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\begin{document}


 







\title[Beilinson-Flach elements and Euler systems II]{Beilinson-Flach elements and Euler systems II: 
\\ The Birch and Swinnerton-Dyer conjecture \\ for Hasse-Weil-Artin $L$-series}

\author{Massimo Bertolini, Henri Darmon and Victor Rotger}

\begin{abstract}
Let $E $ be an  elliptic curve over $\Q$ and let $\varrho$   be
an odd, irreducible two-dimensional 
Artin representation. This article  proves  the 
Birch and Swinnerton-Dyer conjecture in analytic rank zero for the 
Hasse-Weil-Artin $L$-series $L(E,\varrho,s)$, namely, the implication
$$ L(E,\varrho,1) \ne 0\quad  \Rightarrow \quad  (E(H)\otimes \varrho)^{\Gal\!(H/\Q)} = 0,$$
where $H$ is the finite extension of $\Q$ cut out by $\varrho$. 
The proof relies on  $p$-adic families of global  Galois cohomology classes arising
from Beilinson-Flach elements in a tower of products of  
modular curves.  
  \end{abstract}

\address{M.B.: Fakultaet fuer Mathematik, Universitaet Duisburg-Essen, Essen, Germany}
\email{massimo.bertolini@uni-due.de}

\address{H. D.: Department of Mathematics  and Statistics, McGill University, Montreal, 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}
This paper  completes the project undertaken in 
\cite{BDR1} of proving the Birch and Swinnerton-Dyer conjecture in analytic 
rank $0$ for the $\varrho$-isotypic components of Mordell-Weil groups
of elliptic curves, when $\varrho$ is an odd, irreducible two-dimensional
Artin representation.

\medskip
In the early 90's, Kato unveiled  a new approach to the Birch and Swinnerton-Dyer
 conjecture and to the 
cyclotomic Iwasawa main conjecture 
for (modular) elliptic curves, resting on Beilinson elements in the second
 $K$-groups of modular curves and on their $p$-adic deformations.
This approach led to a proof of the implication
$$ L(E,\chi,1) \ne 0 \quad \Rightarrow \quad \Hom_{G_\Q}(\chi, E(\bar\Q)\otimes\C) =0, $$
for all elliptic curves $E$ over $\Q$
and all Dirichlet characters $\chi$ (viewed as Galois characters via class field theory). 
The article \cite{Durham-survey} places Kato's strategy in the broader
framework of ``Euler systems of Garrett-Rankin-Selberg type".  
Roughly speaking, the global objects exploited by Kato  are distinguished elements in the higher Chow group
$\CH^2(X_1(Np^s),2)$ attached to  a pair of modular units, whose logarithmic derivatives are
 Eisenstein series
of weight two.  Replacing either one, or both, of these Eisenstein series by
 cuspidal eigenforms leads   
to the study of   {\em Beilinson-Flach} elements in $\CH^2(X_1(Np^s)^2,1)$ and of
 {\em Gross-Kudla-Schoen} diagonal cycles in the Chow group
$\CH^2(X_1(Np^s)^3)$ of codimension two cycles on the triple product of $X_1(Np^s)$. 
Chapters 2.2 and 2.3. of  \cite{Durham-survey} explain how the images of these  elements under 
$p$-adic
\'etale regulators and Abel-Jacobi maps, when made to vary in $p$-adic families and 
specialised at suitable classical weight one points, might be parlayed into a proof of the implications
\begin{eqnarray}
\label{eqn:BDR}
 L(E,\varrho,1) \ne 0 &  \Rightarrow &  \Hom_{G_\Q}(\varrho, E(\bar\Q)\otimes\C) =0, \\
 \label{eqn:DR}
 L(E,\varrho_1\otimes \varrho_2,1) \ne 0 &  \Rightarrow &  \Hom_{G_\Q}(\varrho_1\otimes \varrho_2, E(\bar\Q)\otimes\C) =0,
 \end{eqnarray}
 where $\varrho$, $\varrho_1$ and $\varrho_2$ are odd, irreducible two-dimensional (complex) Artin representations 
 for which $\varrho_1\otimes\varrho_2$ is self-dual.
 The details of this program are carried out  in \cite{DR2} to obtain the proof of
 \eqref{eqn:DR}. The goal of the present work is to deliver on the other  promise made in
\cite{Durham-survey} by proving the following result:
 
 \medskip\noindent
 {\bf Theorem A}.  {\em Let $E$ be an elliptic curve over $\Q$  and let $\varrho$ 
 be an odd, irreducible, two-dimensional Artin representation.
  Assume that the conductors 
 of $E$ and $\varrho$ are prime to each other. 
 Then
  \eqref{eqn:BDR} holds. }
 
 \medskip\medskip
 Prior to Theorem A, the only irreducible two-dimensional Artin representations for which 
 \eqref{eqn:BDR} had been proved were  those induced from {\em ring class characters} of
 \begin{itemize}
 \item[(i)] imaginary quadratic fields, 
 by a series of works  building
 on the
  methods of Kolyvagin \cite{Ko};
  \item[(ii)]   real quadratic fields, under a mild analytic non-vanishing
  hypothesis, by Corollary A1 of \cite{DR2}.
  \end{itemize}
  In addition to Artin representations with projective image isomorphic to $A_4$, $S_4$ and $A_5$,   
Theorem A also applies to a large class of  two-dimensional representations induced from  
general ray class characters
of  quadratic fields:


\medskip\noindent
 {\bf Theorem B}.  {\em Let $E$ be an elliptic curve over $\Q$  of conductor $N$,  let 
 $K$ be a quadratic field of discriminant $D$, and let 
 $\psi:\Gal(H/K) \lra \C^\times$ be a ray class character of conductor
 $\mathfrak{f} \lhd  \cO_K$.
 Assume that $\gcd(N, D \cdot {\rm norm}(\mathfrak{f}))=1$, and that $\psi$ is of mixed 
 signature if $K$ is real quadratic. 
 Then 
$$ L(E/K,\psi,1) \ne 0 \quad \Rightarrow \quad E(H)^\psi = 0,$$
where 
$$ E(H)^\psi := \left\{ P\in E(H)\otimes \C \quad \mbox{{\rm s.t.} }\quad \sigma P = \psi(\sigma) P, 
\quad \forall \sigma\in\Gal(H/K) \right\}.$$
 }
 

\noindent
Even for imaginary quadratic fields, Theorem B goes well beyond what can be obtained by
the methods of Kolyvagin which apply only to ring class characters.
 For   real quadratic $K$, there is simply no overlap between
Theorem B and 
\cite[Cor.~A1]{DR2}, since ring class characters are either totally even or totally odd.
 
 
It is worth pointing out that
the representation  $\varrho$ arising in Theorem A is typically not
self-dual. Because of this, the root number  arising in the functional equation of
$L(E,\varrho,s)$ does not control the parity of its order of vanishing at $s=1$, 
and 
it is  expected that $L(E,\varrho,1)$  vanishes only in rare, sporadic instances, so that 
the non-vanishing hypothesis in Theorem A is almost always satisfied. Any   
quantitative non-vanishing statement 
 along these lines would
find immediate, unconditional applications to the arithmetic of elliptic curves via Theorems A
and B. It should also be noted that $L(E,\varrho,1)\ne 0$ if and only if
$L(E,\varrho^\vee,1)\ne 0$, where $\varrho^\vee$ is the contragredient representation of $\varrho$. Likewise,  the triviality of the complex vector space $\Hom_{G_\Q}(\varrho, E(\bar\Q)\otimes\C)=(E(\bar\Q)\otimes\varrho^\vee)^{G_\Q}$ is equivalent to that of
$\Hom_{G_\Q}(\varrho^\vee, E(\bar\Q)\otimes\C)$. 

 


\medskip
The  proof of Theorem A  
is based on so-called
{\em Beilinson-Flach elements} whose definition is introduced in \S \ref{sec:beilinson-flach} below, and on their {\em variation in $p$-adic families}. 
For the purpose of the introduction, suffice it  to say that Beilinson-Flach elements
are  distinguished elements in
the higher Chow group $\CH^2(X_1(N)^2,1)$, which can also be interpreted as the motivic cohomology group $H^3_{\cM}(X_1(N)^2,\Q(2))$,
They were first introduced in the 1980's by
 Beilinson \cite[Ch.\,2, \S 6]{beilinson-K1S}, 
who related their image under his complex regulator
 to the value at $s=2$ of the Rankin $L$-series  $L(f\otimes g, s)$ attached to
the convolution of weight two newforms $f$ and $g$ on $\Gamma_1(N)$.
 
 
In the early 1990's, Flach \cite{Fl}  exploited  (a slight variant of) these elements 
to prove the finiteness of the  Selmer group of the symmetric square of an elliptic curve.
Shortly thereafter, these  finiteness results found a  spectacular application
in Wiles' epoch-making proof of the Shimura-Taniyama conjecture and  Fermat's Last Theorem.
 Flach's approach to bounding the Selmer group of the symmetric square
 featured prominently in Wiles' original strategy, but was later 
 obviated by the   
simpler and more flexible approach based on   ``Taylor-Wiles systems"
 introduced
in \cite{taylor-wiles}. 


The 
power and versatility of  Taylor-Wiles systems for
questions surrounding the   deformation theory
of Galois representations
may explain why, with a  few exceptions
(\cite{mazur-weston}, \cite{weston}, $\ldots$),
 Beilinson-Flach elements garnered relatively
less attention in the intervening   decades.
Yet their arithmetic relevance is not confined to the  symmetric square motive
of  a modular form:
guided by an analogy between Beilinson-Flach elements and 
the study of 
Gross-Kudla-Schoen diagonal cycles undertaken in \cite{DR1} and
\cite{DR2},
 the article \cite{BDR1} established  
 a $p$-adic analogue of Beilinson's formula
for Beilinson-Flach elements, 
pointing out a number of 
 arithmetic applications of this formula
 based on  the ``Euler system philosophy"
    propounded by Kato and Perrin-Riou, and
    lying ostensibly outside the  scope of the Taylor-Wiles method.
(Cf. the introduction of loc.cit., and
\cite[Sec.~2.2.]{Durham-survey}.)

The  recent   work 
  of Kings, Lei, Loeffler and Zerbes makes significant strides in furthering the program initiated
  in \cite{BDR1}, applying the ideas of the present work to 
  more general settings and introducing key technical improvements along the way. 
  Thus,   \cite{LLZ1} constructs a full cyclotomic  Euler system of Beilinson-Flach elements and 
 applies it to the study
 of the Selmer group of the tensor
 product of two motives attached to cusp forms, while \cite{LLZ2} applies similar
  techniques to
 study the Iwasawa theory of elliptic curves over the full two-variable 
 $\Z_p$-extension of an imaginary quadratic field. 
The recent preprint \cite{KLZ}  exploits an interpolation of the 
Beilinson-Flach elements in all weights at once
 based on the notion of ``Rankin-Iwasawa classes" 
arising in \cite{kings},
to establish   a general  
 explicit reciprocity law  relating  these families  
 to Hida's three variable $p$-adic Rankin $L$-function, 
 with important applications to the associated three variable Main conjecture. 
It is worth pointing out that
the approach to   Theorem A in the present work rests solely on $p$-adic families of
 Beilinson-Flach elements of minimal level and on their local behaviour at $p$;
it can therefore  be compared to the strategy
 followed by Coates-Wiles  \cite{CoWi} to prove the finiteness of Mordell-Weil groups of
CM elliptic curves. By exploiting the full collection of Beilinson-Flach elements and
 their Euler system properties, notably their ``tame deformations" at auxiliary primes $\ell\ne p$, \cite{KLZ}   also establishes (just as in Thaine and Rubin's strengthening of the  
 Coates-Wiles method)
  the finiteness of certain
 $p$-parts of the relevant Selmer and  Shafarevich-Tate groups
  in the setting of Theorem A.
 
 
Shortly after \cite{BDR1}, Dasgupta 
\cite{Dasgupta}
 had the idea of 
exploiting Beilinson-Flach elements  to factor
 Hida's $p$-adic Rankin $L$-function
of the tensor square  $f\otimes f$  into the Kubota-Leopoldt $p$-adic zeta-function 
and the
Coates-Schmidt $p$-adic $L$-function    attached to  the
 symmetric square of $f$. 
 The main result of \cite{Dasgupta}
 is a deep
manifestation of the Artin formalism for Hida's  $p$-adic $L$-series and rests  on a comparison between
circular units and a new construction of units in number fields arising
 from Beilinson-Flach elements. 
 Dasgupta's approach, which
  is inspired by a  theorem
 of Gross \cite{gross} for the Katz $p$-adic $L$-function of an imaginary quadratic
 field and  builds on the ideas  of the present work,  also
 makes essential use of the improvements described in \cite{KLZ}.

 

  
 \medskip\noindent
{\em Acknowledgements}: The authors are grateful to Samit Dasgupta, 
 Guido Kings, Antonio Lei,
David Loeffler 
and Sarah Zerbes for many stimulating exchanges related to the circle of ideas 
explored in this paper. The work of the second author on this paper
was partially
supported by the Simons Foundation and by the Mathematisches Forschungsinstitut
Oberwolfach. The third author was supported by  MTM20121-34611 during the ellaboration of this work.


 \medskip
   
  \section{Strategy of proof}
 \label{sec:strategy}
Not surprisingly,   modularity plays  a key part in the proof of Theorem A. The elliptic curve $E$ is associated to a weight $2$ newform $f=f_E\in S_2(N_f)$ thanks to \cite{bcdt}, while the Artin representation $\varrho$ is associated to a weight $1$ newform $g=g_\varrho\in S_1(N_g,\chi)$ thanks to \cite{KW}. The Hasse-Weil-Artin $L$-series $L(E,\varrho,s)$ can therefore
  be identified with the convolution $L$-series $L(f\otimes g,s)$, whose analytic continuation follows from   Rankin's method. 
 
Let $L\subset \Q^{\rm ab}$ denote a field of coefficients over which $\varrho$ can be described, and let 
$H$ be the  finite   extension of $\Q$ cut out by $\varrho$ (i.e., the fixed field of the kernel of $\varrho$). Write
$$ E(H)_L^\varrho := \hom_{G_\Q}(V_\varrho, E(H)\otimes L)$$
for the $\varrho$-isotypic part of the Mordell-Weil group of $E$.
Theorem A asserts the triviality of this vector space  when 
$L(E,\varrho,1)\ne 0$. 

The standard approach to bounding $E(H)_L^\varrho$ consists
in   embedding  it in an appropriate 
{\em Selmer group} attached to the choice of a rational prime $p$, which throughout the paper
is assumed to be prime to $N_fN_g$, and {\em ordinary} for $f$, i.e., $p$ does not divide $a_p(f)$. 
To define this Selmer group,
let  
$$V_f:=V_p(E):= H^1(E_{\bar\Q}, \Q_p)(1) =  (\lim_{\leftarrow,n} E[p^n])\otimes \Q_p  $$
 denote Deligne's  $p$-adic representation attached to $f$,
which occurs as a direct factor of the \'etale cohomology
 $H^1_{\et}(X_1(N)_{\bar\Q},\Q_p)(1)$, and let $V_g= V_{g,\sigma}$ denote the Artin representation attached to $g$, viewed as a $p$-adic representation by fixing some embedding $\sigma$ of its coefficient field $L$  into $L_p \subset \bar\Q_p$. 
 The   four dimensional  $L_p$-vector space
 $$ V_p(E) \otimes V_g = V_f \otimes V_g,$$
where the tensor product is taken over $\Q_p$,
 is a $p$-adic representation of $G_\Q$. The  connecting homomorphism of 
 Kummer theory arising from the $p$-power
 descent exact sequence on $E(H)$,  composed with the 
 inverse of the restriction map from $\Q$ to $H$, gives a linear injection of $L_p$-vector spaces
 $$ E(H)_{L_p}^\varrho:=E(H)_L^\varrho\otimes _L L_p \lra  H^1(H,V_p(E)\otimes V_g^\vee)^{\Gal(H/\Q)} \stackrel{\res_H^{-1}}{\lra} H^1(\Q, V_f \otimes V_g^\vee),$$
denoted $\delta$,
 where $V_g^\vee := V_g \otimes \det(V_g)^{-1}$ is the $L_p$-linear dual of the
 representation $V_g$. 
 For each place $v$ of $\Q$ the corresponding map on local points yields an injection
 $$ \delta_v: E(H_v)_{L_p}^\varrho \lra H^1(\Q_v, V_f\otimes V_g^\vee),$$
where $\Gal(H/\Q)$ acts in the natural way on 
the local points $E(H_v):= \oplus_{w|v} E(H_w)$. 
The image of $\delta_v$ in $H^1(\Q_v,V_f\otimes V_g^\vee)$ is called the 
 {\em finite part } of the local cohomology at $v$, and is denoted
 $H^1_{\fin}(\Q_v,V_f\otimes V_g^\vee)$. Likewise, $H^1_{\fin}(\Q_v,V_f\otimes V_g)$ is defined
to be the image of  $E(H_v)^{\varrho^\vee}$ by the appropriate local coboundary map.

