of $K$. The restriction of $r$ to $G_K$ is then 
  equal to the direct
  sum $\psi_0 \oplus \psi_0'$.
   It  follows that the restriction of  $\bar r$ to $G_K$ is identified with  $\psi_0/\psi_0'$,
   and therefore that  $ \psi_0/\psi_0' = \psi$. 
To  show that $\psi_0$ can be chosen to be unramified at the primes of $K$
   dividing  $N_E$, observe that for each prime $\ell|N_E$, the linear representation
   $r$ maps the inertia group at $\ell$ to the subgroup $\C^\times$ of scalar matrices, since ${\bar r}$ is unramified at $\ell$. Now choose a Dirichlet character $\chi:G_\Q\lra \C^\times$ whose restriction to 
   $I_\ell$ agrees with the restriction of $r$, for all $\ell|N_E$, and replace $r$ by $r\otimes \chi^{-1}$. 
   This substitution does not affect the projective representation attached to $r$, but leads to a representation $r$ which is unramified at the primes dividing $N_E$; hence the same is true for
   the resulting character $\psi_0$. 
   \end{proof}
 Note  that, if $K$ is a real quadratic field,
 the character $\psi_0$  is of mixed signature
  (resp.~totally even or odd) 
  when $\psi$ is totally odd (resp.~totally even).
 
 
 By considering the case where  $\alpha$ is further restricted to be a    quadratic character of $K$, it is clear that  the non-vanishing  hypothesis    is satisfied if there are two    quadratic Dirichlet characters $\chi$ (of the two possible signatures) for which 
 \begin{equation}
 \label{eqn:simultaneous}
  L(E,\chi,1) \ne 0, \quad \mbox{ and } \quad L(E,\chi\chi_K,1)\ne 0,
  \end{equation}
 where $\chi_K$ is the (fixed) Dirichlet character attached to $K$.
 Such simultaneous non-vanishing results for pairs of quadratic Dirichlet
 characters with fixed product
  seem to lie just beyond  the reach of  currently available techniques.
 However, a character  $\chi$ satisfying \eqref{eqn:simultaneous}
  is widely expected to exist, and a quick inspection is usually enough
 to produce such a $\chi$, for any given $E$ and $K$. For example, if $E$ is the elliptic curve of smallest conductor $11$ and $K=\Q(\sqrt{5})$ is the real quadratic
  field of smallest discriminant $5$, a cursory inspection of the tables of Cremona reveals that 
  $L(E,\chi,1)\ne 0$ and $L(E,\chi\chi_K,1)\ne 0$  when 
  \begin{enumerate}
  \item $\chi$ is the trivial character;
  \item $\chi$ is the  odd quadratic character of conductor $4$.
  \end{enumerate}
  Hence the pair $(E,K)$ satisfies the non-vanishing hypothesis.
 
  \begin{theorem}
  \label{thm:main-quadratic}
  Let $E$ be an
 elliptic curve over $\Q$ and let $K$ be a quadratic field for which ${\rm sign}(E,K)=1$. 
 When $K$ is real, assume that 
 $(E,K)$ satisfies the non-vanishing hypothesis.
 Then for all ring class characters $\psi\in \Pi_K$,
 $$ L(E/K,\psi,1) \ne  0   \quad\Rightarrow \quad
 E(H)^{\psi}=0.$$   
 \end{theorem}
 \begin{proof} If $\psi$ is a quadratic (genus) character,
  the statement already follows  from the approach of Gross-Zagier 
  and Kolyvagin (or Kato), since the induced representation 
  $V_\psi$ decomposes as a sum of two quadratic Dirichlet characters.
 
 Assume henceforth that $\psi^2\ne 1$, which amounts to
  saying that  $V_\psi$ is irreducible.
  Choose a character $\alpha\in \Pi_{E/K}$ which differs from 
  $\psi$ and $\psi^{-1}$, and for which
  $V_\psi \oplus V_\alpha$ is regular. If $K$ is real, assume
  $\alpha$ to be of opposite signature to $\psi$.
  This choice is possible thanks to the non-vanishing hypothesis, 
  and implies that complex conjugation acts on $V_\psi \oplus V_\alpha$
  with eigenvalues $(1,1,-1,-1)$.
  
