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\begin{document}

\title[Generalised Kato classes]{Elliptic curves of rank two   \\ and generalized Kato classes}

\author{Henri Darmon and Victor Rotger}

\begin{abstract} 
Heegner points play an outstanding role in the study of the 
Birch and Swinnerton-Dyer conjecture, 
providing canonical Mordell-Weil generators whose heights encode  first derivatives of the associated Hasse-Weil $L$-series.
Yet the fruitful connection between Heegner points and $L$-series
 also accounts for their main limitation, namely, that
 they are torsion
 in (analytic) rank $>1$.  
This partly expository article discusses the {\em generalised Kato classes}  
introduced in \cite{BDR2} and \cite{DR2}, stressing their analogy with Heegner
points but explaining why they are expected to give
 non-trivial, canonical elements of the idoneous Selmer group
  in settings where the classical
 $L$-function (of Hasse-Weil-Artin type) that
governs their behaviour has a {\em double zero}  at the center.

The generalized Kato class denoted
$\kappa(f,g,h)$ is associated to
  a triple 
$(f,g,h)$ consisting of an eigenform $f$  of weight two 
and    
 classical $p$-stabilised 
  eigenforms $g$ and $h$ of weight one,
  corresponding   to 
odd two-dimensional Artin representations $V_g$ and $V_h$ of
  $\Gal(H/\Q)$  with $p$-adic coefficients for a suitable number field $H$.
This class is  germane to the Birch and Swinnerton-Dyer 
conjecture  over $H$ for the modular abelian variety $E$ over $\Q$ attached to $f$.  One of the main results of \cite{BDR2} and \cite{DR2} is that   
 $\kappa(f,g,h)$ 
  lies in the pro-$p$
Selmer group of $E$ over $H$ precisely  when $L(E,V_{gh},1)=0$, where 
$L(E,V_{gh},s)$ is the $L$-function of $E$ twisted
by   $V_{gh}:= V_g\otimes V_h$.
In the setting of interest, parity considerations
 imply that $L(E,V_{gh},s)$ vanishes to even order at $s=1$, and
the
Selmer class $\kappa(f,g,h)$ is expected to be trivial when 
$\ord_{s=1}L(E,V_{gh},s) >2$. 
 The 
main new contribution of this article is a conjecture expressing  
 $\kappa(f,g,h)$ as a canonical point
 in $(E(H)\otimes V_{gh})^{G_\Q}$ when
 $\ord_{s=1} L(E,V_{gh},s)=2$. 
 This conjecture strengthens and refines the main conjecture
 of \cite{DLR}, and supplies a framework for understanding  
  the results of \cite{DLR}, \cite{BDR2} and
 \cite{DR2}.
\end{abstract}



\dedicatory{In memory of Robert Coleman}

\address{H. D.: Montreal, Canada}
\email{darmon@math.mcgill.ca}
\address{V. R.: Departament de Matem\`{a}tiques, Universitat Polit\`{e}cnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain}
\email{victor.rotger@upc.edu}

\subjclass{11G18, 14G35}


\maketitle


\tableofcontents

\section{Background and motivation}

The theme of 
 modularity of $p$-adic Galois representations has occupied center
stage in number theory for the last several decades, and  Robert Coleman has been
a major figure in many of its key developments, notably through the theory of 
{\em Coleman families} of $p$-adic modular forms 
and of the
{\em Coleman-Mazur eigencurve} parametrising these families and 
their associated Galois representations. 
By way of background and motivation,
this section explains how much of the  progress achieved on the Birch
and Swinnerton-Dyer conjecture, including the results of
\cite{DLR}, \cite{BDR2} and \cite{DR2}, can be viewed as part of the larger 
program of understanding the modularity of
 ({\em non-semisimple}) $p$-adic Galois representations.
 
 
\medskip
One of the most celebrated modularity results  is the statement  that
all elliptic curves over $\Q$ arise as    quotients of  suitable modular curves:
more precisely, that an elliptic curve $E$ over $\Q$ of conductor $N$ is 
equipped with a surjective  parameterization
\begin{equation}
\label{eqn:modular-par}
\pi_E:  X_0(N)\lra E,
 \end{equation}
  where $X_0(N)$ is the modular curve
attached to Hecke's congruence subgroup $\Gamma_0(N)$.
This was proved in \cite{wiles}, \cite{taylor-wiles}, and \cite{bcdt}
by showing that the $p$-adic representation
$$ H^1(E) := H^1_{\et}(E_{\bar\Q},\Q_p)(1) =  (\lim_{\leftarrow,n} E[p^n])\otimes_{\Z_p} \Q_p  $$
 of
$G_\Q:= \Gal(\bar\Q/\Q)$
  arises as a quotient of the \'etale cohomology group\footnote{The systematic shorthand
$H^i(X) := H^i_{\et}(X_{\bar\Q},\Q_p(i))$
for any variety $X$ over $\Q$ is adopted henceforth  to lighten the notations.}
$$H^1(X_0(N)):=H^1_{\et}(X_0(N)_{\bar \Q},\Q_p(1)).$$ 
The existence of a   Galois-equivariant projection
\begin{equation}
\label{eqn:wiles-thm}
\pi_E: H^1(X_0(N)) \lra H^1(E)
\end{equation}
is the real content of the breakthrough in 
\cite{wiles} and  \cite{taylor-wiles},  
the ostensibly stronger geometric version \eqref{eqn:modular-par} 
being deduced from it by invoking the Tate conjecture
 for curves\footnote{Subsequently,  \eqref{eqn:wiles-thm} has
been generalised
to a host of other   $p$-adic Galois representations, 
while analogues of  \eqref{eqn:modular-par} remain
unavailable in all but the simplest geometric settings.}.

Let $E'$ be an open subvariety of $E$, i.e., the complement  of a zero-dimensional 
subvariety $\Sigma$ of $E$ over $\Q$.
The $p$-adic Galois representation $H^1(E')$ sits in the middle of the short exact
excision
sequence
$$ 0 \lra H^1(E) \lra H^1(E') \lra H^0(\Sigma)_0 \lra 0$$
 of \'etale cohomology groups, where the subscript of $0$  denotes the 
 degree $0$ elements of $H^0(\Sigma)$. 
 By analogy with \eqref{eqn:wiles-thm}, the curve $E'$ is (provisionally) 
 said to be {\em modular}
 if $H^1(E')$ arises as a subquotient of $H^1(Y)$, where $Y$ is an 
 {\em open sub-Shimura variety} of 
 $X_0(N)$---the latter being defined, in the style of 
 La Palice,  as the complement of a closed sub-Shimura variety. 
 
 
 To completely describe the open sub-Shimura varieties of the
modular curve  $X_0(N)$ over $\Q$, note that the latter is the coarse moduli space of elliptic 
 curves $A$  with a marked subgroup scheme of order $N$, and that
 its closed sub-Shimura varieties are  obtained
 by imposing additional endomorphism rings, which can only be equal to 
  orders in   quadratic imaginary fields.
  Given such an  order $\cO\subset K$, 
the associated closed sub-Shimura variety  
 $\Sigma_{\cO}\subset X_0(N)$   consists of $CM$ points for $\cO$, and is
  the coarse moduli space of elliptic curves $A$ with level $N$ structure equipped with an optimal embedding $\iota:\cO\lra \End(A)$ (respecting the level  structure) and acting in a 
 prescribed way on the cotangent space of $A$. 
 By the theory of complex multiplication, the $0$-dimensional variety $\Sigma_{\cO}$ is
  isomorphic over $K$
  (at least, when the discriminant of $\cO$ is prime to $N$)
   to $\phi_K(N)$ copies 
 of $\spec(H_\cO)$, where
 $\phi_K(N)$ is the number of primitive ideals of $K$ of norm $N$ and $H_{\cO}$ is the  ring class field of $K$ attached to $\cO$, whose Galois group over $K$ is canonically identified with the Picard group of 
$\cO$ via global class field theory.
 