  When $v=p$, the finite part of the local cohomology 
     admits an alternate description in terms of the general local conditions attached by
     Bloch and Kato  \cite[\S 3]{bloch-kato}
     to any   $p$-adic representation $V$ 
     of $G_{\Q_p}$:
  \begin{eqnarray}
  \nonumber
  H^1_{\rm exp}(\Q_p, V)  &:=&   \ker\left(H^1(\Q_p, V) \rightarrow H^1(\Q_p, V\otimes 
  \hB_{\cris}^{\phi=1}) \right),    \\
  \label{eqn:def-efg}
  H^1_{\rm fin}(\Q_p, V)  &:=&  \ker\left(H^1(\Q_p, V) \rightarrow H^1(\Q_p, V\otimes \hB_{\cris}) \right)    = 
          \Ext^1_{\cris}(\Q_p,V),  \\
          \nonumber
   H^1_{\rm geom}(\Q_p, V)  &:=&  \ker\left(H^1(\Q_p, V) \rightarrow H^1(\Q_p, V\otimes \hB_{\dR}) \right)   =  \Ext^1_{\dR}(\Q_p,V),         
   \end{eqnarray}
 where $ \hB_{\cris} \subset \hB_{\dR}$ are Fontaine's rings of periods for  cristalline and 
 de Rham representations, respectively, and $\phi$ denotes the cristalline frobenius.
 The subscripts exp, fin, and geom stand for ``exponential", ``finite", and ``geometric" respectively, and
 one has the obvious inclusions
 $$ H^1_{\rm exp}(\Q_p,V) \subset H^1_{\fin}(\Q_p,V) \subset H^1_{\rm geom}(\Q_p,V).$$
 When $V= V_f \otimes V_g$ or $V_f\otimes V_g^\vee$, the condition $p\nmid N_fN_g$ 
 shows, as in the proof of equation  \eqref{eqn:hfin=hexp}, that  the Dieudonn\'e module of $V$ has no $\phi$-invariant vectors (for reasons of weight), and thus
 $$ \delta_p(E(H_p)_{L_p}^\varrho) = H^1_{\rm exp}(\Q_p, V_f\otimes V_g^\vee) = H^1_{\fin}(\Q_p, V_f\otimes V_g^\vee) = H^1_{\rm geom}(\Q_p, V_f\otimes V_g^\vee).$$

 

The cup product   in   Galois cohomology combined with the Weil pairing 
$V_f \times V_f \rightarrow \Q_p(1) $ 
 gives rise to the  perfect local Tate pairing
$$ \langle \ , \ \rangle_v: H^1(\Q_v, V_f\otimes V_g) \times H^1(\Q_v, V_f \otimes V_g^\vee) 
\lra H^2(\Q_v, \Q_p(1)) = \Q_p.$$
For all $v$, the finite subspaces
 $H^1_{\fin}(\Q_v,V_f\otimes V_g)$ and $H^1_{\fin}(\Q_v, V_f\otimes V_g^\vee)$ 
 are orthogonal complements of each other under the local Tate pairing.
 Let
  $$ 
  H^1_{\rm sing}(\Q_v, V_f\otimes V_g):= \frac{ H^1(\Q_v, V_f\otimes V_g) }
  {H^1_{\rm fin}(\Q_v, V_f\otimes V_g)}
  $$ 
  denote the  {\em singular quotient} of the local cohomology at $v$, and write
  $$ \partial_v: H^1(\Q, V_f \otimes V_g) \lra
   H^1_{\rm sing}(\Q_v,V_f\otimes V_g)$$
   for the natural projection, which is referred to as the {\em residue map} at $v$.
  Both  the finite part and the singular quotient of  
$H^1(\Q_p, V_f\otimes V_g)$ are two-dimensional over $L_p$.  
By local Tate duality, similar statements hold for the finite part and the singular quotient of 
$H^1(\Q_p, V_f\otimes V_g^\vee)$. 


The Selmer group attached to $(E,V_g^\vee)$ is the subgroup of 
$H^1(\Q,V_f\otimes V_g^\vee)$, denoted
$H^1_{\fin}(\Q,V_f\otimes V_g^\vee)$, 
 which fits into the cartesian square
$$ \xymatrix{
H^1_{\fin}(\Q,V_f\otimes V_g^\vee) \ar[r]  \ar[d] & H^1(\Q,V_f\otimes V_g^\vee)\ar[d] \\
\prod_v H^1_{\fin}(\Q_v, V_f\otimes V_g^\vee) \ar[r]  & \prod_v H^1(\Q_v, V_f\otimes V_g^\vee). }
$$
The Selmer group attached to $(E,V_g)$ is defined likewise,
 by replacing $V_g^\vee$ by $V_g$.

The standard approach  for bounding $E(H)_L^\varrho$ rests on the following statement whose proof is explained in Prop. 6.3 of \cite{DR2}. 
 \begin{lemma}
 \label{lemma:poitou-tate}
 Assume that the residue map $\partial_p$ attached to the $p$-adic representation $V_f \otimes V_{g,\sigma}$ is a surjective map of $L_p$-vector spaces,
 for all embeddings $\sigma:L\lra L_p$. Then $E(H)_L^\varrho=0$. 
 \end{lemma}
  The proof of theorem A has thus been reduced to the problem of constructing
  two independent  classes in the global 
  cohomology $H^1(\Q, V_f \otimes V_g)$ whose images generate  the
singular quotient $H^1_{\rm sing}(\Q_p, V_f \otimes V_g)$.
 Let $\alpha_g$ and $\beta_g$ be the two  (not necessarily distinct)
 roots of the Hecke poynomial 
  $$x^2-a_p(g) x + \chi(p) = (x-\alpha_g)(x-\beta_g)$$
   attached to $g$ at $p$, and let
  $g_\alpha$ and $g_\beta$ denote the associated $p$-stabilisations of $g$ 
  satisfying 
  $$ U_p g_\alpha = \alpha_g g_\alpha, \qquad U_p g_\beta = \beta_g g_\beta.$$
  When $\alpha_g\ne \beta_g$, these stabilisations are distinct,
  and the unramified representation $V_g$ decomposes
  as a direct sum
  $$ V_g = V_g^\alpha \oplus V_g^\beta$$
   of   one-dimensional eigenspaces for the arithmetic frobenius element at $p$ attached to 
   these two
   eigenvalues. When $\alpha_g=\beta_g$, set $V_g^\alpha:= V_g$. 
  Theorem A will now follow from the following result.
  \begin{theorem} 
  \label{thm:main1} 
 There
  exists a    global  cohomology class
  $$\kappa(f,g_\alpha)  \in   H^1(\Q,V_f\otimes V_g)$$
attached to  to the $p$-stabilised form $g_\alpha$, satisfying:
  \begin{enumerate}
    \item
  The residue  $\partial_p(\kappa(f,g_\alpha))$ belongs to
  $H^1_{\rm sing}(\Q_p, V_f\otimes V_g^\alpha)$;
   \item it   is
   non-zero if and only if $L(f\otimes g, 1) \ne 0$.

     \end{enumerate}
      \end{theorem}
To see why Theorem \ref{thm:main1} implies Theorem A, choose the descent prime $p$ in such a way
that the frobenius element $\sigma_p$ acts on $V_g$ with distinct eigenvalues $\alpha_g$ and $\beta_g$. 
Such a choice is always possible, in view of the Chebotarev density theorem and the fact that the two-dimensional $V_g$ is irreducible   and hence
the image of $\varrho$  contains a non-scalar matrix.
When $L(E,\varrho,1)\ne 0$, Theorem \ref{thm:main1} implies that
the residues of the global classes 
$\kappa(f,g_\alpha)$ and $\kappa(f,g_\beta)$   
generate the two-dimensional $L_p$-vector space
$H^1_{\rm sing}(\Q_p, V_f\otimes V_g)$. Since this reasoning applies to 
the representation $V_{g,\sigma}$ for {\em any} embedding $\sigma$ of $L$ into $L_p$,
Theorem A follows from Lemma \ref{lemma:poitou-tate}.

The ``singular" class $\kappa(f,g_\alpha)$  of Theorem 
\ref{thm:main1} 
is not expected to arise from
 the images of algebraic cycles or elements in $K$-theory under an appropriate \'etale Abel-Jacobi 
or regulator map, since such images are typically cristalline, at least in scenarios of good reduction of the ambient variety.
It is   
constructed  instead as a $p$-adic limit of classes
$\kappa(f,g_x)$ attached to (sufficiently many)   weight two 
specialisations $g_x$ of a Hida family specialising to $g_\alpha$ in weight one.
These ``weight two classes" are obtained via  a geometric construction, as the
images of Beilinson-Flach elements under the $p$-adic \'etale regulator map to 
Galois cohomology.


Section 2 describes the construction of a geometric class $\kappa(f,g)$  attached to 
 a classical
eigenform $g$ of {\em  weight two}  and $p$-power nebentypus character,  and summarises its key properties, including the relation between its $p$-adic logarithm and the values of Hida's $p$-adic Rankin $L$-function attached to $f$ and $g$. This relation extends the main result of \cite{BDR1} to settings where the modular curve has bad reduction at the prime $p$, and relies crucially on the work of Nekovar and Niziol \cite{NN} on syntomic cohomology and $p$-adic regulators for arbitrary semistable varieties, which in turn allowed Besser, Loeffler and Zerbes 
\cite{besser-loeffler-zerbes} to generalize Besser's finite polynomial cohomology to this setting.


Section 3   explains how  the weight two
classes $\kappa(f,g)$, as $g$ ranges over all the  classical  specialisations
of weight two
and $p$-power nebentypus character of a Hida family $\underline{g}$, 
 can be interpolated 
 into a
 $\Lambda$-adic cohomology class 
$\kappa(f,\underline{g})$. 
Since these ordinary weight two
 points are dense in the rigid analytic topology on the relevant
 components of the Coleman-Mazur eigencurve,
it follows that  $\kappa(f,\underline{g})$ is uniquely determined by its weight two specialisations.

The class $\kappa(f,g_\alpha)$ of Theorem \ref{thm:main1} is obtained by letting
$\underline{g}$ be a Hida family specialising to $g_\alpha$ in weight one,
and   specialising the resulting    $\kappa(f,\underline{g})$ 
to $g_\alpha$.
The $p$-adic Beilinson formula of Section 2
is then   parlayed into the explicit reciprocity law relating the Bloch-Kato dual exponential of $\kappa(f,g_\alpha)$ 
to the special value of the same $p$-adic $L$-function at the point attached to 
$g_\alpha$. Since this latter point lies in the range of classical 
interpolation defining the $p$-adic $L$-function, it is directly related to a classical $L$-value, and this is what leads to the proof of Theorem \ref{thm:main1}.






 
   
\section{A $p$-adic Beilinson formula }
\label{sec:beilinson-flach}


\subsection{Siegel units and Eisenstein series}\label{units}

 
Fix a modulus $M = N p^s \geq 3$  with $p\nmid N$ and  $s\geq 0$. Let 
$$Y_s:=Y_1(Np^s) \quad \subset \quad X_s := X_1(Np^s) $$
 be  canonical models over $\Q$ of the  (affine and projective, respectively)
  modular curves  classifying triples $(A,i_N,i_p)$
where $A$ is a (generalised) elliptic curve
and $i_N: \mu_N \ra A$ and $i_p: \mu_{p^s} \ra A$ 
are embeddings of finite group schemes.
 The standard notations  $Y_0(Np)$ and $X_0(Np)/\Q$  are adopted for
  the affine and projective
 modular curves  classifying pairs $(A,C_{Np})$ where $C_{Np}\subset A$ is a cyclic subgroup scheme of $A$ of order $Np$. 
 We will denote by $w_M: X_1(M)\lra X_1(M)$ the standard Atkin-Lehner involution,
and will simply  denote it as  $w$ when the level $M$ is clear from the context.

For $a\in (\Z/M\Z)^\times$, let $\mathfrak{g}_{a;M}$ be the {\em Siegel unit } denoted $ g_{0,a/M}$ 
in \cite{Kato}, whose  $q$-expansion is given by
\begin{equation}\label{def-gaM}
\mathfrak{g}_{a;M} := (1-\zeta_M^a) q^{1/12} \prod_{n=1}^\infty (1-\zeta_M^a q^n) (1-\zeta_M^{-a} q^n), \end{equation}
where $\zeta_M$ is a fixed primitive $M$-th root of unity.
The unit   $\mathfrak{g}_{a;M}$ 
can naturally be viewed as belonging to
$\cO_{Y_1(M)}^\times \otimes \Q$, and its $q$-expansion is defined over $\Q(\mu_M)$, while 
the unit 
$$ \mathfrak{g}_{a;M}^w := w \mathfrak{g}_{a;M}$$
is defined over $\Q$.
If $\chi$ is a primitive even Dirichlet character of conductor $M$, taking values in a field denoted 
$\Q(\chi)$, let
$$
\gauss(\chi) = \sum_{a=1}^M \chi(a) \zeta_M^a
$$
denote the Gauss sum attached to  $\chi$ and the choice of $\zeta_M$, and
 set
\begin{equation} 
\label{eqn:def-g-chi}
\mathfrak{g}_{\chi} =  \frac{1}{\varphi(M)} \cdot \sum_{a=1}^M \chi^{-1}(a)\otimes  \mathfrak{g}_{a;M} \in 
\cO_{Y_1(M)}^\times \otimes \Q(\chi), \qquad
\mathfrak{g}_\chi^w := w \mathfrak{g}_\chi.
\end{equation}
Given a pair $(\chi_1,\chi_2)$ of Dirichlet characters and an integer $k\geq 2$ such that $\chi_1\chi_2(-1)=(-1)^k$, define also the Eisenstein series
\begin{equation}
\label{eqn:def-Ek-chi1-chi2}
 E_k(\chi_1,\chi_2) = 
 \left\{
\begin{array}{cl}
\frac{1}{2} \cdot L(\chi_2, 1-k) & \mbox{ if } \chi_1 = 1 \\
0 & \mbox{otherwise} 
\end{array}
\right\}
+ \sum_{n=1}^\infty  \left(\sum_{d|n} \chi_1(n/d) \chi_2(d) d^{k-1}\right) q^n.
\end{equation}

A direct calculation using \eqref{def-gaM}, \eqref{eqn:def-g-chi}
and \eqref{eqn:def-Ek-chi1-chi2}
shows that the logarithmic derivative of $\mathfrak{g}_{\chi}$ is
\begin{equation}
\label{dlog=h}
d\log \mathfrak{g}_{\chi} =  \frac{2\gauss(\chi^{-1})}{\varphi(M)} \cdot E_2(\chi,1) \frac{dq}{q} \in \Omega^1_{Y_1(M)}.
\end{equation}


 









\subsection{Beilinson-Flach elements}\label{sec:BF}
 
The Beilinson-Flach elements of this section are elements in the higher
 Chow group over $\Q$
of the surface
$$S_s := X_0(Np) \times X_s.$$
For   any field $F$, let $C^2(S_s,1)_F$ denote the group of  finite 
formal linear combinations $\D = \sum_i a_i (Z_i,u_i)$ of pairs 
$(Z_i,u_i)$ with coefficients $a_i \in F$, where $Z_i$ are irreducible curves embedded
 in $S_s$ and $u_i$ are rational functions on $Z_i$. Write $Z^2(S_s,1)_F\subset C^2(S_s,1)_F$ for the subspace of elements $\D$ satisfying 
$$
\mathrm{div}(\D) := \sum a_i \mathrm{div}(u_i)=0.
$$

Any pair $\{ \varphi,\psi\}$ of non-zero rational functions on $S_s$ gives rise to
 the element $\sum_i(Z_i,u_i)$ in $Z^2(S_s,1)_F$, where $\{ Z_i\}$ is the set of irreducible components  in $S_s$ of the divisors of $\varphi$ or $\psi$,
 and, after setting $a_i=\ord_{Z_i}(\varphi)$ and $b_i=\ord_{Z_i}(\psi)$, the rational function $u_i$ on $Z_i$ is given by the tame symbol $u_i = \prod (-1)^{a_i b_i} (\frac{\varphi^{b_i}}{\psi^{a_i}})_{|Z_i}$. Writing $B^2(S_s,1)_F$ for the subspace of $Z^2(S_s,1)_F$ formed by such elements, Bloch's higher Chow group of the surface $S_s$ with coefficients in $F$ is defined as
$$
\CH^2(S_s,1)_F =Z^2(S_s,1)_F/B^2(S_s,1)_F.
$$ 



Letting 
\begin{equation}
\label{eqn:tps}
\tilde\pi_{s}: X_s \lra X_0(Np)
\end{equation}
 denote the natural forgetful projection of modular curves which is compatible with the $U_p$ correspondence acting on both curves, define
\begin{equation}
 \label{eqn:def-Deltaf}
\iota_s = (\tilde\pi_{s},{\rm Id}): X_s  \hookrightarrow S_s,
 \end{equation}
to be the closed embedding given by $\iota_s(x)=(\tilde\pi_{s}(x),x)$. 
Let $$\Delta_s  := \iota_s(X_s) \subset S_s$$ denote the resulting embedded curve in $S_s$.