  
Invoking Lemma \ref{lemma:psi0},
 let $\psi_0$ and $\alpha_0$ be ray class characters of $K$ of conductors prime to $N_E$
 and
 satisfying $\psi_0/\psi_0' = \psi$ and $\alpha_0/\alpha_0' = \alpha$, 
 and set
 $$\psi_g:= \psi_0 \alpha_0, \qquad \psi_h:= (\psi_0'\alpha_0)^{-1}.$$
The two-dimensional representations
$$ \varrho_g := {\rm Ind}_K^\Q \psi_g, \qquad \varrho_h:= {\rm Ind}_K^\Q \psi_h$$
  satisfy the following properties:
\begin{itemize}
\item  $\det(\varrho_g)=\det(\varrho_h)^{-1}$, because the central characters of $\psi_g$ and $\psi_h$ are inverses of each other, 
\item $\varrho_g$ and $\varrho_h$ are odd: this is automatic when $K$ is imaginary; when $K$ is real this follows because 
$\psi$ and $\alpha$ have opposite signature,
hence exactly one of $\psi_0$ and $\alpha_0$ is of mixed signature,
implying that both $\psi_g$ and $\psi_h$ are  of mixed signature.
\item $\varrho_g$ and $\varrho_h$ are irreducible: this amounts to saying that $\psi_g\ne \psi_g'$ and $\psi_h\ne \psi_h'$; 
tracing the definitions, this holds because $\alpha$ has been chosen to be different from $\psi$ and $\psi^{-1}$; 
\item the conductors of $\varrho_g$ and $\varrho_h$ are prime to $N_E$, and hence the same is true for the levels of the associated weight one modular forms.
\end{itemize}
The tensor product $\varrho_{gh}$
  decomposes as
\begin{equation}\label{dec}
V_{gh}=  V_{\psi_g}\otimes V_{\psi_h} = 
 V_{\psi_g\psi_h} \oplus V_{\psi_g \psi_h'} = V_\psi \oplus V_\alpha.
\end{equation}
 Thorem \ref{thm:main-quadratic} now follows from
 the regularity of $V_{g}$ and $V_{h}$  and from
  Theorem \ref{theoremA}, since
 $$L(E,V_{gh},1) = L(E,V_\psi,1) L(E,V_\alpha,1) = L(E/K,\psi,1) L(E/K,\alpha,1) \ne 0.$$
 \end{proof}

We now turn to the proof of Corollary A2 of the introduction:
\begin{theorem} 
\label{thm:mainA5}
Let $E$ be an elliptic curve over $\Q$ and let $K$ be a non-real quintic extension of $\Q$ with 
Galois group $A_5$, of discriminant prime to the conductor of $E$.
Then 
$$  \ord_{s=1} L(E/K,s) = \ord_{s=1} L(E/\Q,s) \quad \Rightarrow \quad
{\rm rank}(E(K)) = {\rm rank}(E(\Q)).  $$
\end{theorem}
\begin{proof} 
Let $\tilde K$ denote the Galois closure of $K$ and fix an embedding of the group $ A_5$ into $ \PGL_2(\C)$. There are exactly two conjugacy classes of such embeddings, and
 the further   choice of an isomorphism $\Gal(\tilde K/\Q)= A_5$ gives rise to a  projective representation
$$\bar\varrho_g: G_\Q \twoheadrightarrow \Gal(\tilde K/\Q)\simeq A_5\subset \PGL_2(\C).$$
 Tate's lifting theorem produces a linear lift $\varrho_g: G_\Q\lra \GL_2(\C)$
of $\bar\varrho_g$, which is odd since $K$ is not totally real. An argument similar to the one 
concluding the proof of  
 Lemma \ref{lemma:psi0}  also shows that $\varrho_g$ can be chosen to unramified 
 at the primes dividing $N_E$.