 
 
 The complements 
 $$Y_{\cO}(N) := X_0(N) -\Sigma_\cO$$
 thus  provide an exhaustive list of the open
 sub-Shimura varieties of $X_0(N)$. Given the modularity of $E$, the modularity of $E'$ amounts to the existence of a Galois-equivariant
 inclusion
 $$i: H^0(\Sigma)_0 \lra H^0(\Sigma_\cO)_0$$
  for suitable $\cO$, realising 
 $H^1(E')$  as a subquotient of $H^1(Y_{\cO}(N))$ via the pushforward under $\pi_E$ and the pullback under $\iota$ of the  first row in the following diagram with exact rows:
 \begin{equation}
 \label{eqn:modularity-open}
 \xymatrix{
 0 \ar[r] & H^1(X_0(N)) \ar[r] \ar[d]_{\pi_E}  & H^1(Y_{\cO}(N)) \ar@{.}[d]_{?} \ar[r] &
 H^0(\Sigma_{\cO})_0 \ar[r] & 0 \\
 0 \ar[r] & H^1(E) \ar[r] & H^1(E') \ar[r] & H^0(\Sigma)_0 \ar[u]_i \ar[r] & 0. 
  }
\end{equation}
 

Consider the simplest non-trivial setting where $\Sigma = \{P_1, P_2\}\subset E(\Q)$ 
consists  of two points defined over $\Q$, so that
$H^0(\Sigma)_0=\Q_p$ with trivial Galois action.
The  resulting extension
\begin{equation}
\label{eqn:deltaP1P2}
 \xymatrix{ 0 \ar[r]& H^1(E) \ar[r] & H^1(E') \ar[r] & \Q_p \ar[r]& 0 }
\end{equation}
encodes the image of the point $P_2-P_1\in E(\Q)$ under the connecting
homomorphism
$$ \delta: E(\Q) \lra H^1(\Q,H^1(E)) := \Ext^1_{G_\Q}(\Q_p,H^1(E))$$
of Kummer theory,
where the $\Ext$ group is taken in the category of continuous $p$-adic
representations of $G_\Q$. 
The following statement, which  gives a  ``modularity criterion" for $E'$
and
 encapsulates many of  the deepest theorems
on the Birch and Swinnerton-Dyer conjecture obtained in the last decades,
is of course expected to hold for all elliptic curves $E$, 
but   the reader  is cautioned 
that the proof of the implication (d) $\Rightarrow$ (a)   currently requires that  $E$ be a semistable 
elliptic curve having at least one odd prime of  nonsplit multiplicative reduction  or  
at  least  two  odd  primes  of split multiplicative reduction. 
\begin{theorem}
\label{thm:gzks} 
Assume that the point $P_2-P_1$ is of infinite order in $E(\Q)$. Then the following are equivalent:
\begin{itemize}
\item[(a)]
The curve $E' = E\setminus \{ P_1,P_2\}$ is modular;
\item[(b)]
the Hasse-Weil $L$-series $L(E,s)$ has a simple zero at $s=1$;
\item[(c)]
the point $P_2-P_1$ generates $E(\Q)\otimes \Q$  and $\sha(E/\Q)$ is finite;
\item[(d)] for all primes $p$,
the group $\Ext^1_{\rm fin}(\Q_p,H^1(E))$ of extensions of $p$-adic 
representations of the
Galois group of $\Q$ that are cristalline at $p$ is one-dimensional over $\Q_p$.
\end{itemize}
\end{theorem}
\begin{proof}[Sketch of proof]
The modularity of $E'$
amounts to the statement that there exists an order $\cO$ in an imaginary quadratic field $K$ such that  the extension \eqref{eqn:deltaP1P2} can be obtained as the pullback of 
\eqref{eqn:modularity-open} 
via an inclusion
$i: \Q_p \lra H^0(\Sigma_{\cO})^{G_\Q}$, whose image contains a degree $0$ divisor
$$D_K\in \Div^0(\Sigma_{\cO})^{G_\Q} \subset \Div^0(X_0(N))(\Q).$$
This means that the point $P_1-P_2\in E(\Q)$ is a non-zero   multiple
of  the {\em Heegner point} $P_{E,K} := \pi_E(D_K)$.


The implication (a)$\Rightarrow$(b) therefore follows from the Gross-Zagier
formula  \cite{GZ}
expressing  the height of $P_{E,K}$  as a non-zero multiple of
$$L'(E/K,1) = L'(E,1) \cdot L(E^K,1),$$
where $E^K$ is the quadratic twist of $E$ by $K$. The existence of
a suitable $K$ for which $L(E^K,1)\ne 0$ follows from a non-vanishing result of Waldspurger or can be deduced from 
analytic number theory techniques (cf.\,\cite{MM}).

The implication (b) $\Rightarrow$ (c) was subsequently proved by Kolyvagin \cite{Ko}, who parlayed the non-triviality of $P_{E,K}$ into a bound on the Mordell-Weil rank and the Selmer group of $E$ over $K$. 

The implication (c) $\Rightarrow$ (d) is a direct consequence of the definitions: in fact (d)  is ostensibly  weaker than (c),   
Selmer groups being  less subtle to control than Mordell-Weil and Shafarevich-Tate 
groups.

 The striking  implication (d) $\Rightarrow$ (a) follows from
Skinner's  ``converse of the Gross-Zagier-Kolyvagin Theorem" \cite{skinner}. This last step is the most recent, and  combines several   new ingredients: the powerful techniques developed by Skinner and Urban to prove the Iwasawa-Greenberg Main Conjecture for elliptic curves over $\Q$ \cite{skinner-urban}, an important variant explored by Xin Wan in his PhD thesis \cite{wan}, and the $p$-adic analogue of \cite{GZ}  formulated and proved in \cite{BDP1}. 

More precisely, choose a prime  $p\geq 5$  of good ordinary reduction for $E$
 such that $E[p]$ is an irreducible $G_\Q$-representation and
  the image of the restriction map $\Sel_p(E) \lra E(\Q_p)/pE(\Q_p)$ does not lie in the
   image of $E(\Q_p)[p]$. 
   A result of Waldspurger ensures the existence of an odd quadratic character $\chi$ 
   such that $L(E,\chi,1)\ne 0$, which can be chosen  so that 
       $\chi(2)=\chi(p)=1$. Let $K$ denote the imaginary quadratic field associated to $\chi$. 
The $p$-adic Selmer group $\Ext^1_{K, {\rm fin}}(\Q_p,H^1(E))$  of $E$ over  $K$
(defined as an Ext group in the category of cristalline representations of $G_K$)
decomposes  as a direct sum of eigenspaces 
$$
\Ext^1_{K,{\rm fin}}(\Q_p,H^1(E)) \simeq \Ext^1_{{\rm fin}}(\Q_p,H^1(E)) \oplus \Ext^1_{K,{\rm fin}}(\Q_p,H^1(E))^{-}
$$
with respect to the action of complex conjugation. Because $L(E,\chi,1)\ne 0$,  the
 results of Kolyvagin (or of Kato)  imply the triviality of
 $\Ext^1_{K,{\rm fin}}(\Q_p,H^1(E))^{-}$. Assumption  (d) therefore implies that $\Ext^1_{K,{\rm fin}}(\Q_p,H^1(E))$ 
is one-dimensional over $\Q_p$. 
One can then argue  as in \cite{skinner}. Namely, the running hypotheses ensure that both Lemma 2.3.2 and Proposition 2.7.3 of loc.cit. apply, and hence, that a 
 $p$-adic $L$-function of the type that occurs in 
 \cite{wan} and \cite{BDP1}  (which interpolates critical values of the $L$-series of the Rankin convolution of the modular form $f$ associated to $E$ with suitable Hecke characters of $K$ of higher infinity-type) 
does not vanish at the trivial point, which lies outside its region of 
classical interpolation. This in turn implies, in light of  \cite[Corollary 2.6.2]{skinner} resting on the variant of the Gross-Zagier formula of \cite{BDP1}, 
 that the Heegner point $P_{E,K}$  has non-trivial $p$-adic formal group logarithm, and 
 is therefore non-torsion.
 As already explained, the non-triviality of $P_{E,K}$   is equivalent to (a), and the implication 
 $(d)\Rightarrow (a)$ follows.
\end{proof}