\begin{lemma}
\label{lemma:negligible}
For any $u\in \cO_{Y_s}^\times$, there
exists
a negligible element $\theta_s\in C^2(S_s,1)_F$ satisfying
\begin{equation}
\label{eqn:div-negligible}
 {\rm div} \, \theta_s = {\rm div}(\Delta_s,u).
 \end{equation}
\end{lemma}
\begin{proof} 
The proof is given in Lemma 3.1. of \cite{BDR1}, and relies on the fact the the group
of degree zero divisors supported at the cusps is torsion in the jacobian of $X_s$
by the  Manin-Drinfeld theorem.
 \end{proof}

Thanks to  Lemma \ref{lemma:negligible}, the class
$(\Delta_s, u) - \theta_s$ lies in $Z^2(S_s,1)_\Q$, and  its image
$$ \mathbf{BF}(u) :=  [(\Delta_s, u) - \theta_s] \quad \mbox{ in } \quad \CH^2_{{\mathrm{neg}}}
(S_s,1)_\Q(\Q)
$$
is independent of the particular choice of $\theta_s$ satisfying \eqref{eqn:div-negligible}. 
It is called the {\em Beilinson-Flach element} attached to the modular unit $u\in \cO_{Y_s}^\times$. 

\begin{definition}
\label{def-Du} 
The  elements $$
\mathbf{BF}(a;Np^s) :=  \mathbf{BF}(\mathfrak{g}_{a;Np^s}^w),
 \qquad
\mathbf{BF}_s:=\mathbf{BF}(1;Np^s) \quad \mbox{ in } \quad \CH^2_{{\mathrm{neg}}}(S_s,1)_\Q(\Q)$$ 
are called the {\em Beilinson-Flach elements of level $Np^s$}. 
\end{definition}



\subsection{Etale and syntomic regulators}
\label{sec:regulators}

Let $\varphi = \sum a_n(\varphi) q^n\in S_k(N_\varphi,\chi)$ be a normalized newform of weight $k$, level $N_\varphi$ and character $\chi$, and let $\omega_{\varphi} := \varphi(q)\frac{{\mathrm{d}}q}{q}$ be the associated regular differential form on $X_1(N_\varphi)$.
Let $K_{\varphi}$ be the finite extension of $\Q_p$ generated by the fourier coefficients of $\varphi$, let $\cO_\varphi$ be its valuation ring, and let $V_\varphi$ denote the two-dimensional Galois representation over $\cO_\varphi$ associated by Deligne to $\varphi$. This Galois representation is pure of weight $-1$ and
is characterized up to isomorphism by the property 
that for any prime $\ell\nmid pN_\varphi$, the characteristic polynomial of an arithmetic Frobenius element $\Fr_\ell$ is equal to the Hecke polynomial $x^2-a_\ell(\varphi)x+\chi(\ell)\ell^{k-1}$.
Write $\alpha_\varphi$ and $\beta_\varphi$ be the roots of this Hecke polynomial at $\ell=p$, ordered so that $\ord_p(\alpha_\varphi)\leq\ord_p(\beta_\varphi)$.


 If $N$ is a multiple of $N_\varphi$ and $\mathbb T_N$ is the Hecke algebra generated by  the good Hecke operators $T_\ell$ with 
 $\ell \nmid N$, the $\varphi$-isotypic component of a $\mathbb T_N$-module $S$ is defined to be
\begin{equation}
\label{eqn:hecke-eigenspace}
S[\varphi]  \ \ := \ \  \{ v\in S \otimes \cO_{\varphi} \ : \   T_\ell(v) = a_\ell(\varphi) v \mbox{ for all } \ell \nmid N \}.
\end{equation} 

Generalising  the setting of \S \ref{sec:strategy}, and modifying slightly the notations therein, let
$f\in S_2(Np)[f_0]$ now denote a normalised eigenform with trivial Nebentypus character arising from a newform $f_0$ on $\Gamma_0(N_f)$ for some $N_f \mid N$. In the context of \S \ref{sec:strategy}, $f_0$ is the eigenform associated to the elliptic curve $E$. 
More specifically, we take 
\begin{equation}
\label{eqn:deff}
f(q)= \sum_{d \mid \frac{Np}{N_f}} \lambda_d f_0(q^d)\in S_2(Np)[f_0]
\end{equation}
 to be an eigenvector for all Hecke operators $U_\ell$, $\ell \mid Np$, 
and assume $f$ is ordinary,
i.e., that $\alpha_f = \alpha_{f_0}$ is a $p$-adic unit.


In addition, for the rest of this section
 let $g\in S_2(Np^s,\chi\cdot \chi_p)$ be an eigenform of weight $2$, level $Np^s$ and nebentype character $\chi \chi_p$, that we have factored in such a way that $$\cond(\chi)\mid N,\qquad\mathrm{cond}(\chi_p)\mid p^s.$$ Assume $g$ arises from a newform $g_0$ of level $N_g p^s$ for some $N_g\mid N$ and that the conductor of $\chi_p$ is equal to $p^s$. In the terminology introduced by Mazur-Wiles \cite{mazur-wiles-mc}, $g$ is said to be
 {\em primitive} at $p$. 
Assume also that $g$ is ordinary at $p$, i.e., that $\alpha_g =a_p(g)$ is a $p$-adic unit.


For any variety $X$ over $\Q$, the notation $\bar{X} := X \times \bar\Q$ will be used
to designate the base change of $X$ to $\bar\Q$. 
Let
\begin{equation}
V_0(Np) = H^1_{\et}(\bar{X}_0(Np),\Z_p)(1) \qquad \mbox{ and} \quad V_s = H^1_{\et}(\bar{X}_{s},\Z_p)(1) \, \mbox{ for } \,s\geq 1
\end{equation}
denote the Galois representation afforded by the Tate module of $X_0(Np)$ and $X_s$, respectively.





Let
$$ \pr_{1,1}: H^2_{\et}(\bar{S}_s,\Z_p)(2) \lra  V_0(Np)\otimes V_s$$
denote the natural projection of $G_\Q$-modules induced by the K\"unneth decomposition, and let 
\begin{equation*}
\reg_{\et}: \CH^2(S_s,1)(\Q) \lra H^1(\Q,H^{2}_{\et}(\bar{S}_s,\Z_p)(2)) \stackrel{\pr_{1,1}}{\lra} H^1(\Q,V_0(Np)\otimes V_s)
\end{equation*}
denote the composition of the $p$-adic \'etale regulator map (as defined for instance in chapter 2 of \cite{Fl}) with the map induced by $\pr_{1,1}$ in Galois cohomology. 

The image under $\reg_{\et}$ of the subgroup generated 
by negligible element vanishes, for the cohomology class of such 
elements take values in the K\"unneth component 
$$H^2_{\et}(\bar{X}_0(Np),\Z_p) \otimes H^0_{\et}(\bar{X}_{s},\Z_p)(2)
 \   \oplus   \ 
 H^0_{\et}(\bar{X}_0(Np),\Z_p)\otimes H^2_{\et}(\bar{X}_{s},\Z_p)(2).$$ 
 Hence the \'etale regulator descends to a map which we continue to denote with the same symbol
\begin{equation}\label{AJet}
\reg_{\et}: \CH^2_{\mathrm{neg}}(S_s,1)(\Q) \lra  H^1(\Q,V_0(Np)\otimes V_s).
\end{equation}



\begin{definition}
\label{eqn:def-kappas1}
The   Beilinson-Flach cohomology classes  of level $Np^s$ are defined to be
\begin{eqnarray*}
\kappa(a; Np^s) &:=&  \reg_{\et}(\mathbf{BF}(a;Np^s))\in H^1(\Q, V_0(Np)\otimes V_s), \\
\kappa_s &:=& \kappa(1;Np^s).
\end{eqnarray*}
%,\\
%\kappa_s(f,g) &:=& \varpi_{f g }(\kappa_s) = 
%  \reg^{f g}_{\et}(\mathbf{BF}_s)
%\in H^1(\Q,V_{f g}).
 \end{definition}
 




%\begin{remark}
%Let $V_{f g}^+$ be Greenberg's $3$-dimensional subspace of $V_{f g}$. The finite subspace
%$H^1_{\fin}(\Q_p,V_{f g})$ is identified with the natural image of $H^1(\Q_p,V_{f g}^+)$ in $ H^1(\Q_p,V_{f g})$.
%\end{remark}

We now turn to providing an analytic description of the restriction to $G_{\Q_p}$ of the classes $\kappa_s$, which will pave the way for establishing its relation to the values of the Hida-Rankin $p$-adic $L$-function associated to Hida families passing through $f$ and $g$. A first step towards this task is the description of  $H^1_{\fin}(\Q_p,V_{f g})$ in terms of the de Rham cohomology of the relevant modular curves. 




Recall   Fontaine's ring $\hB_{\cris}$ of cristalline periods, and let
\begin{equation}
D_f := (\hB_{\cris}\otimes_{\Q_p} V_{f})^{G_{\Q_p}}, \quad 
D_{g}:=(\hB_{\cris}\otimes_{\Q_p} V_{g})^{G_{\Q_p(\mu_{p^s})}}, \quad
D_{f g}:=(\hB_{\cris}\otimes_{\Q_p} V_{f g})^{G_{\Q_p(\mu_{p^s})}}
\end{equation}
be the (potentially crystalline) Dieudonn\'e modules associated to  
 $V_f$, $V_g$ and $V_{fg}$, respectively. Note that $D_{f g} \simeq D_f \otimes D_g$ in the category
of filtered frobenius vector spaces over the field $K$.
Let 
\begin{equation}
\label{eqn:g-star}
g^* = \bar g=\sum \bar{a}_n(g) q^n  \in S_2(Np^s,\chi^{-1} \chi^{-1}_p), \qquad g^w := w g
\end{equation} 
be the modular forms which both belong to the automorphic representation attached to the 
contragredient of $g$;
these modular forms are  eigenvectors for the good  Hecke operators, and
 $$V_{g^*}=   V_g\otimes \chi^{-1}\chi_p^{-1} = V_g^\vee(1).$$
The Dieudonn\'e module of the Kummer dual $V_{fg}^\vee(1) = V_{fg^*}(-1)$ of $V_{fg}$, denoted 
 $D_{fg^*}(-1)$, is canonically
identified with $D_{fg^*}$ but with a shift in the Hodge filtration. The dimensions of the relevant
pieces of the  
  Hodge filtration on $D_{fg}$ and $D_{fg^*}(-1)$ are summarised in the table below:
 $$ \begin{array}{c|ccccc}
            & \Fil^{-2} & \Fil^{-1} & \Fil^0  & \Fil^1 & \Fil^2 \\ \hline
 D_{fg} &     4       &      3      &    1      & 0        & 0 \\
 D_{fg^*}(-1)& 4       & 4            & 3         & 1       & 0. 
 \end{array} 
 $$
 
 The Bloch-Kato logarithm  
$\log_p$  of \cite{bloch-kato} associated  to the Galois representation
$V_{fg}$  gives an isomorphism
 $$
 \log_p: H^1_{\rm exp}(\Q_p(\mu_{p^s}), V_{fg}) \lra 
 \frac{ D_{fg} } {\Fil^0 D_{fg} + D_{fg}^{\phi=1}},$$
 where $H^1_{\rm exp}(\Q_p(\mu_{p^s}), V_{fg})$ is as defined in \eqref{eqn:def-efg}, 
 with $\Q_p$ replaced by $\Q_p(\mu_{p^s})$.
 The cristalline frobenius $\phi$ acts on $D_{fg}$ with eigenvalues 
 \begin{equation} 
 \label{eqn:phi-Vfg}
 \alpha_f \alpha_g,  \qquad 
 \alpha_f \beta_g,   \qquad \beta_f \alpha_g, \qquad 
 \beta_f \beta_g.
 \end{equation}
These eigenvalues are algebraic numbers of absolute value $p$ (relative to any complex embedding of $\bar\Q$)
 and therefore
 $D_{fg}^{\phi=1} = 0$. It follows from  \cite[Cor.~3.8.4]{bloch-kato}
  that 
\begin{equation}
\label{eqn:hfin=hexp}
H^1_{\rm exp}(\Q_p(\mu_{p^s}), V_{fg}) = H^1_{\fin}(\Q_p(\mu_{p^s}), V_{fg}).
\end{equation}
The Bloch-Kato logarithm can therefore be recast in the current context as a map
\begin{equation}
\label{eqn:logp}
\log_p:H^1_{\fin}(\Q_p(\mu_{p^s}), V_{f g})\lra  \frac{D_{fg}}{\Fil^0 (D_{fg})} = 
\Fil^0(D_{f g^*}(-1))^\vee,
\end{equation}
where the last identification arises from the natural duality between $D_{fg}$ and
$D_{fg^*}(-1)$, relative to  which $\Fil^0(D_{fg})$ and $\Fil^0(D_{fg^*}(-1))$ are orthogonal complements
 of each other. 
 
Assume furthermore that none of the eigenvalues in \eqref{eqn:phi-Vfg} are equal to $p$. 
 This assumption  is  satisfied with at most finitely many exceptions
 when $f$ is fixed and $g$ runs over the  weight two specialisations of a
 Hida family, hence will suffice for the applications in this paper. 
 It implies, by another  application of 
 \cite[cor.~3.8.4]{bloch-kato}, that
 $H^1_{\rm exp}(\Q_p(\mu_{p^s}), V_{fg^*}(-1)) = H^1_{\fin}(\Q_p(\mu_{p^s}), V_{fg^*}(-1))$, and hence,
 by duality, that 
\begin{equation}
\label{eqn:hfin=hgeom} 
 H^1_{\rm fin}(\Q_p(\mu_{p^s}),V_{fg}) = H^1_{\rm geom}(\Q_p(\mu_{p^s}),V_{fg}).
\end{equation}
The argument used in the proof of Lemma 2.8 of \cite{Fl} shows that the image of the \'etale regulator 
$\reg_{\et}^{fg}$ consists of extensions of Galois representations occurring in the \'etale cohomology of 
varieties with potentially semistable reduction at $p$, which hence are de Rham at $p$.
 This image is therefore  contained
   in $ H^1_{\rm geom}(\Q_p(\mu_{p^s}),V_{fg})$, and thus  by \eqref{eqn:hfin=hexp}
 in the finite subspace of the local cohomology.

%The {\em $p$-adic syntomic regulator} map $\reg_p^{fg}$ 
%is defined by composing $\log_p$ with  the \'etale regulator $\reg_{\et}^{f g}$ 
%used to construct the Beilinson-Flach classes:
%\begin{equation}
%\label{eqn:regpfg}
%\xymatrix{
%\reg_p^{f g}:\CH^2(S_s,1)(\Q_p(\mu_{p^s}))
%\ar[r]^{\ \ \ \  \ \reg_{\et}^{f g}} &  H^1_{\fin}(\Q_p(\mu_{p^s}),V_{f g}) \ar[r]^{ \ \log_p \quad \quad \quad \quad } &  \Fil^0(D_{f g^*}(-1))^\vee  =   \Fil^{-1}(D_{f g^*})^\vee.     }
%\end{equation}

 
 Let 
\begin{equation}
 D_0(Np) :=  (\hB_{\dR}\otimes V_0(Np))^{G_{\Q_p}}, \qquad 
D_s := (\hB_{\dR}\otimes V_s)^{G_{\Q_p(\mu_{p^s})}}, 
\end{equation}
be the de Rham Dieudonn\'e filtered modules attached to the  Galois representations 
$V_0(Np)$ and $V_s$.
Note that  these $p$-adic Galois representations are $\hB_{\dR}$ but
 not $\hB_{\cris}$-admissible, but that
their $f_0$ and $g_0$-isotypic parts, denoted $V_0(Np)[f]\simeq V_f^i$ and 
$V_s[g] \simeq V_g^j$, are cristalline over $\Q_p$ and $\Q_p(\mu_{p^s})$ respectively.


The \'etale regulator map \eqref{AJet} on $ \CH^2(S_s,1)_{\mathrm{neg}}$ and the Bloch-Kato logarithm associated to the Galois representation $V_0(Np)\otimes V_s$ give rise to a diagram of maps
\begin{equation}
\label{eqn:regp}
\xymatrix{
 \CH^2(S_s,1)_{\mathrm{neg}}(\Q_p(\mu_{p^s}))
\ar[r]^{ \!\!\!\!\!\!\!\!\! \!\!\!\! \reg_{\et}} &  H^1_{\rm geom}(\Q_p(\mu_{p^s}),V_0(Np)\otimes V_s)   &   \\     
  & H^1_{\exp}(\Q_p(\mu_{p^s}),V_0(Np)\otimes V_s) \ar[r]^{ \ \  \ \ \ \ \ \ \ \ \ \  \log_p \quad \quad \quad \quad }  
\ar@{_{(}->}[u] 
&  \Fil^0(D_0(Np)\otimes D_{s}(-1))^\vee.}
\end{equation}


Let 
\begin{equation}
\label{eqn:regp-bis}
\xymatrix{
 \reg_p^{f g}: \CH^2(S_s,1)(\Q_p(\mu_{p^s}))[fg]
  \ar[r]  & \Fil^0(D_0(Np)[f]\otimes D_{s}(-1)[g^*])^\vee}
\end{equation}
be the map obtained by applying the $(f,g)$-isotypic projection to \eqref{eqn:regp}, invoking \eqref{eqn:hfin=hexp}  and   \eqref{eqn:hfin=hgeom} to 
show that the vertical inclusion becomes an isomorphism after restricting to these $(f,g)$-isotypic subspaces.