 Assume that the   field $H$ cut out by $\varrho_g$
is a cyclic extension of $\tilde K$ of   degree a power of $2$.
The image of $\Gal(H/\tilde K)$ under $\varrho_g$
 consists of scalar matrices whose diagonal entries are $2$-power roots of unity.
 Let $\sigma$ be an automorphism of $\C$ which agrees with complex conjugation on the $2$-power roots of unity and sends $\sqrt{5}$ to $-\sqrt{5}$, and let 
 $\varrho_h$ be the Artin representation obtained by applying $\sigma$ to the matrix entries of $\varrho_g$.
 The tensor product $\varrho_{gh}$ is a four-dimensional representation which
 factors through $\Gal(\tilde K/\Q) = A_5$ since it is trivial on 
 $\Gal(H/\tilde K)$.  
 To identify this representation, observe that there are precisely
  two distinct
   irreducible two-dimensional representations, denoted 
   $\varrho_1$ and $\varrho_2$, of 
   the non-trivial central extension 
 $\tilde A_5$ of $A_5$, whose traces lie in $\Q(\sqrt{5})$
 and which are conjugate to each other over this field.
 Viewing $\varrho_1\otimes \varrho_2$ as a representation of
 $\Gal({\tilde K}/\Q)$, one has
 $$ \varrho_{gh} = \varrho_1 \otimes \varrho_2 = (\Ind_{K}^\Q 1) - 1,$$
where the last equality of virtual representations can be
 seen, for instance, by consulting
 the Atlas of finite simple groups.
 The Artin formalism implies that
 $$
  r_{\rm an}(E,\varrho_{gh}) = r_{\rm an}(E/K) - r_{\rm an}(E/\Q), \qquad
    r(E,\varrho_{gh}) = r(E/K) - r(E/\Q).$$
    By hypothesis, $r_{\rm an}(E,\varrho_{gh}) = 0$. 
Theorem \ref{theoremA} can now be invoked to 
conclude 
that  $r(E,\varrho_{gh})=0$, i.e., $r(E,K) = r(E,\Q)$.  
Theorem
\ref{thm:mainA5}  follows.
 \end{proof} 

\subsection{Proof of Theorems B and D}
\label{sec:theoremsBD}
We conclude with the proof of  Theorem D of the introduction, which in turn implies Theorem B.  Assume for  this section that 
\begin{enumerate}
\item[(i)] $L(E,\varrho_{gh},1) = 0$; 
\item[(ii)] the Artin representations $ \varrho_{g}$ and $\varrho_h$ are 
both regular at $p$.
\end{enumerate}
Let $V_{fgh}$  and $V_{fgh}(N) = V_f(Np)\otimes V_g(Np)\otimes V_h(Np)$ 
denote the $K_{fgh}$-vector spaces obtained by specialising
 of $\hV_{f\underline{gh}}$ and $V_{f\underline{gh}}(N)$
 at the weight one point $(y,z)$ attached to
$(f,g,h)$.
%As in \eqref{defSelmer}, the Selmer group of the twist of $E$ by the Artin representation $\varrho_{gh}$ is 
%$$
%\Sel_p(E,\varrho_{gh}) =  \bigcap_\ell \ker\big(H^1(\Q,V_{fgh}) \stackrel{\partial_\ell}{\lra} H^1_{\sing}(\Q_\ell,V_{fgh})\big).
%$$
As discussed right after \eqref{eqn:classes-wt1}, the global cohomology classes
\begin{equation}\label{gclasses}
\kappa(f,g_\alpha,h_\alpha), \quad 
   \kappa(f,g_\alpha,h_\beta) \ \ \in  \ \ H^1(\Q,V_{fgh}(N) )
\end{equation}
  are both locally trivial at primes $\ell\ne p$. 
  Hence the vanishing of $L(E,\varrho_{gh},1)$ implies, by Theorem \ref{main2},  that  $\kappa(f,g_\alpha,h_\alpha)$ and $\kappa(f,g_\alpha,h_\beta)$  belong to the 
 Selmer group $\Sel_p(E,V_{gh}(N))$, which can be identified with a finite sum of copies 
 of the Selmer group $\Sel_p(E,\varrho_{gh})$. 