The Birch and Swinnerton-Dyer conjecture admits an extension to elliptic curves twisted by
Artin representations which arises very naturally in the context of the modularity questions framed above. 
Let
$$\varrho: \Gal(H/\Q) \hookrightarrow  \Aut(V_\varrho) \simeq \GL_n(\bar\Q_p) $$
be an $n$-dimensional representation of the Galois group of a finite extension $H/\Q$, a so-called {\em Artin representation}, viewed as having coefficients in $\bar\Q_p$. The pair $(E,\varrho)$ gives rise to the Hasse-Weil-Artin $L$-series
$$L(E,\varrho,s) := \prod_{\ell} \det(1 - \ell^{-s} (\Frob_\ell^{-1})_{(H^1(E)\otimes V_\varrho)^{I_\ell}})^{-1},$$
where the product is taken over the rational primes $\ell$, the arithmetic frobenius element at
$\ell$ is denoted by $\Frob_\ell$, and $I_\ell$ denotes the inertia group at $\ell$.
The equivariant Birch and Swinnerton-Dyer conjecture for $E$ and $\varrho$,
denoted $\BSD(E,\varrho)$, asserts that
\begin{equation}
\label{eqn:BSDErho}
\ord_{s=1} L(E,\varrho,s) = \dim_{\bar\Q_p} (E(H)\otimes V_\varrho)^{G_\Q}.
\end{equation}
As a first step to understanding $\BSD(E,\varrho)$, it is natural to ask which
$\kappa\in \Ext^1_{\fin}(V_\varrho,H^1(E))$  can be realised
as a subquotient of a suitable $H^1(Y_\cO(N))$. 
The Artin representation $H^0(\Sigma_\cO)_0$ which appears in the upper rightmost term
of the diagram \eqref{eqn:modularity-open}
 is readily analyzed using the theory of complex multiplication.
 Namely,  the slightly larger Artin representation
  $H^0(\Sigma_\cO)$   decomposes as a direct sum
 $$H^0(\Sigma_{\cO}) \otimes \bar\Q_p  =  \oplus_{j=1}^{\phi_K(N)} W_j, \qquad \mbox{ where } 
 W_j = \oplus_\psi V_j(\psi),  \quad \mbox{ with }  V_j(\psi) \subset
V_\psi:=  {\Ind}_K^\Q \psi.$$
In this equation, the second direct sum is taken over the non-trivial, $\bar\Q_p$-valued,
 finite order characters  $\psi$ of $\Gal(H_\cO/K)$ modulo the involution $\psi\mapsto\psi^{-1}$, and 
 $V_j(\psi)$ is a non-trivial irreducible constituent of
 the two-dimensional  representation $V_\psi$ obtained by inducing
the Galois character  $\psi$ from $G_K$ to $G_\Q$. The representation $V_\psi$ is irreducible precisely when $\psi\ne\psi^{-1}$,
and in this case   a non-trivial class 
 $\kappa\in \Ext^1_{\fin}(V_\psi, H^1(E) \otimes\bar\Q_p)$ 
 is expected to be modular if and only if (any of) the analogues of
 conditions (b)-(d) of Theorem \ref{thm:gzks} are satisfied, namely:
 \begin{itemize}
\item[(b')]
The Hasse-Weil-Artin $L$-series $L(E, V_\psi,s)$ has a simple zero at $s=1$;
\item[(c')]
the representation $V_\psi$ occurs with multiplicity one in 
$E(H)\otimes \bar\Q_p$, and the $V_\psi$-isotypic component of the 
 $\sha(E/H)$ is finite;
\item[(d')]
the group $\Ext^1_{\rm fin}(V_\psi,H^1(E)\otimes\bar\Q_p)$  is one-dimensional over $\bar\Q_p$, and generated by 
$\kappa$.
\end{itemize}
Although such a precise result does not seem to appear in the literature, all the ingredients needed to 
prove it seem to be available in principle.

\medskip
The rather narrow notion of modularity described above has a few visible drawbacks:
\begin{enumerate}
\item
Very few Artin representations arise in the cohomology of the
$0$-dimensional Shimura  varieties $\Sigma_\cO$, which
are not even rich enough to capture all of the irreducible two-dimensional
Artin representations of $\Q$.  The  open Shimura
varieties $Y_{\cO}(N)$ thus appear  to   give  no purchase on $\BSD(E,\varrho)$ when
 $\varrho$ is not induced from a ring class character of an
imaginary quadratic field.
\item 
Theorem \ref{thm:gzks} suggests that the 
 modularity of elements of $\Ext^1_{\fin}(V_\psi,H^1(E))$  is 
 purely a  ``rank one phenomenon": if this Ext group has dimension $>1$, none of its
elements are expected to  be realised in  subquotients of any $H^1(Y_{\cO}(N))$.
\end{enumerate}
In order to relate a larger class of non-semisimple Galois representations to modular forms, 
it becomes desirable to relax the notion of modularity.
One way in which one might try to do this  is by replacing the 
curves $Y_{\cO}(N)$ with  more general  ``open Shimura varieties".  
These should include all the 
 varieties whose cohomology (at least, after semisimplification) is 
 directly related to
 automorphic forms via a suitable generalisation of the Eichler-Shimura congruence,
and would eventually encompass 
the complements of sub-Shimura varieties in larger Shimura varieties, as well as
Kuga-Sato varieties and other natural varieties fibered over Shimura varieties, 
 the complements of Heegner cycles in such varieties, and so on.
With this expanded notion of  modularity, the program  
of characterising the non-semisimple Galois representations that are modular
 becomes richer and more subtle. 
See \cite{BDP2} for a   fragment of  experimental mathematics that 
might be  viewed as fitting into this program. 
The following  question seems like it might repay further investigation, 
given the paucity of evidence, both theoretical and experimental, that has been
 gathered around it so far:
\begin{question}
Let $V_1$ and $V_2$ be Galois representations for which 
$\hom(V_1,V_2)$ is irreducible. Suppose that there is a 
non-trivial $\kappa\in \Ext^1_{\fin}(V_1,V_2)$ arising as a subquotient of the 
cohomology of an open Shimura variety. Is
$\Ext^1_{\fin}(V_1,V_2)$ necessarily one-dimensional?
\end{question}
If the answer to this question were ``yes", it would imply that the open curve
$E-\{P_1,P_2\}$ discussed in Theorem 
\ref{thm:gzks} is {\em never } modular when rank$(E(\Q))>1$.  (But see
the inspiring article
\cite{nekovar-scholl}, as well as the striking ongoing
work of Zhiwei Yun and Wei Zhang in the function field case,
for some tantalizing ideas  in the opposite, more optimistic direction.)

A second idea for enlarging the class of $p$-adic Galois representations deemed to be 
modular is to allow {\em $p$-adic limits }  of Galois representations arising in the cohomology of (open) Shimura varieties. 
This idea is very natural in light of the classical work of Deligne-Serre on Artin representations attached to weight one forms, 
whereby such Artin representations are obtained by piecing together the Galois representations attached to modular
forms of higher weights which are realised in the cohomology of Kuga-Sato varieties.
It is via this broader notion of modularity that all odd, irreducible  two-dimensional Artin representations of $\Q$ 
can be related to modular forms. The idea of realising automorphic Galois representations as $p$-adic limits 
has become pervasive in the subject, and led to important advances: for example, it
plays
 a key role  in the   recent construction   \cite{hltt} by
Harris, Lan, Taylor, and Thorne of Galois representations attached to non-self-dual automorphic 
forms on $\GL_n$. 
Even more germane to  this article,
    $p$-adic limits of automorphic Galois representations appear to  capture non-trivial extension 
    classes going beyond settings of ``multiplicity one", as is illustrated by the 
following theorem of Skinner and Urban \cite[Thm.~B]{su-icm}:
\begin{theorem}
\label{thm:su}
Let $E$ be an elliptic curve over $\Q$.
If $L(E,s)$ vanishes to even order $\ge 2$ at $s=1$, 
then the Selmer group
 $\Ext^1_{\fin}(\Q_p, H^1(E))$ of $E$ contains at
  least two linearly independent modular classes.
\end{theorem}
The modular classes in this theorem  are constructed  as $p$-adic limits of geometric 
Galois representations  in the cohomology of  Shimura varieties associated
 to the unitary group $U(2,2)$. Although
 these geometric Galois representations are believed to be semisimple, 
Theorem \ref{thm:su} rests on the fact that this feature need not persist in the limit.