The main goal of the next sections is proving an analytic formula for the value of $\reg_p^{f g}(\mathbf{BF}_s)$ at a suitable vector in $\Fil^0(D_0(Np)[f]\otimes D_{s}(-1)[g^*])$, which we now proceed to describe.


 By the
comparison theorem between \'etale and de Rham cohomology, there are isomorphisms
 of two-dimensional filtered frobenius modules
\begin{equation}\label{comp-iso}
D_f  \simeq H^1_{\dR}(X_0(N_f)_{/\Q_p})(1)[f_0], \quad D_g  \simeq H^1_{\dR}(X_1(N_g p^s)_{/\Q_p(\mu_{p^s})})(1)[g_0]
%D_{\st}(V_0(N p)) \simeq H^1_{\dR}(X_0(Np)_{/\Q_p})
\end{equation}
such that $$\Fil^0 D_f \simeq \Omega^1(X_0(N_f))[f_0] = \langle \omega_{f_0}\rangle \quad \mbox{and} \quad \Fil^0 D_g \simeq \Omega^1(X_1(N_g p^s))[g_0]  =  \langle \omega_{g_0}\rangle .$$



For any projective curve $X/F$, write
$$\langle\ \ ,\ \ \rangle_{X}: H^1_{\dR}(X_{/F})\times H^1_{\dR}(X_{/F})\lra F$$
for the canonical Poincar\'e pairing on $X$. Let $\eta_{f_0}$ be an element in the one-dimensional unit-root subspace of 
$D_f$,
 normalised so that $$\langle \omega_{f_0}, \eta_{f_0}\rangle_{X_0(N_f)} = 1.$$ 

Fix a finite extension $K/\Q_p$ containing $K_f$ and $K_g$, and let $\cO$ denote its valuation ring. For each positive divisor $d$ of $Np/N_f$, write $\pi_d:X_1(Np)\lra X_1(N_f)$ for the standard degeneracy map induced by multiplication by $d$ on the upper half plane. 
Let $$\pi_{d\ast}:V_0(Np)[f_0]\lra V_f,\qquad V_f:=V_0(N_f)[f_0]=H^1_{\et}(\bar X_0(N_f),\cO)(1)[f_0]$$ be the map in \'etale cohomology induced by $\pi_d$ by covariant functoriality.
Our choice of $f$ in  $S_2(Np)[f_0]$ stated in equation \eqref{eqn:deff} determines the projection 
\begin{equation}\label{varpif}
\varpi_f := \sum \lambda_d \pi_{d\ast}: V_0(Np)\otimes \cO\lra V_f.
\end{equation}

Committing a slight abuse of notation,  denote 
by $\varpi_f$ 
the map induced by \eqref{varpif} by functoriality on Dieudonn\'e
modules and the first isomorphism in \eqref{comp-iso},
and write   $\varpi_f^\ast$ for the dual map arising from Poincar\'e duality:
\begin{equation}
\label{eqn:de-Rham-varpi}
\varpi_f: D_0(Np) \lra D_f, \qquad \varpi_f^*: D_f \lra D_0(Np).
\end{equation}


%A similar discussion applies to $g$, and we write $
%\varpi_g: V_s\otimes \cO\lra V_g$ for the projection induced by $g$ as in 
%\eqref{varpif}.



%Define $\varpi_{f g} = \varpi_f \otimes \varpi_g:V_0(Np)\otimes V_s\otimes \cO\lra V_{f g}:=  V_f\otimes_{\cO} V_g$
%and let
%\begin{equation}
%\label{AJetfg} 
%\reg_{\et}^{f g}: \CH^2_{\mathrm{neg}}(S_s,1)(\Q) \stackrel{\reg_{\et}}{\lra} H^1(\Q,V_0(Np)\otimes V_s \otimes \cO) \stackrel{\varpi_{f g }}{\lra} H^1(\Q,V_{f g})
%\end{equation}
%denote the composition of the \'etale regulator map of \eqref{AJet} with $\varpi_{f g}$.




The maps $\varpi_{g}: D_s \lra D_g$ and $\varpi_{g}^*: D_g\lra D_s$ 
are defined similarly, and we  let
$$
\varpi_f^\ast\otimes \varpi_{g}^\ast: D_{fg} \,\lra  D_0(Np)\otimes D_s
$$
denote the map induced by $\varpi_f^*$ and $\varpi_{g}^*$ and the K\"unneth decomposition. 

Define 
$$\eta_f := \varpi_f^* (\eta_{f_0}) \in D_0(Np)[f], \qquad 
\omega_{g} := \varpi_{g}^*(\omega_{g_0}) \in D_s[g], \qquad \omega_{g^w} :=  w \omega_g \in D_s[g^*].$$    
Note that $\eta_f$ is ordinary, by the compatibility of $\varpi_f^*$ with the $U_p$-operator. 
Since $\eta_{f}$ belongs to $ \Fil^{-1}(D_f)=D_f$ and $\omega_{g^w}$ belongs to 
$ \Fil^0(D_{g^*}) \subset \Omega^1(X_s)$, the tensor product
 $\eta_{f}\otimes\omega_{g^w}$ belongs to $\Fil^{-1}(D_0(Np)[f]\otimes D_s[g^*])$.
Because of the assumption on $g$ leading to \eqref{eqn:hfin=hgeom}, the expression
 $$\reg_p^{f g}(\mathbf{BF}_s)(\eta_f \otimes \omega_{g^w}) $$ is defined. The main result of \S \ref{sec:analytic_description} is a formula for
 its value. 
%We now turn to describing the   image of the 
%Beilinson-Flach element $\mathbf{BF}_s$ under the 
%$p$-adic regulator map $\reg_p^{f g}$. 
As a preparation for that, we first recall in \S \ref{sec:cohomology}  
a few needed facts on the rigid geometry of the modular curve $X_1(Np^s)$.




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

This section briefly summarises the description of
 the de Rham cohomology of  $X_s$
 viewed as a curve over a $p$-adic  field, 
following the treatment in \cite[\S 3.1]{DR2},
with suitable modifications aimed at giving a similar 
description of 
 the cohomology of the affine curve $Y_s$.
 We adopt the same notations as in loc.cit., which the reader is encouraged to
 consult for a more detailed description of the objects that are used here.

More specifically, let $F$ be any extension of $\Q_p(\mu_{p^s})$ over which 
$X_s$ admits a semistable model, and over which the closed points of
 $\Sigma := X_s - Y_s$
are defined.  
 
The \'etale cohomology groups of $X_s$ and $Y_s$ are related by
 the Gysin exact sequence of semistable representations of
 $G_F=\Gal(\bar F/F)$:
 \begin{equation}
 \label{eqn:gysin-et}
 0\lra H^1_{\et}(\bar X_s,\Q_p)\lra H^1_{\et}(\bar Y_s,\Q_p)\lra \Q_p(-1)^t\lra
 0,
 \end{equation}
 where $t=|\Sigma|-1$.
%, and is naturally embedded with codimension one in the group $\oplus_{x\in \Sigma} H^1_{\et}(\bar Y_{ x},\Q_p(-1))$.

 Let $D_{\st}$ be Fontaine's comparison functor attached to the ring of semistable periods $\hB_{\st}$. By applying $D_{\st}$
 to the sequence \eqref{eqn:gysin-et}, and invoking the comparison
 theorem between \'etale and de Rham cohomology, one obtains a
 corresponding sequence of filtered Frobenius monodromy (FFM) modules
 \begin{equation}
 \label{eqn:gysin-dR}
 0\lra H^1_{\dR}({X_s}_{/F})\lra^{\!\!\!\!\!\!\!\!\iota}\ \   H^1_{\dR}({Y_s}_{/F})\lra F(-1)^t\lra
 0.
 \end{equation}
 (See for example \cite{CI-hidden}, Proposition 7.6 for details.)
Note that the above FFM modules are isomorphic as filtered Frobenius modules to those obtained by applying the construction of the potentially crystalline Dieudonn\'e module of Section \ref{sec:regulators},
in which $\hB_{\st}$ is replaced by $\hB_{\cris}$. The use of $\hB_{\st}$ equips these modules with the additional structure of a monodromy operator. 


 In particular, it follows from \eqref{eqn:gysin-dR} that the Frobenius operator
 $\Phi$ associated to $Y_s$ acts on the subspace of $H^1_{\dR}({Y_s}_{/F})$
 generated by
 classes of Eisenstein series with eigenvalues of
 absolute value $p$.

 In order to make \eqref{eqn:gysin-dR} more explicit, we recall,
 following \cite{CI-hidden},
 the Maier-Vietoris exact sequences describing the de Rham
 cohomology groups of $X_s$ and $Y_s$.
 Fix a proper semistable model $\cX_s$ of $X_s$ over the ring of integers $\cO_F$ of $F$, whose
 special fiber $C$ is a union of smooth geometrically connected
 components $C_1,\hdots,C_n$ intersecting at ordinary double
 points defined over the residue field $\F$ of $F$. Since
 $s\geq 1$, there are at least two components, i.e.,
 $n\geq 2$. Write $\cG$ for the dual graph of $C$, and $\cV$, resp.\ $\cE$
 for its set of vertices, resp.\ oriented edges. Recall that
 $$\cV=\{C_1,\hdots,C_n\},\qquad \cE=\{(x,C_i,C_j):x\in C_i\cap
 C_j\}.$$ Given $e=(x,C_i,C_j)$ in $\cE$, we say that $s(e):=C_i$ is the source
 of $e$ and $t(e):=C_j$ is the target of $e$. Write
 $$\red:\, X_s(\C_p)\lra C(\bar\F)$$ for the reduction map, and
 identify $X_s(\C_p)$ with the rigid-analytic space over $F$ attached to
 $X_s$. The wide-open space associated to $v\in\cV$ is defined as
 $\cW_v:=\red^{-1}(v)$. Its underlying connected affinoid space $\cA_v$ is
 obtained as the inverse image by reduction of the component $v$ with the singular
 points removed. The wide-open annulus associated to
 $e=(x,v_i,v_j)\in\cE$ is defined as $\cW_e:=\red^{-1}(x)$ (so
 that $\cW_e=\cW_{v_i}\cap\cW_{v_j}$). The collection
 $\{\cW_v\}_{v\in\cV}$ forms an
 {\em admissible covering}  of $X_s(\C_p)$ by wide open spaces. Likewise, an admissible covering of
 $Y_s(\C_p)$ is given by $\{\cW_v'\}_{v\in\cV}$, where
 $\cW_v':=\cW_v-\Sigma$. Note that
 $$\cW_{v_i}'\cap\cW_{v_j}'=\cW_{v_i}\cap\cW_{v_j}=\cW_e,$$ with
 $e=(x,v_i,v_j)\in\cE$.
 Since $H^1_{\dR}(X_s/F)$ and $H^1_{\dR}({Y_s}_{/F})$ are
 identified with the Cech hypercohomology groups of their associated
 admissible coverings, they arise as the middle terms in the following commutative
 diagram originating from the Maier-Vietoris sequences attached to these
 coverings:

 \begin{equation}
 \label{eqn:mv}
 \xymatrix{ 0  \ar[r]^{}
  & \frac{\oplus_{e\in\cE} H^0_{\dR}(\cW_e)}{\delta(\oplus_{v\in\cV}
  H^0_{\dR}(\cW_v))}
  \ar[r]^{i\!\!\!\!\!\!\!\!\!\!}\ar[d]_{=}
  & H^1_{\dR}(X_s/F)
  \ar[r]^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
  \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! r}
  \ar[d]_{\iota}
  & {\ker\left(\oplus_{v\in\cV}H^1_{\dR}(\cW_v)
  \lra^{\!\!\!\!\!\!\!\!\delta}\ \ \oplus_{e\in\cE}
  H^1_{\dR}(\cW_e)\right)}
  \ar[r]^{}\ar[d]_{\bar \iota}
  & 0 \\
  0  \ar[r]^{}
  & \frac{\oplus_{e\in\cE} H^0_{\dR}(\cW_e)}{\delta(\oplus_{v\in\cV}
  H^0_{\dR}(\cW_v))}\ar[r]^{i'\!\!\!\!\!\!\!\!\!\!}
  & H^1_{\dR}({Y_s}_{/F}) \ar[r]^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
  \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!r'}
  & \ker\left(\oplus_{v\in\cV}H^1_{\dR}(\cW_v')
  \lra^{\!\!\!\!\!\!\!\!\delta}\ \  \oplus_{e\in\cE}
  H^1_{\dR}(\cW_e)\right)
  \ar[r]^{}
  & 0  }
 \end{equation}

 In \eqref{eqn:mv}, the various coboundary maps $\delta$ are
 defined by the rule $\delta\{\alpha_v\}(e):
 =\alpha_{t(e)}|_{\cW_e}-\alpha_{s(e)}|_{\cW_e}$,
 and we have used the fact that
 $H^0_{\dR}(\cW_v)=H^0_{\dR}(\cW_v')$, since $\Sigma$ is finite.
 Furthermore, the map $i$ sends the class of a collection $\{\lambda_e\}$ of scalars
with $\lambda_e\in\C_p$  to the class
 of the hypercocycle $(\{0_v\},\{\lambda_e\})$, while the map $r$ sends the
 class of the hypercocycle $(\{\omega_v\},\{f_e\})$ to the class of
 $\{\omega_v\}$. Likewise for $i'$ and $r'$.

\subsection{Analytic description of the syntomic regulator}
\label{sec:analytic_description}

We now give an analytic description of the image under the $p$-adic syntomic regulator of the
Beilinson-Flach element $\mathbf{BF}_s$ attached to the Siegel unit $\mathfrak{g}_{1,Np^s}$.
Recall the modular forms $f\in S_2(Np)$ and  the primitive 
form 
$g\in S_2(Np^s,\chi \chi_p)$ 
considered in \S \ref{sec:regulators}, arising from newforms $f_0$ and $g_0$ of level $N_f$ and $N_g p^s$, respectively. 

\begin{lemma}
\label{lemma:gchi}
We have the equality
$$
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})=
\reg_p^{f g}(\mathbf{BF}(\mathfrak{g}_{\chi^{-1}\chi_p^{-1}}^w))(\eta_f\otimes\omega_{g^w}).
$$
\end{lemma}
\begin{proof}
This follows by noting that 
$$\mathbf{BF}_s=\sum \mathbf{BF}(\mathfrak{g}_{\psi}^w),$$
where the sum is being taken over the even characters $\psi$ of modulus $Np^s$.
It follows from the equivariance of $\reg_p$ under the group of diamond operators acting functorially on both source and target
 that 
$\reg_p^{f g}(\mathbf{BF}(\mathfrak{g}_{\psi}^w))(\eta_f\otimes\omega_{g^w})$ is zero for all $\psi\ne\chi^{-1} \chi^{-1}_p$.
\end{proof}
In light of this lemma, let
\begin{equation}
\label{eq:def-of-h}
\omega_{\cE} :=  \cE(q) \frac{dq}{q} =  \frac{\varphi(Np^s)}{2} \cdot  \dlog {\mathfrak{g}}_{\chi^{-1}\chi^{-1}_p} =  \mathcal{G}(\chi\chi_p)  E_2(\chi^{-1}\chi^{-1}_p,1) \frac{dq}{q}, \qquad \cE   \in M_2(Np^s,\chi^{-1}\chi^{-1}_p) 
\end{equation}
be the differential of the third kind on $X_1(Np^s)$ and the associated
 Eisenstein series introduced in  equation \eqref{dlog=h}
of \S \ref{units}. 



Choose a polynomial $P(x)\in F[x]$ such that the following
 conditions hold:

 \begin{enumerate}
 \item
\label{eqn:frob-annihilation}
 $P(\Phi\times\Phi)$ annihilates the class of
 $\omega_{g^w}\otimes \omega_{\cE^w}$ in
 $H^2_{\dR}({Y_s^2}_{/F})$,
 \item
\label{eqn:frob-invertible}
 $P(\Phi)$ acts invertibly on $H^1_{\dR}({Y_s}_{/F})$.
 \end{enumerate}
 The polynomial $P$ exists because $\Phi\times\Phi$ acts
with
 eigenvalues of complex absolute value $p^{3/2}$
 on the
 subspace of $H^2_{\dR}({Y^2_s}_{/F})$
 generated by the frobenius translates of 
classes of the form $\omega_{g^w}\otimes \omega_{\cE^w}$, while $\Phi$ acts
 on $H^1_{\dR}({Y_s}_{/F})$ with eigenvalues of absolute value
 $1$, $\sqrt{p}$ and $p$ (as is deduced for example by analysing the
 Mayer-Vietoris sequence \eqref{eqn:mv}).