%Denote by 
% $$ \kappa_p^{g_\alpha}(f,g_\alpha,h_\alpha), \quad 
%   \kappa_p^{g_\alpha}(f,g_\alpha,h_\beta)
% \quad \in  H^1_{\fin}(\Q_p, (V_{f} \otimes V_g^{\alpha_g} \otimes V_h)(N))$$
% the natural images of these two Selmer classes in the associated quotient of the local cohomology at $p$.
\begin{lemma}
\label{lemma:main-prelim-bis}
The images of $\kappa(f,g_\alpha,h_\alpha)$ and $\kappa(f,g_\alpha,h_\beta)$ in $H^1_{\fin}(\Q_p, V_{f}(Np) \otimes V_{g}^{\alpha_g}(Np) \otimes V_h(Np))$ belong to $H^1_{\fin}(\Q_p, V_{f}(Np) \otimes V_{g}^{\alpha_g}(Np) \otimes V^{\beta_h}_h(Np))$ and to
 $H^1_{\fin}(\Q_p, V_{f}(Np) \otimes V_{g}^{\alpha_g}(Np) \otimes V^{\alpha_h}_h(Np))$ respectively.
\end{lemma}
\begin{proof}
 This follows directly from Proposition~\ref{prop:the-clincher}.
\end{proof}


\begin{theorem}
  \label{theoremB} 
  Assume that $\Lp^{g_\alpha}(\testf,\testg^*,\testh) \ne 0$ for some
 choice 
 $$ \testf\in S_2(Np)[f], \qquad \testg^*\in S_1(N,\chi^{-1})^\vee[g], \qquad \testh\in S_1(N,\chi^{-1})[h] $$
  of test vectors. 
Then there exist
$G_\Q$-equivariant projections $j_1, j_2: V_{fgh}(N) \ra V_{fgh}$   
for which the global classes
\begin{equation}
\label{eqn:def-kappa-alpha-alpha-alpha-beta}
 \kappa_{\alpha \alpha} := j_1(\kappa(f,g_\alpha,h_\alpha)), \qquad \kappa_{\alpha \beta} := j_2(\kappa(f,g_\alpha,h_\beta)) 
 \end{equation} 
 are linearly independent in $\Sel_p(E,\varrho_{gh})$.
\end{theorem}
\begin{proof} % Let us first show that $\kappa_{\alpha \alpha}$ and $\kappa_{\alpha \beta}$ are non-zero elements of $\Sel_p(E,\varrho_{gh})$. 
% %The  test vectors $\testg^*$ and $\testh$  determine  $\Lambda$-adic test vectors $\htg^*$, $\hth_1$ and $\hth_2$ as in \eqref{eqn:test-vectors2} and \eqref{eqn:dual-test-vectors2} for which
% $$\testg^*_y=\testg^*_\alpha, \qquad \testh_{1,z_1}=\testh_\alpha, \qquad \testh_{2,z_2}=\testh_\beta.$$ Recall from   that 
% $$\Lp^{g_\alpha}(\testf,\testg^*,\testh):= \Lp^{g_\alpha}(\testf,\underline{\testg^*\testh_1})(y,z_1) = \Lp^{g_\alpha}(\testf,\underline{\testg^*\testh_2})(y,z_2).$$ 
% The test vector $\testf$ induces a $G_\Q$-equivariant projection $\varpi_{\testf}: V_f(Np)\ra V_f$ whose associated pullback on de Rham cohomology (via the comparison isomorphisms of \S \ref{blo}) maps $\Fil^0(D_f)$ to the line spanned by $\omega_{\testf}$ in $\Omega^1(X_0(Np))$, and likewise for $\testh_\alpha$ and $\testh_\beta$. The dual test vector $\testg_\alpha^*$ also induces  a $G_\Q$-equivariant projection $\varpi_{\testg_\alpha*}: V_g(Np)\ra V_g$ whose associated pullback on de Rham cohomology maps the unit root subspace of $D_g$ to the line spanned by $\eta_{\testg_\alpha^*}$ in $H^1_{\dR}(X_{s/\Q_p(\zeta_s)})$. They give rise to two $G_\Q$-equivariant projections
%$$j_\alpha=\varpi_{\testf \testg_\alpha^* \testh_\alpha}, j_\beta=\varpi_{\testf \testg_\alpha^* \testh_\beta}: V_{fgh}(N) \ra V_{fgh}.$$
Let  $\hg$ and $\hh$   be Hida families  specialising to $g_\alpha$ and to $h_\alpha$ 
respectively in weight one, i.e., for which $g_y=g_\alpha$ and $h_{z}=h_\alpha$ 
for suitable weight one points $(y,z) \in \Omega_{\hg} \times \Omega_{\hh}$. 
By Proposition \ref{LZ-Lambda-expdual0} combined with  Theorem \ref{pL-PR},
the non-vanishing of 
 $$ \Lp^{g_\alpha}(\testf,\testg^*, \testh) := \Lp^{g_\alpha}(\testf,\testhg^*,\testhh)(y,z)$$
implies that the local class $\kappa_p^g(\testf,\testg_\alpha^*,\testh_\alpha)$ has non-zero $p$-adic logarithm.
It follows a fortiori  from   \eqref{eqn:ohta-classes} and   \eqref{eqn:specialised-ohta-classes}
that the local class
$\kappa_p^{g}(f,\underline{gh})(y,z)$ is non-zero, and therefore, that the natural image of 
 the global class $\kappa(f,g_\alpha,h_\alpha)$ 
 in 
 $H^1(\Q_p, V_{f}(Np) \otimes V_{g}^{\alpha_g}(Np) \otimes V_h(Np))$ is non-zero as well.
 The same argument in which $h_\alpha$ is replaced by $h_\beta$ implies 
 (in light of equation \eqref{Lpfgh} of  the introduction) the same conclusion for
 the global class $\kappa(f,g_\alpha,h_\beta)$. 
 