\medskip
The primary goal of this article is to discuss a  different  approach
 for    constructing  {\em  canonical}  extension classes of $\varrho$ by 
$H^1(E)$ for a large class of self-dual Artin representations $\varrho$ of dimension $4$ (and their lower-dimensional subrepresentations, in case $\varrho$ is reducible) arising as the tensor product $\varrho=\varrho_1\otimes \varrho_2$ of a pair of odd, two-dimensional Artin representations.
The construction of these classes is one of the main results of \cite{DR2} (resp.\,\cite{BDR2}) when both $\varrho_1$ and $\varrho_2$ are irreducible (resp.\,when exactly one of $\varrho_1$ and $\varrho_2$ is irreducible), and is based on $p$-adic limits of {\em non-semisimple}, 
but ``geometrically modular" Galois representations.
These limit classes are referred to as {\em generalised Kato classes} because their construction is 
inspired by the seminal work \cite{kato} of Kato (cf.\,also \cite{scholl}, \cite{BD14}) on $\BSD(E,\chi)$ for $\chi$ a Dirichlet character.
Like Heegner points in the setting of $\BSD(E,V_\psi)$,  generalised Kato classes enjoy close relations
 to ($p$-adic) Hasse-Weil-Artin $L$-functions  attached to $E$ and $\varrho$, 
 but unlike Heegner points, they are expected to generate a  non-trivial subgroup of the Selmer group 
 attached to $E$ and $\varrho$ 
precisely  when $\ord_{s=1}L(E,\varrho,s) = 2$. 
The   formulae of  \cite{DR2}  (cf.~Corollary \ref{cor:DR2} below)
relating the linear independence of two
 generalised Kato classes to the non-vanishing  of certain $p$-adic $L$-series 
can thus be regarded as a $p$-adic Gross-Zagier formula ``in analytic rank two". 
 
The main new contribution of this article is a conjecture expressing  
the same generalised Kato classes as   canonical elements 
 in $(E(H)\otimes V_{\varrho})^{G_\Q}$ when
 this latter space is two-dimensional. 
 This conjecture strengthens and refines the ``elliptic Stark conjecture"
 of \cite{DLR}, and provides a framework for understanding  
  the results of  \cite{DLR}, \cite{BDR2} and
 \cite{DR2}.
The  settings in which $\varrho$ is reducible often take on special arithmetic interest and are described
in detail in the last chapter.


 \section{Hida families and periods for weight one forms}
\label{sec:periods}

This section provides background on certain canonical
 structures  associated to a 
weight one form $g$, arising from the Hida families specialising 
in weight one to (a $p$-stabilisation of)
$g$.  These  
are important for   the conjectures of 
Section \ref{sec:conjecture}, 
but   Section \ref{sec:periods}
can be skipped on a first reading by the reader  
 wishing to  get a quick feeling for the  generalised Kato classes described
 in  Sections 
 \ref{sec:definition} and \ref{sec:properties}. 
 On the other hand, it is also worth noting that
  Section \ref{sec:periods} is entirely self-contained.   Conjecture \ref{conj:stark},
  which can be viewed as a $p$-adic analogue of the Stark conjecture for the adjoint of the Galois representation attached to a weight one form, appears to be new and may be of independent interest.
 
  
 Let  $g\in S_1(N,\chi)$ be a newform of weight one and level $N$ with Fourier coefficients in   
  a field $L$,
 and let 
$$\varrho: G_\Q  \lra \Aut(V) \simeq \GL_2(L)$$
be the Artin representation associated to it  by the construction of Deligne and Serre. 
We view $\varrho$ as  acting on a two-dimensional $L$-vector space $V$,
where $L\subset \C$ can be chosen to be contained in a cyclotomic field. 

Let $H$ be the number field cut out by $\varrho$, so that $\varrho$ factors through $\Gal(H/\Q)$.   
Fix a rational prime $p$ and choose a prime $\fp$ of $H$ above $p$. The latter
determines a canonical inclusion 
$$H\subset H_p\subset \bar\Q_p$$
of $H$ in its completion $H_p$ at $\fp$. Assume  that the pair 
$(\varrho,p)$ satisfies the following conditions:
\begin{enumerate}
\item[(I)] The prime $p$  splits completely in $L/\Q$, so that 
$L$ is equipped with an embedding into $\Q_p$ which will be fixed from now on.
This assumption, which  is made solely  to lighten the notations and could easily be dispensed with,
allows $\varrho$ to be viewed as a $\Q_p$-linear representation via the natural action  of $G_\Q$
on
the $\Q_p$-vector space
 $V\otimes_L\Q_p$. 



 \item[(II)]
 The representation
$V$  is unramified at $p$. There is then a well defined arithmetic
 frobenius 
element 
$$\Frob_p\in\Gal(H/\Q)$$
 acting canonically on $V$, and the characteristic polynomial of $\varrho(\Frob_p)$
is equal to the  Hecke polynomial
 $$ 
 x^2-a_p(g) x + \chi(p)  =: (x-\alpha_g)(x-\beta_g)  $$
 attached to $g$.
\item[(III)] 
  The modular form $g$ is   {\em regular} at $p$, i.e., 
   $\alpha_g\ne \beta_g$.
   After possibly enlarging $L$, it may also be assumed that  
   this coefficient field contains
   the roots  of unity 
   $\alpha_g$ and  $\beta_g$.
 \item[(IV)]  The representation $\varrho_g$ is not induced from a character of a real quadratic
 field $K$ in which the prime $p$ splits. The rationale for this condition,
 which  seems to be essential for a number of the 
 constructions and conjectures   proposed in this paper, is explained in 
 \cite[\S 1.1]{DLR}.
 \end{enumerate}
  The $p$-stabilisations of $g$   at $p$ are the   normalised eigenforms of
  weight one  with Fourier coefficients in $L$ 
  defined by
  $$ g_\alpha := g(z) - \beta_g g(pz), \qquad g_\beta := g(z) - \alpha_g g(pz).$$
  They are eigenvectors for the $U_p$-operator satisfying
  $$ U_p g_\alpha = \alpha_g g_\alpha, \qquad U_p g_\beta = \beta_g g_\beta.$$
The Artin representation $V$ decomposes naturally as a direct sum
  $$ V = V^\alpha \oplus V^\beta$$
  into one-dimensional eigenspaces for   $\Frob_p$, 
  with eigenvalues $\alpha_g$ and $\beta_g$ respectively.
  
By a theorem of Hida, 
there exists a finite flat extension  $\Lambda_g$  of the Iwasawa algebra $\Lambda$
 and a Hida family $\hg \in \Lambda_\hg[[q]]$ 
  of tame level $N$ and tame character $\chi$   
passing through the $p$-stabilised weight one eigenform  $g_\alpha$. When $g$ is cuspidal, the regularity hypothesis imposed on $g$ implies that such a Hida family is unique, thanks to a recent result of Bella\"iche and Dimitrov \cite{BeDi}. 