Just as in \cite[\S 3.2]{DR2}, condition \eqref{eqn:frob-annihilation}
 in the choice of $P$ means that $P(\Phi\times\Phi)
 \left(\omega_{g^w}\otimes \omega_{\cE^w}\right)$ is exact and
  hence there exists a system of rigid one-forms
 $$\rho_{_{\!P}}(g^w,\cE^w) = \{ \rho_{_{\!P}}^{v_1,v_2}(g^w,\cE^w)\in\Omega^1_{\rig}(\cW_{v_1}'\times \cW_{v_2}') \}$$
 indexed by pairs $(v_1,v_2) \in \cV\times \cV$, 
 satisfying
 \begin{equation}
 \label{eqn:coleman-besser}
 d\rho_{_{\!P}}^{v_1,v_2}(g^w,\cE^w)=P(\Phi\times\Phi)
 \left(\omega_{g^w}\otimes \omega_{\cE^w}{\ |_{\cW_{v_1}'\times \cW_{v_2}'}}\right).
 \end{equation}

Note that the choice of the system of one-forms $\rho^{v_1,v_2}_{_{\!P}}(g^w,\cE^w) $ is not unique, as  it is  well-defined only up to systems of closed rigid  one-forms on
 $\cW_{v_1}'\times \cW_{v_2}'$. Nevertheless, we shall use this system to construct a canonical class in $H^1_{\dR}(X_s/F)$ associated to $g^w$ and $\cE^w$, independent of all
  the choices made.



 For any  $a\in (\Z/Np^s\Z)^\times$, let
  $\langle a\rangle \in \Aut(Y_s)$ denote
 the  diamond operator associated to $a$ on $Y_s$, and for any pair $(a,b)$ of such elements let
$\langle a,b\rangle $ denote the automorphism of $Y_s\times Y_s$ given by the product of $\langle a \rangle$ and $\langle b \rangle$ acting on the first and second variable, respectively. 

Fix an $a\in (\Z/Np^s\Z)^\times$ such that $\chi \chi_p(a)\not=1$ for the rest of the section, and define the linear homomorphism
\begin{equation}\label{eqn:correspondence}
\delta^{\ast}_a: \oplus_{v_1,v_2}\Omega_{\rig}^1(\cW_{v_1}'\times\cW_{v_2}')\lra\oplus_v\Omega_{\rig}^1(\cW_v'), \quad  \delta_a^* =  \diag^* \circ ({\rm Id}-\langle a,1\rangle^*)\circ({\rm Id}-\langle 1,a\rangle^*)
 \end{equation}
where $\diag^*$ denotes the map induced by pull-back under the diagonal
 embedding $\cW_v'\lra \cW_v'\times \cW_v'$. 

\begin{lemma}
The map $\delta_a^\ast$ sends collections of closed $1$-forms to collections of exact $1$-forms.
\end{lemma}
\begin{proof} 
By the K\"unneth formula, there is a decomposition $$H^1_{\dR}({\cW_v'\times\cW_v'}) \simeq (H^0\otimes H^1) \, \oplus \, (H^1\otimes H^0),$$ where we use the shorthands
$H^0 = H^0_{\dR}({\cW_v'})$ and $H^1 = H^1_{\dR}({\cW_v'})$.
Since the endomorphism ${\rm Id}-\langle a,1\rangle^*$ vanishes on the first factor, and ${\rm Id}-\langle 1,a\rangle^*$ vanishes on the second, the claim follows.
\end{proof}

\begin{corollary}
The image $\Xi_{_{\!P,a}}(g^w,\cE^w)$ of the system of closed rigid $1$-forms $\delta_a^\ast(\rho_{_{\!P}}(g^w,\cE^w))$  in $\oplus_v H^1_{\dR}(\cW_v')$ does not depend on the choice of solution of the differential equation \eqref{eqn:coleman-besser}.
\end{corollary}



As the notation indicates, the class $\Xi_{_{\!P,a}}(g^w,\cE^w) $ does depend on both our choices of
 $P$ and $a$. In order to eliminate this ambiguity, define the class
 \begin{equation}
 \label{eqn:xiP}
 \Xi_{_{\!P}}(g^w,\cE^w)
 :=(1- \chi\chi_p(a))^{-1}(1-\chi^{-1}\chi^{-1}_p(a))^{-1}
 \Xi_{_{\!P,a}}(g^w,\cE^w)\in
 \oplus_{v\in\cV}H^1_{\dR}(\cW_v').
 \end{equation}
This class still depends on $P$, but  is independent of the
 choice of $a$ as above. Indeed, this follows from the very definition of $ \Xi_{_{\!P}}(g^w,\cE^w)$ and the fact that $\langle a \rangle$ acts on the $g^w$-isotypic (resp.\,$\cE^w$-isotypic) component of $H^1_{\dR}({Y_s}_{/F})$ as multiplication by $\chi^{-1}\chi_p^{-1}(a)$ (resp.\,$\chi \chi_p(a)$).

Following \cite[Def.~\!3.1]{DR2},
define the subspace $H^1_{\dR}({X_s}_{/F})^{(1)}$
of {\em pure classes of weight one}  of $H^1_{\dR}({X_s}_{/F})$
by the condition that $\Phi$ acts via eigenvalues of complex absolute value $\sqrt{p}$ on its elements. Likewise, define the subspace $\oplus_v H^1_{\dR}(\cW_v')^{(1)}$ of $\oplus_v H^1_{\dR}(\cW_v')$ by the same condition. Weight considerations show that 
$\oplus_v H^1_{\dR}(\cW_v')^{(1)}$ is contained in the kernel of the map $\delta$
 appearing in the commutative diagram \eqref{eqn:mv}, 
and that the map $r'\circ \iota$ defined in the same diagram
 induces an isomorphism between the above spaces of pure weight one classes. Denote by 
$${\rm spl}_{X}:\oplus_v H^1_{\dR}(\cW_v')^{(1)}\lra H^1_{\dR}({X_s}_{/F})^{(1)}$$ 
the (Frobenius-equivariant) inverse isomorphism. In light of condition \eqref{eqn:frob-invertible}
in our choice of $P$, define the 
 class
\begin{equation}\label{def-xi} 
\Xi(g^w,\cE^w):=P(\Phi)^{-1}{\rm spl}_{X}
 \left(\Xi_{_{\!P}}(g^w,\cE^w)\right)\in H^1_{\dR}(X_s/F)^{(1)}.
\end{equation}
It is immediate to check that $\Xi(g^w,\cE^w)$ not depend on our choice of $P$.

Let $\pi_s$ and $\varpi_s$ be the natural degeneracy maps arising in the diagram
$$\xymatrix {
\tilde\pi_s:X_s\ar[r]^{\!\!\!\!\!\!\pi_s}&
X_0(Np^s)\ar[r]^{\varpi_s}& X_0(Np),
}$$
where $\tilde\pi_s$ is the morphism appearing in equation \eqref{eqn:tps}.
The degeneracy map $\varpi_s$ that we choose is the unique one which is compatible with the $U_p$ operators in
level $Np^s$ and $Np$. 
Let
$$\eta_{f,s}:=\varpi_s^\ast \eta_f,\qquad \tilde\eta_{f,s}:=\pi_s^\ast\eta_{f,s}.$$
The class $\eta_{f,s}$ is ordinary, by the ordinariness of $\eta_f$ combined with the compatibility of $\varpi_s$ with $U_p$. 
\begin{proposition}
\label{prop:poincare}
The equality 
\begin{equation}
\label{formula:poincare}
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})=\frac{2}{\varphi(Np^s)}
\langle  \tilde\eta_{f,s},\Xi(g^w,\cE^w)\rangle_{X_s}
\end{equation}
holds.
\end{proposition}
\begin{proof} This formula follows by applying the same argument as in the proof of \cite[Proposition 3.3]{BDR1}, with the following difference. While in loc.\,cit.\, we work with the modular curve $X_1(N)$ which admits a smooth model over $\Z_p$, here the modular curve $X_s$ has bad reduction at $p$ and we are bound to work with a semistable model of this curve. Besser's theory of finite polynomial cohomology has been extended recently to the setting of semistable varieties by Besser, Loeffler and Zerbes in \cite{besser-loeffler-zerbes}, building on previous work of Nekovar and Niziol \cite{NN}. 
The proof of \cite[Proposition 3.3]{BDR1}  applies verbatim after replacing 
Besser's \cite[Prop.\,6.3]{besser-K1S} with \cite[Proposition 3.12]{besser-loeffler-zerbes},  taking into account that
$$
\omega_{\cE^w} =  \frac{\varphi(Np^s)}{2} \cdot d\log (\mathfrak g_{\chi^{-1}\chi^{-1}_p}^w).
$$
\end{proof}

The fact that the nebentype characters of $g^w$ and $\cE^w$ are inverse one of another implies that 
the class $\Xi(g^w,\cE^w)$ arises as the pull-back of a class -- denoted with the same symbol by abuse of notation -- in the cohomology of $X_0(Np^s)$ under $\pi_s$. 
\begin{proposition}
\label{prop:poincare-bis}
For all $s\ge 1$,
\begin{equation}
\label{formula:poincare-bis}
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})=  
\langle  \eta_{f,s},\Xi(g^w,\cE^w)\rangle_{X_0(Np^s)}.
\end{equation}
\end{proposition}
\begin{proof}
Because the degree of $\pi_s$ is $\frac{1}{2}\varphi(Np^s)$, and 
the pullback and pushforward maps are adjoint to each other
relative to the Poincar\'e pairing,
\begin{equation}\label{i1}
\langle \tilde\eta_{f,s},\Xi(g^w,\cE^w)\rangle_{X_s} = \frac{1}{2}\varphi(Np^s) \langle \eta_{f,s},\Xi(g^w,\cE^w)\rangle_{X_0(Np^s)}.
\end{equation}
The result follows. 
\end{proof}

It will be convenient to consider the anti-ordinary counterparts of the classes $\eta_f$ and $\eta_{f,s}$:
$$\eta_f^w:=w \eta_f,\qquad \eta_{f,s}^w:=w \eta_{f,s},$$
where we recall that $w$ refers in the above equations to the  
 Atkin-Lehner involutions $w_{Np}$ and $w_{Np^s}$ of levels   $Np$ and $Np^s$ respectively.
 

Let $\{\cW_\infty,\cW_0\}$ denote the standard admissible covering of $X_0(Np)$ by wide-open
 neighbourhoods of the Hasse domain of $X_0(N)$, as described by Coleman in \cite{coleman-bu}. 
% In general, for any class $\omega \in  H^1_{\dR}(X_0(Np))$ we shall write $\omega_0\in H^1_{\rig}(\cW_0)$  and $\omega_\infty\in H^1_{\rig}(\cW_\infty)$ for the restriction of $\omega$ to $\cW_0$ and $\cW_\infty$, respectively. 
By construction, arguing as in \cite[Lemma 4.6]{DR2}, the ordinary unit root 
 class $\eta_f\in H^1_{\dR}(X_0(Np))$ is supported on $\cW_0$, and has vanishing residues at the supersingular annuli, while the anti-ordinary unit root class $\eta_f^w$ is supported 
 on $\cW_\infty$.
%, that is to say, $\eta_{f,0}=0$.


Let $U_p^w = w  U_p w $ be the adjoint operator of $U_p$ with respect to the Poincar\'e pairing on $X_0(Np^s)$, and let 
$$
e_{\ord}= \lim U_p^{n!} \quad \mbox{ and } \quad e^w_{\ord}= \lim (U^w_p)^{n!}=w e_{\ord}w
$$ 
denote Hida's ordinary and anti-ordinary projectors, respectively. 
Pull-back under $\varpi_s$ gives rise to an isomorphism between the ordinary subspaces of $H^1_{\dR}(X_0(Np^s))$ and $H^1_{\dR}(X_0(Np))$. 
Hence $e_{\ord} \Xi(g,\cE) $ arises as the pull-back under 
$\varpi_s$ of some ordinary class in  $H^1_{\dR}(X_0(Np))$, that we again denote with the same symbol. In the context of the next theorem, 
$e_{\ord}\Xi(g,\cE)$ is to be understood in the latter sense.
Since $e_{\ord}\Xi(g,\cE) $ belongs to the subspace of the cohomology 
which is pure of weight one, its  restriction to the wide open\ $\cW_\infty$ has vanishing residues along the supersingular annuli. Therefore, it 
also makes sense to pair it with the (restriction of) 
the anti-ordinary class $ \eta_f^w$.
\begin{theorem}
\label{eqn:conceptual} 
For all $s\geq 1$, 
$$ 
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})=  \alpha_f^{s-1} 
 \cdot \langle  \eta_f^w ,e_{\ord}\Xi(g, \cE) \rangle_{\cW_\infty}.
$$
\end{theorem}
\begin{proof}
Since the class 
$\eta_{f,s} $ 
is ordinary, it is fixed by $ e_{\ord}$ and thus
\begin{eqnarray}
\label{i2}
\langle \eta_{f,s},\Xi(g^w,\cE^w)\rangle_{X_0(Np^s)} & = &  \langle e_{\ord}\eta_{f,s}, \Xi(g^w,\cE^w)\rangle_{X_0(Np^s)} 
\\ \nonumber
&=&  \langle  \eta_{f,s}, e_{\ord}^w \Xi(g^w,\cE^w)\rangle_{X_0(Np^s)} \\
 \nonumber & = &  \langle  \eta_{f,s}^w, e_{\ord}\,  \Xi(g,\cE)\rangle_{X_0(Np^s)},
\end{eqnarray}
where  the self-adjointness of the involution $w$ has been exploited to derive the last equation.

 
But since $e_{\ord} \Xi(g,\cE)$ arises as a pullback under 
$\varpi_s$, we can write
\begin{eqnarray}\label{i3}
 \langle  \eta_{f,s}^w, e_{\ord}\,   \Xi(g,\cE)\rangle_{X_0(Np^s)}  & = &  
\langle  \varpi_{s }  (\eta_{f,s}^w), e_{\ord}\,   \Xi(g,\cE)\rangle_{X_0(Np)}
\\   \nonumber
&=& 
\langle  \varpi_{s} w  \varpi_{s}^*(\eta_f), e_{\ord}\,  \Xi(g,\cE)\rangle_{X_0(Np)}
\\
\nonumber
&=&  \langle   w \circ U_p^{s-1}(\eta_f), e_{\ord}\,   \Xi(g,\cE)\rangle_{X_0(Np)}  \\ 
\nonumber
& = &  \alpha_f^{s-1} \langle   \eta_{f}^w, e_{\ord}\, \Xi(g,\cE)\rangle_{\cW_\infty},
\end{eqnarray}
where the last identity follows from the fact that the anti-ordinary unit root class $\eta_f^w$ is 
supported on $\cW_\infty$. The theorem now follows from Proposition \ref{prop:poincare-bis} combined with \eqref{i2} and \eqref{i3}. 
\end{proof}

We are now in a position to state the main result of this section, namely, the evaluation of 
the $p$-adic syntomic regulator attached to $\mathbf{BF}_s$ in terms of  the $q$-expansions of concrete
(overconvergent) $p$-adic modular forms.
For a $p$-adic modular form $\varphi=\sum a_n(\varphi)q^n$, let 
$$\varphi^{[p]}=\sum_{p\nmid n} a_n(\varphi) q^n$$ denote its ``$p$-deprived'' counterpart.