 
 After choosing a basis   of $\Hom_{G_\Q}(V_{fgh}(N), V_{fgh})$ (of cardinality $t$, say),
 the resulting isomorphism
 $V_{fgh}(N) \lra \oplus_{i=1}^t V_{fgh}$ induces an isomorphism  
 $$ H^1(\Q_p, V_{f}(Np) \otimes V_{g}^{\alpha_g}(Np) \otimes V_h(Np)) \lra \oplus_{i=1}^t
 H^1(\Q_p, V_{f} \otimes V_{g}^{\alpha_g} \otimes V_h)$$ 
on the quotients of the local cohomology groups at $p$. 
Hence there are $G_\Q$-equivariant homomorphisms 
 $j_1, j_2: V_{fgh}(N) \rightarrow V_{fgh}$ for which the global
 classes $ \kappa_{\alpha \alpha}$ and $ \kappa_{\alpha \beta}$
 defined in \eqref{eqn:def-kappa-alpha-alpha-alpha-beta}
 have non-trivial image in the quotient $H^1(\Q_p, V_{f} \otimes V_{g}^{\alpha_g} \otimes V_h)$.

As argued above right after \eqref{gclasses}, the global classes $\kappa(f,g_\alpha,h_\alpha)$ and $\kappa(f,g_\alpha,h_\beta)$ lie in the Selmer group $\Sel_p(E,V_{gh}(N))$ and hence $\kappa_{\alpha \alpha} $ and $\kappa_{\alpha \beta}$ likewise belong to $\Sel_p(E,\varrho_{gh})$. 
Lemma \ref{lemma:main-prelim-bis}  further implies that their natural images 
 in 
$H^1(\Q_p, V_{f} \otimes V_{g}^{\alpha_g} \otimes V_h)$ belong to the complementary subspaces 
$H^1_{\fin}(\Q_p, V_f \otimes V_g^{\alpha_g} \otimes V_h^{\beta_h})$ and
  $H^1_{\fin}(\Q_p, V_f \otimes V_g^{\alpha_g} \otimes V_h^{\alpha_h})$ respectively. 
These images are therefore linearly independent, and the theorem follows.
\end{proof}


 