The Hida family $\hg$ comes equipped with the following canonical structures:
 \begin{itemize}
 \item[(a)]  There is a  locally free $\Lambda_g$-module $\hV_{g}$
  of rank two, affording
 Hida's 
  ordinary $\Lambda$-adic Galois representation
$$
\varrho_{\hg}: G_\Q \lra  \Aut_{\Lambda_g}(\hV_{g}) 
$$
which is realised in the inverse limit of ordinary \'etale cohomology groups
associated to the tower $X_1(Np^r)$ of modular curves. This representation  
interpolates the Galois representations associated by Deligne
 to the classical specializations of $\hg$.
 \item[(b)]
The restriction of $\hV_{g}$ to $G_{\Q_p}$ admits a stable filtration 
\begin{equation*}
\label{lambdadic-fil}
0 \lra  \hU_{g}  \lra  \hV_{g}  \lra  \hW_{g} \lra  0,
\end{equation*}
where both $\hU_{g} $ and $\hW_{g}$ are flat  $\Lambda_\hg[G_{\Q_p}]$-modules that are
  locally
free of rank one over $\Lambda_\hg$, and the quotient
$\hW_{g}$  is unramified, with $\Frob_p$ acting on $\hW_{g}$ as multiplication by
the $p$-th  Fourier coefficient $a_p(\hg)$. 
\item[(c)]
Let ${\Q_p^{\nr}}$ denote the maximal unramified extension of $\Q_p$ and let 
 ${\widehat \Q_p^{\nr}}$ denote its $p$-adic completion.
In \cite{ohta}, Ohta constructs a canonical $\Lambda_g$-adic period
\begin{equation*}
%\label{Ohta-w}
\omega_{\hg} \in D(\hW_{g}) := ( {\widehat \Q_p^{\nr}} \hat\otimes \hW_{g})^{G_{\Q_p}},
\end{equation*}
corresponding to the normalised $\Lambda$-adic eigenform $\hg$ under the isomorphism
in Theorem (A) of the introduction of \cite{ohta}.
%  characterized by the fact that $y_{k,\epsilon}(\omega_{\hg})$  is the image in $D(W_{g_{k,\epsilon}})$ of the normalized regular differential form naturally associated to $g_{k,\epsilon}$.
\item[(d)]  There is a natural perfect Galois-equivariant duality, given in Theorem (B) of the introduction of
\cite{ohta},
$$  \hU_{g} \times \hW_{g} \lra \Lambda_g(\det(\varrho_{\hg})),$$
where $G_\Q$ acts on the module $\Lambda_g$ of the right-hand side
via the determinant of $\varrho_{\hg}$.
\end{itemize}
Let
$$
y_g: \Lambda_g \lra \Q_p
$$
be the specialization map
attached to the $p$-stabilised weight one form  $g_\alpha$.
By specialising the structures above attached to $\hg$ via the map $y_g$, we obtain 
\begin{itemize}
\item[(a')]
 A {\em non-canonical}  isomorphism of $\Q_p[G_\Q]$-modules
$$
\Phi_{g_\alpha}:    V_g:= \mathbb{\hV}_{g} \otimes_{y_g} \Q_p   \stackrel{\sim}{\lra}   V\otimes_L \Q_p.
$$
\item[(b')]
A non-trivial
$G_{\Q_p}$-stable filtration 
\begin{equation*}
\label{eqn:es-specg}
0 \lra  U_{g}  \lra  V_{g}  \lra  W_{g} \lra  0
\end{equation*}
of $V_g$ by one-dimensional subspaces,
where  $U_{g} := \hU_{g}\otimes_{y_g} \Q_p$ and $W_{g} := \hW_{g}\otimes_{y_g} \Q_p$. The Frobenius element
$\Frob_p$ acts on $W_{g}$ and $U_g$  as multiplication by
$\alpha_g$ and $\beta_g$ respectively.  Since these eigenvalues are assumed to be distinct,
the  exact sequence  above
%\eqref{eqn:es-specg}
splits canonically,
leading to the  identifications
\begin{equation*}
   U_g = V_g^\beta, \qquad W_g = V_g^\alpha, \qquad
   V_g = U_g \oplus W_g =  V_g^\beta\oplus V_g^\alpha.
  \end{equation*}
  \item[(c')]
Specialising  Ohta's period  leads to a canonical element 
\begin{equation}
\label{Ohta-w}
\omega_{g_\alpha} := y_g(\omega_{\hg}) \in D(V_g^{\alpha}) := (\Q_p^{\nr} \otimes 
V_{g}^\alpha)^{G_{\Q_p}} = (H_p \otimes V_g^\alpha)^{G_{\Q_p}}.
\end{equation}
  \item[(d')]
  The duality in (d) above specialises via $y_g$ to 
  a canonical  pairing of $\Q_p$-vector spaces
  \begin{equation*}
   \langle \ , \rangle:  V_{g}^\beta \times V_{g}^\alpha \lra \Q_p(\chi),
   \end{equation*}
  which induces a pairing by functoriality (denoted by the same symbol by a slight abuse of notation):
   \begin{equation}
  \label{eqn:pairing-g} 
  \langle\  , \ \rangle: D(V_g^\beta) \times D(V_g^\alpha) \lra D(\Q_p(\chi)).
\end{equation}
  When this pairing is perfect, it can be used to 
 define a period $\eta_{g_\alpha} \in D(V_g^\beta)$, as the  
 unique element satisfying
  $$ \langle \eta_{g_\alpha}, \omega_{g_\alpha} \rangle =  \gauss(\chi) \otimes 1,$$
  where 
  $$ \gauss(\chi) := \sum_{j=1}^{\mf_\chi)} \chi(j) e^{2\pi i j/\mf_\chi}$$ 
  is the Gauss sum attached to the Dirichlet character $\chi$, viewed as an element
  of $H_p$ by assigning an $\mf_\chi$-th root of unity in $H_p$ to the complex number
  $e^{2\pi i/\mf_\chi}$.
  \end{itemize}

Making use of the above arsenal we now turn to introduce certain $p$-adic periods associated to $g$ and the choice of a $L$-structure on $V_g$. We assume for simplicity in the sequel that $g$ is a {\em cusp} form, and thus $V_g$ is irreducible.

Fix a $G_\Q$-equivariant isomorphism
 $j_g: V \otimes_L \Q_p \lra V_g$ and let
 $V_g^{L}:= j_g(V)$ denote the associated $L$-rational structure on $V_g$, 
 which by Schur's lemma is well-defined up to scaling by $\Q_p^\times$. Since $j_g$ induces isomorphisms $V_g^\alpha \simeq V^\alpha \otimes_L \Q_p$ and $V_g^\beta \simeq V^\beta \otimes_L \Q_p$, we may choose $L$-bases $v_g^\alpha$ and $v_g^\beta$ for $V_g^L \cap V_g^\alpha$ and 
 $V_g^L \cap V_g^\beta$ respectively, so that $$V_g^L \cap V_g^\alpha = \langle v_g^\alpha\rangle_L\quad \mbox{ and } \quad V_g^L \cap V_g^\beta = \langle v_g^\beta\rangle_L.$$



Define $p$-adic periods
 \begin{equation}
 \label{eqn:periods-forg}
 \Omega_{g_\alpha} =  \Omega_{g_\alpha}(V_g^L)\in H_p^{\Frob_p = \alpha_g^{-1}}, \qquad\quad
 \Xi_{g_\alpha} = \Xi_{g_\alpha} (V_g^L)\in H_p^{\Frob_p = \beta_g^{-1}}
 \end{equation}
 by setting
 $$ \Omega_{g_\alpha} \otimes  v_g^\alpha =   \omega_{g_\alpha}, \qquad
 \Xi_{g_\alpha} \otimes v_g^\beta = \eta_{g_\alpha}.$$
 These periods depend on the choice of the basis $(v_g^\alpha, v_g^\beta)$ for 
$V_{g}^L$ , but only up to multiplication by $L^\times$. 


 Furthermore, 
 for all $\mu\in \Q_p^\times$,
 $$\Omega_{g_\alpha}(\mu V_g^L) = \mu^{-1} \cdot \Omega_{g_\alpha}(V_g^L), \qquad\quad
   \Xi_{g_\alpha}(\mu\cdot V_g^L) = \mu^{-1} \cdot  \Xi_{g_\alpha} (V_g^L).$$
 It follows  that the ratio
\begin{equation}
\label{eqn:period-g}
 \mathcal{L}_{g_\alpha} := \frac{\Omega_{g_\alpha}}{\Xi_{g_\alpha}} \in (H_p)^{\Frob_p = \frac{\beta_g}{\alpha_g}} 
 \end{equation}
is a number in $H_p^\times$ that is well-defined up to multiplication by elements in $L^\times$. 