The Atkin-Serre operator $d= q\frac{d}{dq}$ 
sends $p$-adic  overconvergent modular forms of weight $0$ to overconvergent modular forms
of weight two.
The image of the $p$-depleted classical weight two Eisenstein series
$E_2^{[p]}(\chi^{-1}\chi_p^{-1},1)$ under $d^{-1}$ is the $p$-adic overconvergent Eisenstein series
of weight $0$:
$$ d^{-1} E_2^{[p]}(\chi^{-1}\chi_p^{-1},1) = E_0^{[p]}(1,\chi^{-1}\chi_p^{-1}).$$
The product $g^{[p]} \times E_0^{[p]}(1,\chi^{-1}\chi_p^{-1})$ is therefore  an overconvergent cusp form of
weight two with trivial nebentypus character, and hence its image under the ordinary projection 
$e_{\ord}$ corresponds to a 
classical weight two cusp form on $\Gamma_0(Np)$, which can be viewed as an element of the 
first de Rham 
  cohomology of $X_0(Np)$. The inner product $\langle \ , \ \rangle_{X_0(N)} $
 in the theorem below denotes the 
usual Poincar\'e duality on this de Rham cohomology group.
\begin{theorem}
\label{thm:eq6} 
For all $s\geq 1$, 
$$ 
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})= 
 \mathcal{G}(\chi \chi_p) \cdot \frac{\alpha_f^{s-1}   }{1-\chi^{-1}(p)\alpha_f ^{-1}\alpha_{g}}\cdot 
 \langle  \eta_f^w,  e_{\ord}(g^{[p]}
\cdot E^{[p]}_0(1,\chi^{-1}\chi_p^{-1}))
\rangle_{X_0(Np)}.
$$
\end{theorem}
\begin{proof} 
 The restriction to $\cW_\infty$ of  $e_{\ord} \Xi(g, \cE)$ is the cohomology class of a specific  overconvergent  weight two 
$p$-adic modular form representing a rigid analytic differential form on $\cW_\infty$ 
with vanishing annular residues.  
%By  Thm.  2.1. of  \cite{atkin-li},  we have
%\begin{equation}
%\label{eqn:atkin-li}
%g^w =  \mathcal{G}(\chi^{-1}_p) \alpha_g^{-s} g^\iota,
%\end{equation}
%and hence
%\begin{equation}
%\label{eqn:wp-action}
%e_{\ord} \Xi(g^w,h^w)=   \mathcal{G}(\chi^{-1}_p)   \alpha_g^{-s}  e_{\ord} \Xi(g^\iota, h^w).
%\end{equation}  
 The class $e_{\ord} \Xi(g,\cE)$ 
 is defined by exactly the same analytic recipe as the class denoted 
$e_{\ord} \xi(\omega_{\testg}, \omega_{\testh})$ in equation (85) of \cite{DR2}, 
after replacing $\testg$ by $g$ 
and setting
$$ \testh =  w_{p^s} \cE \ \ \ \ (=\mathcal{G}(\chi) E_2(\chi^{-1},\chi_p)).$$
In \cite{DR2}, 
the form $\testh$ was assumed to be cuspidal 
while $w_{p^s} \cE$ is an Eisenstein series, but the calculations of 
loc.cit. remain valid in the latter setting.
In particular,
the expression
$$ w_{p^s}\testh = \mathcal{G}(\chi_p^{-1}) \alpha_h^{-s} \omega_{\testh^\iota}$$
that appears in the statement of 
Prop. 4.13 of  \cite{DR2}   can be replaced by $\omega_\cE$, yielding
\begin{eqnarray}
\label{eqn:what-we-wish-was-in-DR2}
e_{\ord} \Xi_P(g, \cE) &=&
e_{\ord} \xi_P(\omega_g, \omega_{w_{p^s}\cE}) =
-2 e_{\ord}( g\times d^{-1} \cE^{[p]})  \\ 
&=& 
-2 \mathcal{G}(\chi\chi_p) e_{\ord}( g \times   E_0^{[p]}(1, \chi^{-1}\chi_p^{-1})),
\end{eqnarray}
where $P(x)$  is the linear polynomial $2(1-p^{-1}\chi^{-1}(p)\alpha_{g} x)$,
and  $\Xi_P(g ,\cE)$ is defined as in \eqref{eqn:xiP}.
Combining equation  \eqref{eqn:what-we-wish-was-in-DR2}  
with Theorem \ref{eqn:conceptual}, we obtain
\begin{equation}
\label{eqn:almost-there}
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})= - 2 \mathcal{G}(\chi \chi_p) \alpha_f^{s-1}  \cdot \langle  \eta_f^w, P(\Phi)^{-1}
e_{\ord}(g 
\cdot E_0^{[p]}(1,\chi^{-1}\chi_p^{-1}))\rangle_{\cW_\infty}.
\end{equation}
Note that $g$ can be replaced by $g^{[p]}$ in equation \eqref{eqn:almost-there}, as a direct calculation shows that $$(g^{[p]} - g) \cdot E_0^{[p]}(1,\chi^{-1}\chi_p^{-1})$$ belongs to the kernel of $U_p$, and a fortiori of 
$e_\ord$. (See for instance Lemma 2.17 of \cite{DR1}.)  
Finally, for any class $\Xi$ in $H^1_{\dR}(\cW_\infty)$, we have 
$$\langle\eta_f^w,\Phi  \Xi\rangle_{\cW_\infty}=\alpha_f^{-1}\langle\Phi \eta_f^w,\Phi\Xi\rangle_{\cW_\infty}=p\alpha_f^{-1}\langle\eta_f^w,\Xi\rangle_{\cW_\infty}=
\beta_f\langle\eta_f^w,\Xi\rangle_{\cW_\infty},$$ which in turn implies
\begin{equation}
\label{eqn:last-nonsense}
\langle\eta_f^w,P(\Phi)^{-1}\Xi\rangle_{\cW_\infty}=
P(\beta_f)^{-1}\langle\eta_f^w,\Xi\rangle_{\cW_\infty}.
\end{equation}
In light of the explicit description of $P(x)$ given above, 
combined with  equations
\eqref{eqn:almost-there} and \eqref{eqn:last-nonsense},
$$ 
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})= 
 \mathcal{G}(\chi \chi_p) \cdot \frac{\alpha_f^{s-1}   }{1-\chi^{-1}(p)\alpha_f ^{-1}\alpha_{g}}\cdot 
 \langle  \eta_f^w,  e_{\ord}(g^{[p]}
\cdot E^{[p]}_0(1,\chi^{-1}\chi_p^{-1}))
\rangle_{\cW_\infty}.
$$
Proceeding just as in the proof of Theorem \ref{eqn:conceptual},
Theorem \ref{thm:eq6} follows 
from the compatibility between the de Rham pairings on
$\cW_\infty\subset X_0(Np)$ and on $X_0(Np)$,  using the fact that
$\eta_f^w$ is    anti-ordinary and supported on $\cW_\infty$.
  \end{proof}
 





 \subsection{Hida's $p$-adic $L$-function}


A point of the  rigid analytic space $$\Omega:= \Hom_{\cts}(\Z_p^\times,\C_p^\times) $$ is  said to be 
{\em arithmetic} if it is of the form 
$$ \nu_{k,\epsilon}(x) := x^{k-1} \epsilon(x), \qquad \mbox{ (for all $x\in \Z_p^\times$)},$$
where $k\ge 2$ is an integer,  and $\epsilon$ is a finite order character factoring through 
$(\Z/p^s\Z)^\times$. The space $\Omega$ is referred to as the {\em weight space} and plays a basic role in the theory of $p$-adic familes of modular forms.


 A {\em Hida family} of  tame character $\chi:(\Z/N\Z)^\times \lra \C_p^\times$
 is a pair $(\Omega_\varphi,\underline{\varphi})$, where
\begin{enumerate}
\item
$\Omega_\varphi$ is a rigid analytic space equipped with a finite  morphism 
$$ \wt: \Omega_\varphi \lra \Omega,$$
\item 
$\underline{\varphi} = \sum_{n=1}^\infty a_n(\underline{\varphi}) q^n$ is a formal $q$-series with coefficients in the ring $\Lambda_\varphi :=\cA(\Omega_\varphi)$ of rigid analytic functions on 
$\Omega_\varphi$,
\end{enumerate}
such that, for all   points $x\in \Omega_\varphi$ of arithmetic weight $\wt(x) = \nu_{k,\epsilon}$,
the specialisation 
$$ \varphi_x := \sum_{n=1}^\infty a_n(\underline{\varphi})(x) q^n$$
is the $q$-series of a normalised eigenform of weight $k$, level $Np^s$ and character $\chi_{\varphi_x} = \chi\epsilon \omega^{1-k}$, where $\omega$ is the Teichm\"uller character of conductor $p$.

Let $g$ be the weight one modular form considered in Section \ref{sec:strategy}, and
let $g_\alpha$ be one of its ordinary $p$-stabilisations. A theorem of Hida ensures the existence of 
a Hida family, denoted $\hg$,  of tame character $\chi$ which
specialises to  $g_\alpha$ at a suitable point $y_0\in \Omega_{\hg}$ lying above 
$\nu_{1,1}$. 
For any  arithmetic point $y\in \Omega_{\hg}$  above   $\nu_{\ell,\epsilon}$, for some 
$\ell\ge 1$ and some  
 character $\epsilon$ of $p$-power conductor, denote by  
  $g_y\in  S_\ell(Np^s,\chi \epsilon \omega^{1-\ell})$ the specialisation of $\hg$  at $y$. 
  
  
 Write $\underline{E}(1,\chi^{-1})$ for the  ordinary Hida family of 
  Eisenstein series,  whose
 specialisation at the point $\nu_{\ell,\epsilon}$ in weight space is equal to the classical  Eisenstein series 
$$ \underline{E}(1,\chi^{-1})_{\nu_{\ell,\epsilon}} = 
E_\ell(1,\chi^{-1}\epsilon \omega^{1-\ell})$$ 
of weight $\ell$ and character $\chi^{-1} \epsilon \omega^{1-\ell}$. 
The Iwasawa algebra admits an involution inducing the map $\nu_{\ell,\epsilon} \mapsto  \nu_{2-\ell,\epsilon^{-1}}$ 
on weight space. Applying the change of scalars to  $\underline{E}(1,\chi^{-1})$ via this involution yields a 
$\Lambda$-adic modular form , denoted $\underline{E}^*(1,\chi^{-1})$,
 whose  specialisation at  the arithmetic point
 $\nu_{\ell,\epsilon}$ is the overconvergent modular form
$$ \underline{E}^*(1,\chi^{-1})_{\nu_{\ell,\epsilon}}  = E_{2-\ell}(1,\chi^{-1} \epsilon^{-1} \omega^{\ell-1}).$$
 
The expression $ g_y^{[p]} \cdot  E^{[p]}_0(1, \chi^{-1}\chi_p^{-1})$ appearing in 
Theorem \ref{thm:eq6} 
 is the specialisation at the point $y$ of weight $\nu_{2,\chi_p \omega}$ of the 
 $\Lambda$-adic modular form 
 $$ \underline{g} \underline{E}^{*[p]}(1,\chi^{-1}),$$
 whose specialisation at a point $y$ of weight $\nu_{\ell,\epsilon}$ is given by
 \begin{equation}
\label{eqn:specell}
\left\{\hg^{[p]}\cdot \underline{E}^{*[p]}(1,\chi^{-1})\right\}_x =  g_y^{[p]} \cdot  E_{2-\ell}^{[p]}(1, \chi^{-1} \epsilon^{-1} \omega^{\ell-1}).
\end{equation} 
Note that for all arithmetic  $y$, this specialisation 
is an overconvergent modular form of weight $2$ and trivial character, whose ordinary projection is therefore classical of level $Np$; hence, 
$$ e_{\ord}(\underline{g}^{[p]} \cdot \underline{E}^{*[p]}(1,\chi^{-1}) \mbox{ belongs to } 
S_2(\Gamma_0(Np), \Q_p) \otimes \Lambdag.$$
The reader is referred to Section 4.5 of \cite{DR2} for details on dual families of modular forms, which motivate the definition of the $\Lambda$-adic modular form of equation \eqref{eqn:specell}.

Let $L_p(\hf,\hg)(x,y,j)$ be the $3$-variable  Rankin $p$-adic $L$-function on $\Omega_{\hf}\times\Omega_{\hg}\times\Omega$
introduced in \cite[\S 2.2]{BDR1}, where $\hf$ is the Hida family passing through $f$.
It interpolates the algebraic part of the critical values $L(f_x\otimes g_{y},j)$, where $f_x$ and $g_y$ are classical specialisations of $\hf$ and $\hg$ whose weights $k$ and $\ell$ satisfy 
\begin{equation}\label{klj}
k > \ell, \qquad \frac{k+\ell-1}{2}\leq j\leq k-1.
\end{equation}
Consider the restriction of $L_p(\hf,\hg)(x,y,j)$ to the set of  points $(x_0,y,\nu_{\ell,\epsilon})$, where $x_0$ is such that $\hf_{x_0}=f$, and $y\in\Omega_{\hg}$ is a point lying above $\nu_{\ell,\epsilon}$. 
In the notations of \cite[\S 2.2.1]{BDR1}, this amounts to considering only the values of
$L_p(\hf,\hg)(k,\ell,j)$ when  $k=2$, $j=\ell$, 
$t=1-\ell$ and $m=\ell$. 
In light of equations (27) and (24) in loc.cit., combined with the equality
$$d^{1-\ell}E^{[p]}_{\ell}(\chi^{-1}\epsilon^{-1}\omega^{\ell-1},1)=
E_{2-\ell}^{[p]}(1,\chi^{-1}\epsilon^{-1}\omega^{\ell -1}),$$
this restriction agrees, up to a non-zero fudge factor which depends only on the fixed form $f$, with the ($p$-adic analytic) function described for all  such $y$ by
\begin{equation}
\label{eqn:interpolation}
L_p(f,\hg)(y): =\langle  \eta_f^w,e_{\ord}(g_y^{[p]} \cdot  E_{2-\ell}^{[p]}(1,\chi^{-1}\epsilon^{-1}\omega^{\ell-1}))\rangle_{X_0(Np)}.
\end{equation}
Note that the points of the form $(x_0,y,\nu_{\ell,\epsilon})$ lie outside the range of classical interpolation for $L_p(\hf,\hg)$, unless   $g_y$ is classical and of weight one.
The next result relates the values of  
$L_p(f,\hg)$ at arithmetic points $y$ of weight $\nu_{2,\epsilon}$ 
to the $p$-adic regulators of Beilinson-Flach elements:
\begin{theorem}
\label{cor:wt2value} 
Let $\epsilon$ be a character of conductor $p^s$, let $y\in \Omega_{\hg}$ be a point over $\nu_{2,\epsilon}$ and let $g = g_y\in  S_2(Np^s,\chi \epsilon \omega^{-1})$ denote the specialization of $\hg$ at $y$. Then
$$ 
\reg_p^{f g}(\mathbf{BF}_s)(\eta_f\otimes\omega_{g^w})=  \mathcal{G}(\chi \epsilon  \omega^{-1})\cdot \frac{\alpha_f^{s-1}}{1-\chi(p)^{-1}\alpha_f^{-1} \alpha_{g}}\cdot  L_p(f,\hg)(y).
$$
\end{theorem}
\begin{proof}
This
follows directly by combining equation \eqref{eqn:interpolation} with Theorem \ref{thm:eq6}.
\end{proof}


Another important property of $L_p(f,\hg)$ pertains to its specialisations at classical weight one points attached to the Hida family $\hg$.


\begin{theorem}
\label{thm:weight-one-specialisation}
Let $y\in\Omega_{\hg}$ be a point over $\nu_{1,1}$, corresponding to a classical $p$-stabilised weight one form
$g=g_y\in S_1(Np,\chi)$. Then
$$L_p(f,\hg)(y)\ne 0 \quad \Leftrightarrow\quad L(f\otimes g,1)\ne 0.$$
\end{theorem}
\begin{proof} By 
equation \eqref{eqn:interpolation}, 
\begin{eqnarray}
\nonumber
L_p(f,\hg)(y)&=& \langle  \eta_f^w,e_{\ord}\left(g_y^{[p]} \cdot  E^{[p]}_1(1,\chi^{-1})\right)\rangle_{X_0(Np)} \\ 
\label{eqn:valuex1}
&  = &  \langle  \eta_f^w,  g_y^{[p]} \cdot  E^{[p]}_1(1,\chi^{-1})\rangle_{X_0(Np^2)} \pmod{\bar\Q^\times},
\end{eqnarray}
where the fact that $g_y^{[p]} E^{[p]}_1(1,\chi^{-1})$ is a classical modular form of level $Np^2$
has been  used  to deduce the second equality. 
The expression in \eqref{eqn:valuex1} 
is an explicit non-zero multiple of the value 
of the classical complex Rankin-Selberg $L$-function 
$L(f\otimes g,s)$ of the convolution of $f$ and $g$ at $s=1$. (See equation (27) and the one immediately following it in \cite[\S 2.2.2]{BDR1}.)
The theorem follows.
\end{proof}



\section{$\Lambda$-adic classes}
\subsection{Norm-compatible systems of elements}
For all $s\ge 0$, the natural $U_p$-compatible projections
\begin{equation}
\label{eqn:pi-on-Xs}
\pi_{s+1,s}: X_{s+1} \ra X_s
\end{equation}
 of modular curves
 give rise to   maps
\begin{equation}
\label{eqn:pi-on-Ss}
\pi_{s+1,s}: S_{s+1} \lra S_s
\end{equation}
 on the associated surfaces, 
  that we continue to denote with the same symbol by a slight abuse  of notation. 
 Write also $\pi_{s+1,s}:\cO_{X_{s+1}}^\times\lra \cO_{X_s}^\times$ 
 for the norm maps on units induced by $\pi_{s+1,s}$.
\begin{proposition}
\label{prop:norm} 
For all $s\ge 1$,
$$ \pi_{s+1,s }(\mfg^w_{a;Np^{s+1}}) = \mfg^w_{a;Np^s}. $$
\end{proposition}
\begin{proof} 
Lemma 2.12 of \cite{Kato} implies that the units $g_{a;Np^{s+1}}$ are compatible relative to the pushforward
via the maps $\pi'_{s+1,s}: X_{s+1} \lra X_s$ that are compatible with the operator
$U_p'$. 
% The $U_p$-compatible degeneracy map $\pi_{s+1,s}$ is described analytically, in terms of the
% identification of $Y_1(Np^s)(\C)$ with the quotient $\Gamma_1(Np^s)\backslash {\cH}$,
% as the map sending $z\in \cH$ to $z$, while its dual $\pi_{s+1,s}' := w_{p^s} \pi_{s+1,s} w_{p^{s+1}}$
% is given by the map sending $z$ to $pz$. 
%The formula for the $q$-expansion of $\mfg_{a;Np^s}$ can be used to verify the distribution
% relation
% \begin{equation}
% \prod_{j=0}^{p-1}\mfg_{a+jNp^s;Np^{s+1}}(z)=\mfg_{a;Np^s}(pz).
%\end{equation}
This compatibility of the units $\mfg_{a;Np^s}$ relative to  the $\pi_{s+1,s}':= w_{p^s} \pi_{s+1,s} w_{p^{s+1}}$ 
implies the $\pi_{s+1,s}$-compatiblity of the units $\mfg_{a;Np^s}^w$.
See also \S 2.11 and \S 2.13 of \cite{Kato}, or
Thm.~2.24 of \cite{LLZ1}, for a convenient summary of these and further  calculations in  the more general setting treated in \cite{Kato}.
(Recall that the unit $ \mfg_{a;Np^s}$ is denoted $g_{0,\frac{a}{Np^s}}$ in \cite{Kato} and \cite{LLZ1}.)
\end{proof}