 % \begin{proof}   The classes $\kappa_{\alpha \alpha}$ and $\kappa_{\alpha\beta}$ lie in  $\Sel_p(E,\varrho_{gh})$ for any choice of $G_\Q$-equivariant projections $j_1, j_2$.  Let us prove that $j_1$ and $j_2$ can be chosen in such a way that they are linearly independent. Since $V_{f g h}^{G_{\Q_p}} = 0$, it follows from   \eqref{H1f} and the long exact sequence in Galois   cohomology associated to the ordinary filtration   $V_f^+ \subset V_f$ that there is an isomorphism\begin{equation}\label{iso1}H^1_{\fin}(\Q_p,V_{f g h}) \simeq H^1(\Q_p, V_{f}^+ \otimes V_{g h}).\end{equation}
%and thus  the local class $\kappa_p(f,g_\alpha,h_\alpha)$ can be regarded
% as a natural element in $H^1(\Q_p, V_{f}^+ \otimes V_{g h}(N))$.




%As in the introduction, let $H_p$ denote the completion of $H$ with respect to the embedding $\bar\Q \hookrightarrow \bar\Q_p$ fixed at the outset. Arguing exactly as in the paragraph preceeding \eqref{eqn:bloch-kato-logarithm-bis}, Bloch-Kato's logarithm associated to the $p$-adic representation $V_{f g h}$ gives rise to a linear isomorphism\begin{equation}\label{BK1}\log_p: H^1(\Q_p, V_{f}^+ \otimes V_{g h}) \lra \left(D(V_f^-)\otimes D(V_{gh})\right)^\vee,\end{equation}where $D(V_f^-)=(V_f^-\otimes \hat{\Q}_p^{\mathrm{nr}})^{G_{\Q_p}}$ and $$D(V_{gh}) := (V_{gh}\otimes H_p)^{\Gal(H_p/\Q_p)} \simeq \oplus_\lambda D(V_{gh}^{\lambda})$$where  $\lambda\in\{(\alpha_g,\alpha_h), (\alpha_g,\beta_h), (\beta_g,\alpha_h), (\beta_g,\beta_h) \}$ ranges over the (distinct, by the regularity assumption on $p$) pairs of eigenvalues of $\Frob_p$ acting on $V_{g}$ and $V_h$. 

%Recall from \S \ref{sec-Hida} that we view $V_f^-$ as an unramified $G_{\Q_p}$-module of rank $1$ over $\cO_{fgh}$ and hence
 %$D(V_f^+)$, we may regard \eqref{BK1} as a linear map
%\begin{equation}\label{BK2}
%\log_p: H^1_{\fin}(\Q_p,V_{f g h}) \lra D(V_{gh})^\vee.
%\end{equation}
%Namely, given a local class $\kappa_p\in H^1_{\fin}(\Q_p,V_{f g h})$ and an element $v\in D(V_{gh})$, %we have
%$$
%\log_p(\kappa_p)(v) := \log_p(\kappa_p)(\omega_f \otimes v).
%$$

%Note that there are natural isomorphisms
%\begin{equation}\label{BK0}
%H^1(\Q_p, V_{f}^+ \otimes V_{g h}) \simeq (\Sel_p(E/H_p) \otimes V_{g h})^{\Gal(H_p/\Q_p)} \simeq \oplus_\lambda (\Sel_p(E/H_p) \otimes V_{g h}^\lambda)^{\Gal(H_p/\Q_p)},
%\end{equation}

 


% The Dieudonn\'e modules $D(V_f^-)$ and $D(V_{gh}^{\lambda})$ are one-dimensional over $K_{fgh}$. The class $\omega_{f}$ gives rise to a natural generator of $D(V_f^-)$; choose likewise for each $\lambda$ a generator $v_\lambda$ of $D(V_{gh}^{\lambda})$, that we regard as elements in $V_{gh}^\lambda\otimes H_p$ that are fixed by the action of $\Gal(H_p/\Q_p)$. 