This expression is a canonical
  $p$-adic period attached to the    eigenform $g_\alpha$, and 
  can be viewed as a $p$-adic avatar of the Petersson norm of   $g$. 

  
Definition 1.8 of \cite[\S 1.2]{DLR}  associates in many cases a canonical {\em $p$-adic Stark unit} $u_{g_\alpha}$ attached to 
$g_\alpha$ as follows. Let $V_{\rm ad}  := \End_0(V)$ be the three dimensional
adjoint representation attached to $V$ consisting of trace zero endomorphisms 
of $V$. Since complex conjugation acts with eigenvalues $-1$, $-1$ and $1$ on $V_{\rm ad}$, 
it follows that $\hom_{G_\Q}(V_{\rm ad}, (\cO_H^\times)_L) = L \cdot \varphi_{\rm ad}$
for a suitable generator $\varphi_{\rm ad}$. 
If one further assumes that $\alpha_g \ne \pm \beta_g$, then the subspace of
$V_{\rm ad}$ on  which $\Frob_p$ acts as multiplication by $\frac{\beta_g}{\alpha_g}$
is one-dimensional; after choosing an $L$-basis $v_{g_\alpha}$ for it, one   lets
 \begin{equation}
 u_{g_\alpha} := \varphi_{\rm ad}(v_{g_\alpha}).
 \end{equation}
 This  element  is well defined up to multiplication
 by $L^\times$, and hypothesis (IV) above guarantees that it is a non-zero vector of  the $V_{\rm ad}$-isotypic
  subspace of $(\cO_H^\times)_L$.
 See  \cite[\S 1.2]{DLR} for further details. 
  \begin{conjecture}
  \label{conj:stark}
The period in \eqref{eqn:period-g} satisfies
  $$ \mathcal{L}_{g_\alpha} = \log_p(u_{g_\alpha}) \pmod{L^\times}.$$
  \end{conjecture}
  \begin{remark}
  It would be interesting to  test this conjecture numerically.
  To the extent that $\mathcal{L}_{g_\alpha}$ is  a $p$-adic avatar of the 
  Petersson norm of $g$, Conjecture \ref{conj:stark}
  can be viewed as a $p$-adic analogue of the Stark conjecture for the 
  $L$-function attached to the adjoint of $g$, in the   form in which it is illustrated, for
  example, in the concluding paragraphs of \cite{stark}.
  \end{remark}
  
  

 
 
 
 
\section{Generalized Kato classes}\label{sec:Kato}
\subsection{Definition}
\label{sec:definition}
Let $E$ be an elliptic curve over $\Q$ and let
$$\varrho_1,\varrho_2: \Gal(H/\Q)  \lra \GL_2(L)$$
be odd, irreducible two-dimensional  
Artin representations of  $G_\Q$ satisfying
$$ \chi := \det(\varrho_1) = \det(\varrho_2)^{-1},$$
where  $L$ and $H$ are finite extensions of $\Q$ 
(and $L$ is chosen, as before, to be contained in a cyclotomic field).
Let $V_{1}$ and $V_{2}$ be  $L[\Gal(H/\Q]$-modules which are two-dimensional over $L$
and 
realise $\varrho_1$ and $\varrho_2$ respectively. Observe that
$$V_{12} := V_{1} \otimes_L V_{2} $$ 
is  a four-dimensional $L$-linear representation of
$\Gal(H/\Q)$ with real traces, i.e., it is isomorphic to its contragredient representation.

Fix a rational prime $p$ and continue to assume that hypotheses (I-IV) of the previous section hold for both the pairs $(\varrho_1,p)$ and $(\varrho_2,p)$.

The  progress in modularity realised over the last two decades 
implies the existence of cusp forms $f$, $g$, and $h$ attached to
$E$, $\varrho_1$ and $\varrho_2$ respectively, whose associated  $p$-adic representations,
 denoted
$V_f$, $V_g$ and $V_h$, satisfy
\begin{equation}
\label{eqn:isom-reps}
 H^1(E) = V_f, \qquad   V_{1} \otimes_L \Q_p \simeq V_g, \qquad  V_{2} \otimes_L \Q_p \simeq V_h.
 \end{equation}
  It is important to keep in mind that the last two
  isomorphisms,  whose existence is proved by comparing traces on both sides, are only well-defined up to
  multiplication by a scalar in $\Q_p^\times$ (by Schur's lemma), 
   and that the $\Q_p$-vector spaces
   $V_g$ and $V_h$ therefore admit no natural $L$-rational structure.
 Let 
 $$   V_{gh} := V_g \otimes V_h,  \qquad V_{fgh} := V_f \otimes V_{gh}   $$
 denote the tensor products of $\Q_p$-linear representations of $G_\Q$, and write 
   \begin{equation}
 \label{eqn:non-canonical-isom}
j_{gh}: V_{12} \otimes_L \Q_p \lra V_{gh}
 \end{equation}
 for the isomorphism induced from \eqref{eqn:isom-reps}. Let
 \begin{equation}
 \label{eqn:L-ratgh}
 V_{gh}^L := j_{gh}(V_{12})
 \end{equation}
  denote the resulting $G_\Q$-stable 
 $L$-rational structure on $V_{gh}$, which is well-defined up to multiplication by a scalar in 
 $\Q_p^\times$, even when $V_{gh}$ is reducible, because $V_g$ and $V_h$ themselves are
 irreducible.
 
 
  

 
 
Because of \eqref{eqn:isom-reps}, the Hasse-Weil and Artin $L$-functions
 attached to $E$, $\varrho_1$ and $\varrho_2$ 
  are
equal to the 
Hecke $L$-functions attached to $f$, $g$ and $h$ respectively:
$$ L(E,s) = L(f,s), \qquad  L(\varrho_1,s) = L(g,s), \qquad  L(\varrho_2,s) = L(h,s),$$
and therefore  admit 
functional equations and analytic continuations to the entire complex plane. By the theory of   
Rankin-Selberg and Garrett, the same is true of the degree $8$   $L$-function 
$L(V_{fgh},s)$ attached to the convolution of $f$, $g$ and $h$.

 
Let $N=\mathrm{lcm}(N_f,N_g,N_h)$ denote the least common multiple of the conductors of $E$, $\varrho_1$ and $\varrho_2$ and assume further that $p$ does not divide $N$.
As in the previous section,  let
 $$ 
 x^2-a_p(g) x + \chi(p)  =: (x-\alpha_g)(x-\beta_g), \qquad
  x^2-a_p(h) x + \chi^{-1}(p)  =: (x-\alpha_h)(x-\beta_h) $$
  be the Hecke polynomials at $p$
   attached to $g$ and $h$ respectively, and assume that the coefficient field
    $L$ contains
   the roots  of unity 
   $\alpha_g$, $\beta_g$, $\alpha_h$ and $\beta_h$.
Denote as before by
  $$ g_\alpha := g(z) - \beta_g g(pz), \quad g_\beta := g(z) - \alpha_g g(pz), \qquad h_\alpha := h(z) - \beta_h h(pz), \quad
  h_\beta := h(z) -\alpha_h h(pz)$$
  the relevant $p$-stabilisations of $g$ and $h$. 
  
  One of the running assumptions of \cite{DR2} that is also enforced in this article
 is that the Artin conductor of $V_{gh}$ is relatively prime to the conductor of $E$.
 Under this assumption, \cite[Theorem 1.4]{prasad} implies that the local root numbers that govern the sign in the functional equation for
 $L(E,V_{gh},s)$ are equal to $1$ at all  places of $\Q$,  and the Hasse-Weil-Artin $L$-function  
 attached to $E$ and $V_{gh}$ therefore vanishes to {\em even order} at the symmetry point 
 $s=1$ for its functional equation.