 Write
$$
\pi_{s+1,s}: \CH^2(S_{s+1},1) \ra \CH^2(S_s,1)
$$
for the norm maps on higher Chow groups induced by push-forward under the maps 
$\pi_{s+1,s}$ of \eqref{eqn:pi-on-Ss}. Note that $\pi_{s+1,s}$  preserves
 the subspaces of negligible classes, and hence descends to a well-defined map
\begin{equation}
\label{push}
\pi_{s+1,s}: \CH^2_{{\mathrm{neg}}}(S_{s+1},1) \ra \CH^2_{{\mathrm{neg}}}(S_s,1).
\end{equation}
The norm compatibilities of the units $\mfg_{a;Np^s}$ are inherited by the
associated Beilinson-Flach elements:
\begin{proposition}
\label{prop:compatibility-ddot}
For all $s \ge 1$, 
\begin{equation}
\label{Up-comp-eqn}
\pi_{s+1,s}(\mathbf{BF}(a;Np^{s+1}))  =
 \mathbf{BF}(a;Np^s), \qquad
\pi_{s+1,s}(\mathbf{BF}_{s+1})  =
 \mathbf{BF}_s \quad \mbox{ in } \quad \CH^2_{{\mathrm{neg}}}(S_s,1)(\Q).
\end{equation}
\end{proposition} 

\begin{proof} 
This follows directly from 
Proposition \ref{prop:norm} in light of the fact that the map
$\pi_{s+1,s}$ of \eqref{eqn:pi-on-Ss} maps the diagonally embedded
$X_{s+1} \simeq \Delta_{s+1} \subset S_{s+1}$ to
$\Delta_s$, and is identified
 with the projection 
$\pi_{s+1,s}$ of \eqref{eqn:pi-on-Xs}. 
\end{proof}

\begin{corollary}
The Beilinson-Flach cohomology classes $\kappa_s$ of
Definition \ref{eqn:def-kappas1} are compatible under the norm maps, i.e.,
$$ \pi_{s+1,s}(\kappa_{s+1}) = \kappa_s.$$
\end{corollary}
In particular, the classes 
$\kappa_s$ can be packaged together into the inverse limit class
$$ \kappa_\infty := (\kappa_s)_{s\ge 1} \in H^1(\Q, V_0(Np) \otimes \hV_\infty),$$
where
$$ \hV_\infty := \lim_{\leftarrow,s} V_s.$$


\subsection{The $\Lambda$-adic Beilinson-Flach cohomology class}

The module $\hV_\infty$ of the previous section becomes more manageable when replaced by its image under the ordinary projection. Let 
\begin{equation*}
V_{s}^{\ord} :=   e_{\ord}  V_s, 
\qquad \hV_\infty^{\ord} := e_{\ord} \hV_\infty = \lim_{\leftarrow,s} e_{\ord} V_s.
\end{equation*}
  Note that the action of the group $D_s$ of diamond operators on $X_s$   endows the $G_\Q$-module $\hV_{\infty}^{\ord}$ with a natural structure of module over the Iwasawa algebra $\LLambda = \Z_p[[D_\infty]] := \invlim \Z_p[D_s]$. A theorem of Hida asserts that this module is finitely generated and locally free over this algebra. Its Hecke eigenspaces realise the $\Lambda$-adic representations 
  attached to ordinary families of eigenforms.
  
The   $\Lambda$-adic Beilinson-Flach cohomology class is defined by setting
\begin{eqnarray*}
  \kappa^{\ord}_{s} & := & e_\ord  \kappa_s \in H^1(\Q, V_0(Np)\otimes V_s^{\ord}), \\
  \kappa^{\ord}_\infty &:= & e_{\ord} \kappa_\infty = 
  (\kappa_s^{\ord})_{s\ge 1} \in H^1(\Q, V_0(Np) \otimes \hV_\infty^{\ord}).
  \end{eqnarray*}
 

 
Recall the cusp form $f=q + \sum_{n\geq 2} a_n(f)q^n\in S_2(N p)$ introduced in \S \ref{sec:regulators}, on which $U_p$ acts with eigenvalue $\alpha_f$. 
%The $p$-adic Galois representation $V_f$ is characterized by the property that the characteristic %polynomial of an arithmetic Frobenius element $\Fr_\ell$ at a prime $\ell\mid N_f$ is $P_\ell(T)=T^2-%a_\ell(f)T+\ell$. The $G_{\Q}$-module $V_0(Np)$ contains $2^{\sharp \{d\geq 1, d\mid \frac{N p}{N_f}\}}$ copies of $V_f$, and the eigenform $f$ singles out in a natural way a $G_\Q$-equivariant projection
%$$
%\varpi_f: V_0(Np) \lra V_f.
%$$
%Assume $f$ is ordinary at $p$ and let $\alpha_f$ denote the single $p$-adic unit root of $P_p(T)$. 
Define 
$$
\psi_f: G_{\Q_p} \lra \cO^\times
$$
to be the unramified character of $G_{\Q_p}$ such that $\psi_f(\Fr_p) = \alpha_f$ and let $\epsilon_{\cyc}: G_{\Q_p} \lra \Z_p^\times$ denote the cyclotomic character. There is an exact sequence of $G_{\Q_p}$-modules
\begin{equation}\label{eqn:local-triangular-class} 
0 \, \ra \, V_f^+ \, \ra \, {V_f} \, \ra \, V_f^- \, \ra \, 0
\qquad \mbox{with\ \ } V_f^+ \simeq \cO(\psi_f^{-1} \epsilon_{\cyc} )
 \mbox{\ \ and\ \ }
V_f^- \simeq \cO(\psi_f)
\end{equation} as $\cO[G_{\Q_p}]$-modules.

Let 
$$\underline{\epsilon}_{\cyc} : G_{\Q_p}\lra \Lambda^\times$$ denote the $\Lambda$-adic cyclotomic character satisfying the interpolation property 
$$
\nu_{\ell,\epsilon} \circ \uepsilon = \epsilon \epsilon_{\cyc}^{\ell-1}   \omega^{1-\ell}
$$
for any $\ell\geq 1$ and for any Dirichlet character $\epsilon$ of $p$-power conductor.

Let $\hg = \sum_{n\geq 1} \mathbf{a}_n(\hg) q^n$ be a $\Lambda$-adic cuspidal eigenform of tame level $N$ and tame character $\chi$, arising from a $\Lambda$-adic newform of level $N_{\hg}$, and let $\Lambdag$ denote the finite flat extension of the Iwasawa algebra $\Lambda := \Z_p[[1+p\Z_p]]$ generated by the coefficients $\mathbf{a}_n(\hg)$. Hida and Wiles associated to $\hg$ a two-dimensional Galois representation $\mathbb{V}_{\hg}$ over $\Lambdag$, again characterized up to isomorphism by the property that the characteristic polynomial of $\Fr_\ell$ is $T^2-\mathbf{a}_\ell(\hg)T+\underline{\epsilon}_{\cyc}(\ell)$ for primes $\ell\nmid Np$. 


Wiles \cite{W} proved a $\Lambda$-adic analogue of \eqref{eqn:local-triangular-class}, which asserts that (cf.\,also \cite{Oh00})
\begin{equation}\label{Wiles-ord}
0 \, \ra \, \mathbb{V}_{\hg}^+ \, \ra \, {\mathbb{V}_{\hg}}  \, \ra \, \mathbb{V}_{\hg}^- \, \ra \, 0,
\qquad \mbox{with\ \ } 
\mathbb{V}_{\hg}^+ \simeq \Lambdag(\psi_{\hg}^{-1} \chi \underline{\epsilon}_{\cyc} ) \mbox{\ \ and\ \ }
\mathbb{V}_{\hg}^- \simeq \Lambdag(\psi_{\hg})
\end{equation}
 as $\Lambdag[G_{\Q_p}]$-modules.
Here
$$
\psi_{\hg}: G_{\Q_p}\lra \Lambdag^\times 
$$
 is the unramified character sending   $\Fr_p$  to  
  $\mathbf{a}_p(\hg)$. 

Fix once and for all a choice of the isomorphisms in \eqref{Wiles-ord}. Similarly to the previous sections, the $\Lambda$-adic form $\hg$ gives rise to an epimorphism
 $$\varpi_{\hg}: \hV_{\infty}^{\ord} \lra \mathbb{V}_{\hg}$$
 of $\tilde\Lambda[G_\Q]$-modules.
 Set
$$
\mathbb{V}_{f,\hg}: = V_f\otimes \mathbb{V}_{\hg}, \qquad   \quad \varpi_{f,\hg} := \varpi_f \otimes \varpi_{\hg}: V_0(Np) \otimes \hV_{\infty}^{\ord} \lra \mathbb{V}_{f,\hg}.
$$

\begin{definition} The $\Lambda$-adic Beilinson-Flach cohomology class associated to the pair $(f,\hg)$ is
$$\kappa(f,\hg) = \varpi_{f,\hg} ( \kappa^{\ord}_{\infty})\in H^1(\Q,\mathbb{V}_{f,\hg}).$$
\end{definition}

Let $\ell\geq 1$ be an integer and $\epsilon$ be a Dirichlet character of conductor $p^s$ for some $s\geq 1$. Let $y$ be a point in $\Omega_{\hg} = \mathrm{Spf}(\Lambdag)$ such that $\mathrm{wt}(y)=\nu_{\ell,\epsilon}$, and assume that the specialization $g_{y}$  of $\hg$ at $y$ is a {\em classical} eigenform (which is always the case if $\ell\geq 2$). Then $g_y$ belongs to $M_\ell(Np^s,\chi\epsilon\omega^{1-\ell})$, and is actually a cuspform if $\ell\geq 2$. Define
\begin{equation}\label{def-kappa-fgy}
\kappa(f,g_y)  \in H^1(\Q,V_{f g_y})
\end{equation}
to be the specialization at $y$ of the $\Lambda$-adic class $\kappa(f,\hg)$.
Let 
$$
\kappa_p(f,\hg) \in  H^1(\Q_p,\mathbb{V}_{f \hg}), \quad \kappa_p(f,g_y) \in H^1(\Q_p,V_{f g_y})
$$
denote the restriction to $G_{\Q_p}$ of the $\Lambda$-adic class $\kappa(f,\hg)$ and of its specialization $\kappa(f,g_y)$.

Since the $p$-adic syntomic regulator map is the composition of the \'etale regulator with Bloch-Kato's logarithm map, Theorem \ref{cor:wt2value} translates into the following statement:
 
 \begin{theorem}\label{prop:bloch-kato-log-in-families}
Let $y\in \Omega_{\hg}$ be an arithmetic point of
weight-character  $\nu_{2,\epsilon}$ for some character $\epsilon$ of conductor $p^s$.
%, and let $g=g_{y}\in S_2(Np^s,\chi\epsilon\omega^{-1})$ be the specialization of $\underline{g}$ at $y$. 
Then
 $$ \log_p \,\kappa_p(f,g_y) (\eta_f \otimes \omega_{g_y^w}) =\mathcal{G}(\chi \epsilon \omega^{-1})\cdot \frac{\alpha_f^{s-1} }{1-\chi(p)^{-1}\alpha_f^{-1} \alpha_{g_y}^{-1}}
 \cdot  L_p(f,\hg)(y).
 $$
 \end{theorem}
 


\subsection{The explicit reciprocity law}

The $\Lambdag$-module
  $\mathbb{V}_{f \hg}$ admits a natural $G_{\Q_p}$-stable 
  filtration, given by
$$
\mathbb{V}_{f \hg}^{++} := V^+_f\otimes \mathbb{V}^+_{\hg} \quad \subset \quad \mathbb{V}_{f \hg}^{+} :=   V_{f}\otimes \mathbb{V}^+_{\hg} \, + \, V^+_f\otimes \mathbb{V}_{\hg}  \quad \subset \quad \mathbb{V}_{f \hg}.
$$  

Note that $\mathbb{V}_{f \hg}^{++}$ and $\mathbb{V}_{f \hg}^{+}$ have rank $1$ and $3$ over $\Lambdag$, respectively. 
\begin{lemma}\label{isos}
\begin{enumerate}
\item[(i)]  There is an isomorphism $\mathbb{V}_{f \hg}/\mathbb{V}_{f \hg}^{+} \, \simeq \, \Lambda_g(\psi_f \psi_{\hg})$ of $\Lambdag[G_{\Q_p}]$-modules.
\item[(ii)]The quotient $\mathbb{V}_{f \hg}^{+}/\mathbb{V}_{f \hg}^{++}$ decomposes as a $\Lambdag[G_{\Q_p}]$-module as
$$
\mathbb{V}_{f \hg}^{+}/\mathbb{V}_{f \hg}^{++} \, \simeq \, \mathbb{V}_{f \hg}^{f} \,\oplus \,\mathbb{V}_{f \hg}^{g} \, \mbox{ where }  \mathbb{V}_{f \hg}^{f} \, = \, \Lambdag(\psi_f \psi_{\hg}^{-1} \chi \cdot  \underline{\epsilon}_{\cyc}) \, \mbox{ and } \,\mathbb{V}_{f \hg}^{g} \, = \, \Lambdag(\psi_f^{-1} \psi_{\hg}\epsilon_{\cyc}) .
$$
\end{enumerate}
\end{lemma}

\begin{proof}  The lemma follows directly from the definitions
 of $\mathbb{V}_{f \hg}^{++}$ and $\mathbb{V}_{f \hg}^{+}$, and
  the formulae described in \eqref{eqn:local-triangular-class} and \eqref{Wiles-ord}.
\end{proof}

The natural inclusion $\mathbb{V}^+_{f \hg} 
\hookrightarrow \mathbb{V}_{f \hg}$ induces a homomorphism 
$H^1(\Q_p,\mathbb{V}^+_{f \hg})\, \hookrightarrow \, H^1(\Q_p,\mathbb{V}_{f \hg})$, which is injective because $H^0(\Q_p,\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg}) = 0$.
Thus  $H^1(\Q_p,\mathbb{V}^+_{f \hg})$ is identified 
with a subgroup of $H^1(\Q_p,\mathbb{V}_{f \hg})$.


Define $\xi_{fg} := 1-\alpha_f a_p(\hg)\in \Lambdag$. For the remainder of the article, we replace the ring $\Lambdag$ and all modules over it by their localization at the multiplicative set generated by the powers of $\xi_{fg}$. Since this element vanishes at no classical point of weight $\ell\geq 1$, this modification does not affect any of the subsequent arguments concerning the specialization at these points.  

\begin{lemma}
The local class $\kappa_p(f,\hg)$ belongs to $H^1(\Q_p,\mathbb{V}^+_{f \hg}) \, \subset \, H^1(\Q_p,\mathbb{V}_{f \hg})$.
\end{lemma}

\begin{proof} It follows from known properties of the \'etale regulator map, as explained in \cite{Ne2} and \cite[\S 2]{Fl}, that for all points $y$ of weight $\nu_{2,\epsilon}$, the specializations $\kappa_p(f,g_y)$ belong to the image of $H^1(\Q_p,V^+_{f g_y})$ in $H^1(\Q_p,V_{f g_y})$. Since these points form a dense subset of  $\Omega_{\hg}$ for the rigid analytic topology, we conclude that $\kappa_p(f,\hg)$ belongs to the kernel of the map
$$
H^1(\Q_p,\mathbb{V}_{f \hg}) \, \lra  \, H^1(I_p,\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg}).
$$
%arising from the natural projection $\mathbb{V}_{f \hg} \,\ra \,\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg}$ and restriction
In order to prove the lemma, it thus suffices to show that the kernel of the restriction map $$H^1(\Q_p,\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg}) \, \stackrel{\res}{\lra} \, H^1(I_p,\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg})$$ is trivial. Note that such kernel is a quotient of $H^1(\Q_p^{\mathrm{ur}}/\Q_p,(\mathbb{V}_{f \hg}/\mathbb{V}^+_{f \hg})^{I_p})$, which by Lemma \ref{isos} (i) is isomorphic to $H^1(\Q_p^{\mathrm{ur}}/\Q_p,\Lambdag(\psi_f \psi_{\hg}))$. This module is $\xi_{gh}$-torsion. In view of the redefinition of $\Lambdag$, this implies the sought-after claim.
\end{proof}



Set $\underline{\Psi} := \psi_f \psi_{\hg}^{-1} \chi$.
 The two previous lemmas allow us to define
$$
\kappa^f_p(f,\hg) \in H^1(\Q_p, \Lambda_g(\underline{\Psi} \cdot \underline{\epsilon}_{\cyc} ))
$$
as the projection of $\kappa_p(f,\hg)$ to the first factor $\mathbb{V}_{f \hg}^{f}$.