% We claim that the class $\kappa_{\alpha \alpha} = j_1(\kappa(f,g_\alpha,h_\alpha))$ satisfies  \begin{enumerate}  \item[(i)] $\log_p(\res_p \kappa_{\alpha \alpha})(\omega_{f} \otimes v_{\beta_g \beta_h}) = 0$.  \vspace{0.1cm} \item[(ii)] $\log_p (\res_p  \kappa_{\alpha \alpha})(\omega_{f} \otimes v_{\beta_g \alpha_h})$  is non-zero for a suitable choice of $j_1$.  \end{enumerate}



%In order to prove (i), note that Bloch-Kato's logarithm associated  to $V_f\otimes V_{gh}^{\alpha_g \alpha_h}$ gives rise as in \eqref{BK1} to an isomorphism $$H^1(\Q_p,V_f^+\otimes V_{g h}^{\alpha_g \alpha_h}) \stackrel{\sim}{\lra} D(V_f^+)\otimes D(V_{gh}^{\alpha_g \alpha_h}) \stackrel{\sim}{\lra} \left(D(V_f^-)\otimes D(V_{gh}^{\beta_g \beta_h})\right)^\vee. $$

% Since (i) is equivalent to saying that $\log_p(\res_   \kappa_{\alpha \alpha})$ vanishes on $D(V_f^-)\otimes D(V_{gh}^{\beta_g \beta_h})$, the above isomorphisms shows that this is equivalent to proving that the natural image of $\kappa_{\alpha \alpha} $ in $H^1_{\fin}(\Q_p, V_f \otimes V_{gh}^{\alpha_g\alpha_h})$ is trivial. But it follows from \eqref{iso1} that $\res_p \kappa_{\alpha \alpha}$ belongs to $H^1(\Q_p, V_{f}^+ \otimes V_{g h})$, while Corollary  \ref{cor:cristalline-lambda} asserts that the $\Lambda$-adic class    $\kappa_p(f,\underline{g h})$ lies in    $H^1(\Q_p,\mathbb{V}_{f \underline{g} \underline{h}}^+(N))$. From this one concludes that, independently of the choice of $j_1$, the local class   $\res_p \kappa_{\alpha \alpha}$ lies in $H^1(\Q_p,V_{f g h}^+ \cap V_f^+ \otimes V_{gh})$, and hence its projection to $H^1_{\fin}(\Q_p, V_f \otimes V_{gh}^{\alpha_g\alpha_h})=H^1(\Q_p,V_f^+ \otimes V_g^- \otimes V_h^-)$   is zero, as claimed.
  
% For (ii), note similarly as above that the Bloch-Kato logarithm induces an isomorphism$$\log_p: H^1(\Q_p,V_f^+\otimes V_{g h}^{\alpha_g \beta_h}) \stackrel{\sim}{\lra} \left(D(V_f^-)\otimes D(V_{gh}^{\beta_g \alpha_h})\right)^\vee.$$

%Since $\omega_f \otimes v_{\beta_g\alpha_h}$ is a generator of $D(V_f^-)\otimes D(V_{gh}^{\beta_g\alpha_h})$, it follows that $\log_p(\res_p  \kappa_{\alpha \alpha})(\omega_f \otimes v_{\beta_g \alpha_h})$ is non-zero if and only if the natural image of $\kappa_{\alpha \alpha}$  in $H^1_{\fin}(\Q_p, V_f \otimes V_{g h}^{\alpha_g \beta_h})$ is non-zero. 