  
The article  \cite{DR2} describes the  construction of four 
canonical
 (a priori non-trivial, and distinct) {\em generalised Kato classes}
\begin{equation}
\label{eqn:gkc}
\kappa(f,g_\alpha,h_\alpha),  \  \kappa(f,g_\alpha,h_\beta), \ 
\kappa(f,g_\beta,h_\alpha), \  \kappa(f,g_\beta,h_\beta) \quad \in \quad
H^1(\Q,V_{fgh}).
\end{equation}
These classes are essentially obtained as {\em $p$-adic limits} 
\begin{equation}
\label{eqn:gen-kato-class}
 \kappa(f,g_\alpha, h_\alpha) := \lim_{k\lra 1} \kappa(f,g_k,h_k),
 \end{equation}
as $(g_k,h_k)$ range over the classical specialisations of weight $k\ge 2$ of Hida families 
$\hg$ and $\hh$ specialising to $g_\alpha$ and $h_\alpha$ respectively in weight one, and
$\kappa(f,g_k,h_k)$ arises from a geometric construction whereby it 
is realised in the $p$-adic \'etale cohomology of some (open) variety over $\Q$. 


More precisely, let $V_f(N)$ denote the $f$-isotypic component of $H^1(X_0(N))$, which is 
(non-canonically) isomorphic  to a finite number of copies of $V_f$, indexed by
 the positive divisors of $N/N_f$. 
Let $V_g(N)$ and $V_h(N)$ denote the similar spaces occurring as the 
 weight one specialisations of the $\hg$ and $\hh$-isotypic parts of the inverse limits of the ordinary quotients of
 $H^1(X_1(Np^s))$, which are abstractly isomorphic to  a direct sum of finitely many
 copies of $V_g$ and $V_h$ respectively, 
 endowed with all the structures  described in (a')-(d') 
 of Section \ref{sec:periods}.
The classes in (14) of \cite{DR2} take values in 
 the Galois representation 
 $$ V_{fgh}(N) = V_f(N) \otimes V_g(N) \otimes V_h(N)$$
 and the classes of \eqref{eqn:gkc} are obtained   by  applying to them a suitable
 surjective $G_\Q$-equivariant projection
 \begin{equation}
 \label{eqn:choice-pi}
  \pi: V_{fgh}(N) \lra V_{fgh}
  \end{equation}
 compatible with the $L$-structure, filtration,   Ohta periods and dualities described in 
 (a')-(d'). 
 The dependence of $\kappa(f,g_\alpha,h_\alpha)$ on the choice of $\pi$ is supressed 
 from the notations but should be kept in mind.


The generalised Kato classes belong to the global cohomology group
 $H^1(\Q,V_{fgh}) = \Ext^1_{G_\Q}(\Q_p, V_{fgh})$, where $\Q_p$ stands for the one-dimensional
$p$-adic representation of
$G_\Q$ with trivial action and the Ext group is taken in the category of 
finite dimensional $\Q_p$-vector spaces equipped with a continuous $G_\Q$-action
 (whose restriction to $G_{\Q_p}$ need not be  de Rham).

When $\hg$ and $\hh$ are cuspidal Hida families,
the ``weight two" classes
$\kappa(f,g_2,h_2)$ attached to   weight two 
specialisations $g_2$ and $h_2$ of $\hg$ and $\hh$ are obtained from
 the $p$-adic \'etale Abel-Jacobi image of  a {\em Gross-Kudla-Schoen diagonal cycles}
 in the Chow group
of null-homologous codimension two cycles in the triple
product  of the modular curve $X_1(Np^s)$. 
It is worth noting that when passing from $k=1$ to  $k>1$, the local root number at $\infty$ attached
to $L(V_{fg_kh_k},s)$ changes sign (while the other root numbers stay the same), so that
this $L$-function vanishes to {\em odd order} at its center. The presence of Gross-Kudla-Schoen diagonal cycles in this range is consistent  with the Beilinson-Bloch conjecture for
$L(V_{fg_kh_k},s)$ and in fact provides  evidence for it. (Cf.\,the preprint \cite{YZZ} of Yuan-Zhang-Zhang, 
where the case $k=2$ is studied.)
The fact that the extension 
$\kappa(f,g_\alpha,h_\alpha)$  does not arise directly in   $p$-adic 
\'etale cohomology, but only as a $p$-adic limit of  
 geometric Galois representations, explains 
  why $\kappa(f,g_\alpha,h_\alpha)$ need not be cristalline at $p$ in general. 
  %As we recall below in Theorem \ref{thm:DR2-C}, Theorem C of \cite{DR2} is a reciprocity law which ensures that $\kappa(f,g_\alpha,h_\alpha)$ is cristalline at $p$ if and only if $L(E,\varrho_{gh},1)=0$. 

The analogy with the work of Kato \cite{kato}, \cite{scholl} arises when the cuspidal Hida families $\hg$ and $\hh$ 
are replaced by Hida families of Eisenstein series.  
A global class $\kappaBK(f,g_\alpha,h_\alpha)$, designated as the
{\em Beilinson-Kato class} attached to $(f,g_\alpha,h_\alpha)$,  is then  defined as in 
\eqref{eqn:gen-kato-class},  but replacing the \'etale Abel-Jacobi images 
$\kappa(f,g_2,h_2)$ 
 by   
 $p$-adic \'etale regulators of  {\em Beilinson elements} in 
 the higher Chow group
$K_2(X_1(Np^s))=\CH^2(X_1(Np^s),2)$
attached to  a pair of modular units whose logarithmic derivatives 
give rise to $g_2$ and $h_2$. We refer the reader to \cite{BD14} for more details in this setting.


In the intermediate setting where exactly one of $\hg$ and $\hh$ 
(say, $\hg$) is cuspidal (and thus $\hh$ is Eisenstein), global classes
$\kappa(f, g_2,h_2)$ can be   constructed geometrically as 
  $p$-adic \'etale regulators of  suitable {\em Beilinson-Flach elements} 
in  the higher
Chow group $K_1(X_1(Np^s)^2)=\CH^2(X_1(Np^s)^2,1)$ 
 attached to a  modular unit  whose logarithmic derivative is $h_2$. 
 The limit cohomology class arising in \eqref{eqn:gen-kato-class} is then denoted
$\kappaBF(f,g_\alpha,h_\alpha)$,
and called the {\em Beilinson-Flach class}
  attached to the triple 
$(f,g_\alpha, h_\alpha)$. 
The  Beilinson-Flach classes in
$p$-adic families
were introduced and studied  in  \cite{durham}, \cite{BDR1} and \cite{BDR2}. See  also 
\cite{LLZ} and   \cite{KLZ} for more recent work 
leading to  substantial  extensions and  refinements  of the  results of
loc.cit. in the setting of Beilinson-Flach elements.
 
 

The $p$-adic Selmer group
\begin{equation}
\label{eqn:selmer-grp}
H^1_{\fin}(\Q, V_{fgh}) := \Ext^1_{\cris}(\Q_p, V_{fgh}) \end{equation} 
 attached to
 $E$ and $V_{gh}$ are 
 the group of  extensions of $\Q_p$ by
$V_{fgh}$   in the category of $\Q_p$-linear representations
 of $G_{\Q}$ that are cristalline at $p$.
Let 
 $$  (E(H)_L \otimes V_{12})^{\Gal(H/\Q)}, \qquad   \mbox{ where  } E(H)_L := E(H)\otimes_{\Z} L $$
 denote the  $\varrho_{12}$-isotypic part of the Mordell-Weil group of $E$.
  It is a finite-dimensional $L$-vector space by the
 Mordell-Weil theorem, and is equipped with a natural inclusion
 \begin{equation}
  \label{eqn:L-structure}
   (E(H)_L \otimes V_{12})^{\Gal(H/\Q)} \subset H^1_{\fin}(H, V_p(E) \otimes_{\Q_p} V_{gh})^{\Gal(H/\Q)} = H^1_{\fin}(\Q,V_{fgh})
      \end{equation}
    induced from the connecting homomorphism $\delta$ of Kummer theory for $E(H)$ and the map
   $j_{gh}$ of \eqref{eqn:L-ratgh}.  