Let $y\in \Omega_{\hg}$ be a classical point of weight $\mathrm{wt}(y)=\nu_{\ell,\epsilon}$ with $\ell \geq 2$ and $\epsilon \ne \omega^{\ell-1}$, and let $K_y/\Q_p$ denote the residue field of $y$. 
 Write 
$\Psi_y:G_{\Q_p}\lra K_y^\times$ for the specialisation at $y$
 of the unramified $\Lambda$-adic 
character $\underline{\Psi}$. 
The classical eigenform $g_y \in S_\ell(Np^s,\chi \epsilon \omega^{1-\ell})$ is
 new at $p^s$, and the specialization of $\kappa^f_p(f,\hg)$ at $y$ yields a local class 
$$\kappa^f_p(f,g_y)\in H^1(\Q_p,K_y(\Psi_y \cdot \epsilon \cdot  \epsilon^{\ell-1}_{\cyc} \cdot \omega^{1-\ell} )).$$
\begin{lemma} 
\label{lemma:one-dim-local}
 For all arithmetic points $y$ of weight $\nu_{\ell,\epsilon}$ with $\ell\ge 2$, 
the local cohomology group $H^1(\Q_p,K_y(\Psi_y \cdot \epsilon \cdot  \epsilon^{\ell-1}_{\cyc} \cdot \omega^{1-\ell} ))$ is one-dimensional over $K_y$ and equal to 
$H^1_{\exp}(\Q_p, K_y(\Psi_y \cdot \epsilon \cdot  \epsilon^{\ell-1}_{\cyc} \cdot \omega^{1-\ell} ))$.
\end{lemma}
\begin{proof} 
Write simply $V=V^f_{fg_y}$ for the specialization at $y$ of 
$ \mathbb{V}_{f \hg}^{f}$ and $D=D_{\dR}(V^f_{fg_y})$ for its de 
Rham Dieudonn\'e module. By \cite[Def.\,3.10]{bloch-kato}, 
\cite[Ex.\,2.19]{bellaiche-notes}, the Bloch-Kato logarithm map 
is an isomorphism
\begin{equation}\label{logBK}
\log_p: \, H_{\exp}^1(\Q_p,V) \, \ra \, D/(\Fil^0 D + D^{\phi=1}).
\end{equation}
% By Lemma \ref{isos} (ii), $V \simeq K_y(\Psi_y \cdot \epsilon\cdot\epsilon^{\ell-1}_{\cyc} \cdot\omega^{1-\ell})$. 
By \cite[\S 2.1.3]{bellaiche-notes}, $\dim_{K_y} H^1(\Q_p,V) = e_0+ e_1 +1$ where $e_0=\dim H^0(\Q_p,V)$ and $e_1$ is the multiplicity of $K_y(1)$ as a quotient of $V$. Since $\ell \geq 2$, $e_0=0$. Since $\epsilon \ne \omega^{\ell-1}$, $e_1=0$ and thus $H^1(\Q_p,V)$ is one-dimensional over $K_y$. 
Moreover, since the Hodge-Tate weight of $V$ is $1-\ell \leq -1$, the subspace of $D$ on which frobenius acts as the identity is $D^{\phi=1}=0$, and $\Fil^{1-\ell}(D) \supsetneq \Fil^{2-\ell}(D)$. Since \eqref{logBK} is an isomorphism and $\dim H_{\exp}^1(\Q_p,V) \leq \dim H^1(\Q_p,V)=1$, it follows that in fact $\Fil^{1-\ell}(D)=D \supset \Fil^0(D)=0$, and thus \eqref{logBK} gives rise to an isomorphism
\begin{equation}
\label{logBK2}
\log_p: \, H_{\exp}^1(\Q_p,V) = H^1(\Q_p,V) \lra D
\end{equation}
of one-dimensional $K_y$-vector spaces. The result follows.
\end{proof}
For each arithmetic point $y\in\Omega_{\hg}$ 
of weight $\nu_{\ell,\epsilon}$ with $\ell \ge 2$, the choice of a  
period 
$\mho_y\in D_{\dR}(V^{g_y^*}_{fg_y^*})$
attached to the dual $V^{g_y^*}_{fg_y^*}$ of $V_{fg_y}^f$  determines an 
  isomorphism
   \begin{equation}
\label{logs}
\log_{\mho_y}: H^1(\Q_p,K_y(\Psi_y \cdot \epsilon\cdot\epsilon^{\ell-1}_{\cyc} \cdot\omega^{1-\ell} )) \lra  
K_y, \qquad \log_{\mho_y}(\kappa):= \log_p(\kappa)(\mho_y).
\end{equation}
Fix a choice of $\mho_y$ by  defining, for all arithmetic 
points $y$ of weight $\nu_{2,\epsilon}$ with
 $\epsilon\ne 1$ of conductor $p^s$, 
\begin{eqnarray}
\label{eqn:def-period-Jy}
\mho_y & :=&   \alpha_f^{1-s}      
 \gauss(\chi \epsilon \omega^{ -1})^{-1} \cdot (\eta_f \otimes \omega_{g_y^w}).
%     \\   \nonumber
%     & :=&  \alpha_f^{-s} \alpha_{g_y}^s \chi(p)^{-s}   
%    \cdot (\eta_f \otimes \omega_{g_y^*},
  \end{eqnarray}
Ohta's $\Lambda$-adic Eichler-Shimura isomorphism can be used to show that these periods agree with those
arising in   Perrin-Riou's  generalised  Coleman maps attached to families of
cyclotomic twists  of a fixed Galois representations and their generalisations  to   twists by unramified $\Lambda$-adic characters, as described in  
\cite[Thm.~4.15]{LZ}. 
 (Cf.~the discussions in \cite[\S 5.2, \S 5.3]{DR2}, or in 
 \cite[\S 7.4, \S 7.6, Thm.~7.6.1]{KLZ}.) 
It follows that,  for all $\kappa\in H^1(\Q,\hV_{f\hg}^f)$,
  the function 
 $y\mapsto \log_{\mho_y}(\kappa)$ interpolates  to an element of the fraction field
 of $\Lambdag$, 
 denoted ${\cL}(\kappa)$.
 
 
% where 
%\begin{itemize}
%\item the $\Gamma$-factor is given by $$ \Gamma^*(2-\ell) = \left\{ \begin{array} {ll}\frac{1}{(\ell-2)!} & \mbox{ if } \ell\ge 2, \\ (1-\ell)!   & \mbox{ if } \ell \le 1; \end{array} \right.$$
% \item the term $\mathfrak{E}_y$ is given by $$ \mathfrak{E}_y = \left\{  \begin{array}{ll} 1 & \mbox{ if } \epsilon \ne \omega^{\ell-1};\\(1-p^{1-\ell}  \alpha_f \alpha_{g_y}^{-1} \chi(p)^{-1})(1-p^{\ell-2}  \alpha_f \alpha_{g_y}^{-1} \chi(p))^{-1} & \mbox{ if } \epsilon = \omega^{\ell-1} \end{array} \right.$$
%\end{itemize}
%Note that $\mathfrak{E}_y \ne 0$ when $\ell=1$ and $ \alpha_f \alpha_{g_y}^{-1} \chi(p)\ne 1$. 



 
 
% Let $V_{g_y}^-$ denote the  unramified quotient of $V_{g_y}$, arising in the counterpart for $g_y$ of the exact sequence \eqref{eqn:local-triangular-class}, which can be obtained as the specialisation at $y$ of \eqref{Wiles-ord}.Write $$D(V_{g_y}) = (V_{g_y} \otimes \mathbf{B}_{\dR})^{G_{\Q_p}}$$ for the (de Rham) Dieudonn\'e module associated to  $V_{g_y}$. It is a filtered module which becomes isomorphic, after extending scalars from $\Q_p$ to $\Q_p(\mu_{p^s})$,  to the module $D_{\cris}(V_{g_y})$, viewed in Section \ref{sec:regulators}as a filtered Frobenius module.
%, which may be identified with a submodule of the de Rham cohomology group $H^1_{\dR}(X_s,\cL_\ell)$ of $X_s$ with values in a suitable sheaf $\cL_\ell$.  

% Let $\omega_{g_y} \in D(V_{g_y})$ denote the normalized regular differential form attached to $g_y$, which spans the non-trivial step in the Hodge filtration of $D(V_{g_y})$, and let $\omega_{g_y^w} \in D(V_{g_y^*})$ be the similar element associated to $g_y^*$.  

% As it is shown by Ohta in \cite{Oh00},  there exists a canonical  $\Lambdag$-module generator  $$ \omega^-_{\hg} \quad \mbox{ of } \quad D(\mathbb{V}_{\hg}^-) := (\mathbb{V}_{\hg}^-\,\hat\otimes \,\hat{\Z}_p^{\ur})^{G_{\Q_p}} = (\Lambda_g({\psi_{\hg}})\,\hat\otimes \,\hat{\Z}_p^{\ur})^{G_{\Q_p}} \subset \Lambda_g\,\hat\otimes \,\hat{\Z}_p^{\ur}(\psi_{\hg}^{-1}), $$  where the second equality follows from \eqref{Wiles-ord}. 
% Ohta's element $\omega^-_{\hg}$ is characterized by the property that  for all classical points $y\in \Omega_{\hg}$ of weight $\nu_{2,\varepsilon}$,  the specialisation of $\omega^-_{\hg}$ to $V^-_{g_y} \otimes \hat \Z_p^{\ur}$   agrees with the natural projection of $\omega_{g_y}$ to  $$D_{\dR}(V^-_{g_y}) := (V^-_{g_y} \otimes \mathbf{B}_{\dR})^{G_{\Q_p}}   = (V^-_{g_y} \otimes \hat{\Z}^{\ur}_p)^{G_{\Q_p}}, $$ where the last equality follows from the fact that $V^-_{g_y}$ is unramified.  For any classical point $y\in \Omega_{\hg}$ of weight  $\mathrm{wt}(y)=\nu_{\ell,\epsilon}$ with $\ell \geq 2$ and $\epsilon \ne \omega^{\ell-1}$, write $\omega^-_{g_y}\in  D_{\dR}(V^-_{g_y})$   for the specialisation of $\omega^-_{\hg}$ at $y$. In general, when $\ell \geq 3$,   $\omega^-_{g_y}$ is just a non-zero multiple of the projection of $\omega_{g_y}$   to $D_{\dR}(V^-_{g_y})$.

% The element $\eta_f \otimes \omega^-_{g_y^w}$ is a natural basis of  $D_{\dR}(V^g_{f g*_y})$ over $K_y$. We use this choice of generator  it  fix the  isomorphism $D_{\dR}(V^f_{fg_y}) \simeq K_y$, in such a way  that the logarithm map of \eqref{logs} is now given by
% \begin{equation} \label{kappaf} \log_p \, \kappa^f_p(f,g_y) \, = \,  \log_p \,\kappa_p(f,g_y) (\eta_f \otimes \omega_{g_y^w}). \end{equation}

Recall the point $y\in \Omega_g$ of weight-character $\nu_{1,1}$ satisfying $\hg_{y} = g_\alpha$, and let $\Lambdag'$ denote the localisation of $\Lambdag$ at the prime
 ideal $\ker(y)$. 




\begin{theorem}
\label{pL-PR} 
 For the map $\cL$ as above,
\begin{equation}
\label{pL-PR-eq}
  \cL(\kappa_p^f(f,\hg))  
= L_p(f,\hg) 
\pmod{(\Lambdag')^\times}.
 \end{equation}
\end{theorem}
\begin{proof}
Theorem \ref{prop:bloch-kato-log-in-families}  implies that the elements 
$$ \cL(\kappa_p^f(f,\hg)), \qquad (1-\chi(p)^{-1} \alpha_f^{-1} \alpha_{\hg}^{-1})^{-1}  L_p(f,\hg)$$
agree on all points $y\in \Omega_g$ of weight $\nu_{2,\epsilon}$, as $\epsilon$ ranges over all the characters of $p$-power conductor.
Since  both elements belong to the fraction field of $\Lambda_g$, it follows that they must be  equal.   Theorem \ref{pL-PR} follows after noting that the factor $(1-\chi(p)^{-1} \alpha_f^{-1} \alpha_{\hg}^{-1})$ does not lie in the kernel of the weight one point $y$
attached to $g_\alpha$  (for reasons of weight, since $\alpha_f$ has complex absolute $\sqrt{p}$ while $\alpha_{g_{y}}$ is a root of unity), and hence 
belongs to $(\Lambdag')^\times$. 
\end{proof}

% When $y \in \Omega_{\hg}$ is a point of weight $\nu_{2,\epsilon}$, with $\epsilon\omega^{-1}$ of conductor $p^s$, $s\ge 1$, equation \eqref{variation-log} gives
% \begin{equation} \label{variation-log-bis} \log_p \kappa_p^f(f,g_y)  =  \gauss(\omega^{-1}\epsilon) \alpha_f^s \alpha_{g_y}^{-s}   \cL(\kappa^f_p(f,\hg))(y). \end{equation} Combining \eqref{variation-log-bis} with the equality $$ \log_p \,\kappa_p^f(f,g_y)  = \mathcal{G}(\chi\epsilon^{-1} \omega)\cdot (1-\chi(p)^{-1} \alpha_f^{-1}\alpha_g^{-1})^{-1}\cdot \alpha_f^{s-1} \alpha_{g_y}^{-s} \cdot  L_p(f,\hg)(y) $$
% provided by Theorem \ref{prop:bloch-kato-log-in-families} in light of equation \eqref{logs}, one finds
% $$
% L_p(f,\hg)(y) = \mathcal{G}(\chi)^{-1} \cdot \alpha_f(1-\chi(p)^{-1}\alpha_f^{-1}\alpha_{g_y}^{-1}) \cL(\kappa^f_p(f,\hg))(y) ,$$
% for infinitely many points $y$ of weight $\nu_{2,\epsilon}$.
% Since a non-zero element of $\Lambda_g$ can only vanish at  finitely many points of $\Omega_{\hg}$, the theorem follows. \end{proof}



 % The work \cite{LZ} of Loeffler and Zerbes can be used to show (cf.~for instance \cite[\S 5.3, 5.4]{DR2}) that there  exists an element $\cL(\kappa_p^f(f,\hg))\in \Lambdag$  satisfying, for all points $y \in \Omega_{\hg}$ with $\mathrm{wt}(y)=\nu_{\ell,\epsilon}$:
% \begin{equation}\label{variation-log}
% \cL(\kappa_p^f(f,\hg))(y) =  \left\{ \begin{array}{ll}
% \mho_y\mho_{\Psi_y}   \cdot \log_p\,\kappa^f_p(f,g_y), & \mbox{ if } \ell \ge 2, \\
% \mho_y\mho_{\Psi_y}   \cdot  \exp_p^*\,\kappa^f_p(f,g_y), & \mbox{ if } \ell \le 1,
%  \end{array}\right.
% \end{equation}
% where $\mho_{\Psi}$ is a fixed generator for the free rank one $\Lambda_g$-module
% $(\Lambdag(\Psi^{-1})\hat\otimes\hat\Z_p^{\ur})^{G_{\Q_p}}$ (cf.\ Prop.\ 3.3 of \cite{LZ}) and $\mho_{\Psi_y}$ denotes its specialisation 
% at $y$.

At the  weight one point $y$ attached to $g=g_\alpha$, we have 
$H^1_{\fin}(\Q_p, V_{f g}) = 0$, and hence the class 
$\kappa_p^f(f,g_{\alpha})$ can only be cristalline if it is $0$.
Let
$$ \exp_p^*: H^1(\Q_p, V_{f g})  
\lra 
\Fil^0 D(V_{f g})$$
denote the dual exponental map of Bloch and Kato.
\begin{theorem}
\label{thm:last-result}
Let $y=\in\Omega_{\hg}$ be the  point over $\nu_{1,1}$, corresponding to the classical $p$-stabilised weight one form
$g_y= g_\alpha\in S_1(Np,\chi)$. Then
$$\exp^\ast_p(\kappa_p^f(f,g_\alpha))\ne 0 \quad \Leftrightarrow\quad L(f\otimes g,1)\ne 0.$$
\end{theorem}
\begin{proof}
Perrin-Riou's local reciprocity law and its extension in 
\cite[Thm.~4.15]{LZ}  asserts that, at the weight one point $y$ attached to 
$g_\alpha$, 
$$ \cL(\kappa_p(f,\hg))(y_0) \sim  \exp^* \kappa^f_p(f,g_{\alpha}),$$
where  the symbol $\sim$ denotes agreement up to an (explicit) non-zero
element of $\Fil^0 D_{\dR}(V_{f g_{\alpha}})$. (The Euler factor  that arises in the $p$-adic interpolation process is non-vanishing at $y$, since the associated Galois representation
$V_{f g}$ is pure of weight $-1$.)
The result now  follows by combining Theorems  \ref{pL-PR}
and  \ref{thm:weight-one-specialisation}.
\end{proof}
Theorem \ref{thm:last-result}
 implies that the class $\kappa(f,g_y) = \kappa(f,g_\alpha)$ of equation \eqref{def-kappa-fgy} satisfies the two properties listed in Theorem \ref{thm:main1}. 
This concludes the proof of Theorems A and B of the Introduction.

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\end{document}