%Let $j_1=\varpi_{\testf \testg^* \testh}: V_{fgh}(N)=V_f(Np)\otimes V_{gh}(N) \ra V_{fgh}$ be the projection induced by the choice of test vectors, satisfying that\footnote{V to H: Do you think the sentence " the natural image of $\kappa_{\alpha \alpha} = j_1(\kappa(f,g_\alpha,h_\alpha))$ in $H^1(\Q_p,V_f^+\otimes V^{\alpha_g \beta_h}_{gh})$ coincides with the specialisation $\kappa_p^g(\testf,\testg^*_y,\testh_z)$ of $\kappa^g_p(\testf,\underline{\testg^*\testh})$ at the classical point $x=(y,z)$ of weight one" needs some more justification, or can be taken as an obvious consequence of the definitions? I didn't have time yet to go through the details of the construction of $\kappa_p^g(\testf,\testg^*_y,\testh_z)$ in order to verify that. note that $\varpi_{\testf \testg^* \testh}$ is not a notation that had been introduced previously, and is taken to be here as the "natural map" induced by the test vectors. There is one thing that puzzles me  bit: since $\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha) = \Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\beta)$, this seems to imply that we could perhapsc take $j=j_1=j_2$ to be the projection induced by the same test vectors $\testf$,  $\testg^*$,  $\testh$.} the natural image of $\kappa_{\alpha \alpha} = j_1(\kappa(f,g_\alpha,h_\alpha))$ in $H^1(\Q_p,V_f^+\otimes V^{\alpha_g \beta_h}_{gh})$ coincides with the specialisation $\kappa_p^g(\testf,\testg^*_y,\testh_z)$ of $\kappa^g_p(\testf,\underline{\testg^*\testh})$ at the classical point $x=(y,z)$ of weight one. Hence (ii) amounts to showing that the local cohomology class $\kappa_p^g(\testf,\testg^*_y,\testh_z)$ is not trivial.



% Note that there are natural isomorphisms $$ H^1(\Q_p, V^+_f \otimes V_{g h}^{\alpha_g \beta_h}) \simeq H^1(\Q_p, V^+_f \otimes V_g^- \otimes V_h^+) \simeq H^1(\Q_p, \Q_p(\Upsilon_{g}^+)),$$  where $\Upsilon_g^+ = \Psi_f^{-1} \Psi_{g_\alpha} \Psi_{h_\alpha}^{-1} \times \chi^{-1}  \times \epsilon_{\cyc}$ and the last isomorphism follows from Corollary \ref{cor:4-step-filtration}. 




% Set $\Psi := \Psi_{f}^{-1}\Psi_{\hg}   \Psi_{\hh}^{-1}  \chi^{-1}$.The character $\Upsilon_g^+$ is specialisation  at $x=(y,z)$ of the $\Lambda$-adic character$\heta_{\hg}^+ = \Psi \times \epsilon_{\cyc}$ that appears in the corresponding row of the table in Lemma\ref{lemma:4-step-filtration}. As recalled at the beginning of \S \ref{subsec:PR}, the class $\ukappa^g := \kappa^g_p(\testf,\underline{\testg^*\testh})$ takes values in $\Lambda_{f \underline{g h}}\{ \Psi\}(1)$ and Proposition \ref{LZ-Lambda-expdual0} and Theorem \ref{pL-PR} apply. Namely, the combination of the two statements assert that$$\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha) := \Lp^{g_\alpha}(\testf,\underline{\testg^*\testh})(y,z) = \cL(\kappa^g_p(\testf,\underline{\testg^*\testh}))(y,z) = \log_p \kappa_p^g(\testf,\testg^*_y,\testh_z).$$Since we are assuming that $\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha) \ne 0$, (ii) follows.

%As explained in \eqref{Lpfgh} in the introduction, we have $\Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\alpha) = \Lp^{g_\alpha}(\testf,\testg_\alpha^*,\testh_\beta)$, which is why in Theorem D we refer to this value simply as $\Lp^{g_\alpha}(\testf,\testg^*,\testh)$. Hence the above argument also applies to $\kappa_{\alpha \beta}$, showing that (i) $\log_p(\res_p \kappa_{\alpha \beta})(\omega_{f} \otimes v_{\beta_g \alpha_h}) = 0$ and (ii) $\log_p (\res_p \kappa_{\alpha \beta})(\omega_{f} \otimes v_{\beta_g \beta_h})$  is non-zero for a suitable choice of $j_2$.
% All together implies that the linear map $$ \kappa \mapsto (\log_p(\res_p \kappa)(\omega_f\otimes v_{\beta_g\beta_h}), \log_p(\res_p \kappa)(\omega_f\otimes v_{\beta_g\alpha_h}))$$  sends the classes $\kappa_{\alpha \alpha}$ and $\kappa_{\alpha \beta}$  to vectors of the form $(0,a)$ and $(b,0)$ respectively, where $a$ and $b$ are both non-zero. The theorem follows.  \end{proof}



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\end{thebibliography}
\end{document}