 When $L(E,\varrho_{gh},s)$ has a double zero at $s=1$, Conjecture
 $\BSD(E,\varrho_{gh})$ described in the previous section
  predicts that the associated Mordell-Weil      and Selmer group  
  in \eqref{eqn:L-structure}
 are  $2$-dimensional over $L$ and $\Q_p$ respectively.
The finiteness of the relevant Shafarevich-Tate group
furnishes the Selmer group  with a natural $L$-rational structure  
$$ \Hom_{G_\Q}(V_{12}, E(H)_L) \subset H^1_{\Sel}(\Q, V_{fgh}).$$
  
As mentioned above, the vanishing of the central critical value  $L(E,\varrho_{gh},1)$ implies that the generalised Kato classes in \eqref{eqn:gkc} are cristalline at $p$ and thus belong to $H^1_{\fin}(\Q,V_{fgh})$. The main goal of this article is to give a precise conjectural description 
   of the position of the generalised Kato classes 
   in $H^1_{\fin}(\Q,V_{fgh})$ 
 relative to the $L$-structure given by  
  \eqref{eqn:L-structure}, in a way that recovers older conjectures of Kato and Perrin-Riou
in the setting of Beilinson-Kato  classes when $g$ and $h$ are  Eisenstein series, 
and is consistent with the theorems and conjectures
of \cite{DR2} and \cite{DLR}. 


\subsection{Basic properties}
\label{sec:properties}
In this section we recall some of the main properties of the generalised Kato classes  
already established in 
\cite{DR2} and \cite{DLR}. 

Restricting \eqref{eqn:selmer-grp} to $G_{\Q_p}$,
let   $$H^1_{\fin}(\Q_p, V_{fgh}) := \Ext^1_{\fin,\Q_p}(\Q_p, V_{fgh}) $$ denote 
 the group of cristalline extensions of $\Q_p$ by
$V_{fgh}$ in the category of $\Q_p$-linear representations of $G_{\Q_p}$, and let 
$H^1_{\sing}(\Q_p,V_{fgh}) := H^1(\Q_p, V_{fgh}) / H^1_{\fin}(\Q_p, V_{fgh})$ denote
the ``singular quotient" of the local cohomology at $p$. 
Recall that $L$ has been  chosen to be large enough to
 contain the frobenius eigenvalues $\alpha_g$, $\beta_g$, $\alpha_h$ and $\beta_h$, 
 which therefore belong to $\Q_p$. Let
\begin{equation}
\label{eqn:subspaces-Vg-Vh}
 V_g^{\alpha}, V_{g}^\beta \subset V_g, \qquad V_h^\alpha, V_h^\beta \subset V_h
 \end{equation}
be the  eigenspaces in $V_g$ and $V_h$ respectively associated to these eigenvalues,
and set
\begin{equation}
\label{eqn:subspaces-Vgh}
V_{gh}^{\alpha\alpha} := V_g^\alpha \otimes V_h^\alpha,  \qquad 
 V_{gh}^{\alpha\beta} := V_g^\alpha \otimes V_h^\beta,  \qquad 
  V_{gh}^{\beta\alpha} := V_g^\beta \otimes V_h^\alpha,  \qquad 
  V_{gh}^{\beta\beta} := V_g^\beta \otimes V_h^\beta 
  \end{equation}
  of $V_{gh}$.  
 Even though $V_g$ and $V_h$ are both assumed to be regular at $p$,
  the same need not be true
 for $V_{gh}$, and in this case
 \begin{equation}
 \label{eqn:frob-decomp}
  V_{gh} = V_{gh}^{\alpha\alpha} \oplus V_{gh}^{\alpha\beta} \oplus V_{gh}^{\beta\alpha} 
 \oplus V_{gh}^{\beta\beta}.
 \end{equation}
gives a strict refinement of the decomposition of $V_{gh}$  into frobenius eigenspaces.
  In fact, some of the most interesting arithmetic applications of the generalised Kato classes
   (notably those spelled out in sections  \ref{sec:cm-case}, \ref{sec:rm-case}, and
   \ref{sec:adjoint-case}) 
  arise when
    $V_{gh}$ is not regular at $p$.
  
The first basic result extends Kato's explicit reciprocity law (corresponding to the case where
$g$ and $h$ are both Eisenstein series) to the setting
where both $g$ and $h$ are 
cuspidal (Theorem C  
of \cite{DR2}) as well as to the intermediate Beilinson-Flach setting (Theorem 3.10 of \cite{BDR2}).
 
 \begin{theorem}
\label{thm:DR2-C}
The natural image of $\kappa(f,g_\alpha,h_\alpha)$ in $H^1_{\sing}(\Q_p, V_{fgh})$ 
belongs to $H^1_{\sing}(\Q_p, V_f \otimes V_{gh}^{\beta\beta})$, and analogously for the remaining classes of \eqref{eqn:gkc}. 
Moreover, the
 following are equivalent:
\begin{enumerate}
\item
For all choices of  $\pi$ in \eqref{eqn:choice-pi}, 
the generalised Kato classes of \eqref{eqn:gkc} belong  to the Bloch-Kato  Selmer group
of $V_{fgh}$, i.e., their images in $H^1_{\sing}(\Q_p, V_{fgh})$ are trivial;
\item
the central critical value
 $L(E,V_{gh},1)$ vanishes. 
\end{enumerate}
\end{theorem}
\noindent
Assume from now on that   $L(E,V_{gh},1) = 0$, so that 
\begin{enumerate}
\item
the $L$-series $L(E,V_{gh},s)$ has a zero of even order $\ge 2$ at $s=1$;
\item
the generalised Kato classes  of 
\eqref{eqn:gkc}  belong  to the Selmer group
attached to $E$ and $V_{gh}$.  
\end{enumerate}
 One is then naturally interested in a formulating non-vanishing criterion for these Selmer classes:
\begin{conjecture}
\label{conj:non-van}
The generalised Kato classes in \eqref{eqn:gkc} generate a non-trivial
subgroup of the Selmer group of $V_{fgh}$ for a suitable choice of 
$\pi$ in \eqref{eqn:choice-pi},  if and only if the following equivalent 
conditions are satisfied:
\begin{itemize}
\item[(a)] The $L$-series 
$L(E,V_{gh},s)$ has a double zero at $s=1$;
\item[(b)] the Mordell-Weil group $(E(H)_L\otimes V_{12})^{G_Q}$ is two-dimensional over
$L$;
\item[(c)] the Selmer group $H^1_{\fin}(\Q,V_{fgh})$ is two-dimensional over $\Q_p$.
\end{itemize}
\end{conjecture}
\begin{remark}
Although the equivalence of conditions (a), (b) and (c) certainly
lies very deep, it
is part of a well-established conjecture, namely $\BSD(E,V_{gh})$. 
The main novelty of Conjecture 
\ref{conj:non-van}
 is in providing a criterion for the non-triviality of the space
generated by the generalised Kato classes.
Note that Conjecture \ref{conj:non-van} does not predict that all four of the classes
in \eqref{eqn:gkc} are non-trivial, nor even that these four classes generate the Selmer group,
when (a), (b), and/or (c) are satisfied. These stronger 
conclusions are expected to be false in general, as illustrated by some of
the examples in Chapter \ref{sec:cases}.
\end{remark}

 
Let 
$$\kappa_p(f,g_\alpha,h_\alpha) = \res_p(\kappa(f,g_\alpha,h_\alpha))$$ 
denote the  image of the global class $\kappa(f,g_\alpha,h_\alpha)$ 
   in the local cohomology group
 $$H^1_{\fin}(\Q_p, V_{fgh}) = (H^1_{\fin}(H_p, V_f) \otimes V_{gh})^{\Gal(H_p/\Q_p)} = (E(H_p) \otimes V_{gh})^{\Gal(H_p/\Q_p)}.$$
As we describe more explicitly below, Theorem D of \cite{DR2} asserts that this image is controlled  by suitable $p$-adic avatars
of  the {\em second derivative} of the classical $L$-series $L(f, V_{gh},s)$ at the central critical point $s=1$.

These $p$-adic values were defined and explored in \cite{DR2} and \cite{DLR} and are denoted
\begin{equation}
\label{eqn:Lvalues}
\Lp^{g_\alpha}(\testf,\testg^*,\testh), \qquad \Lp^{g_\beta}(\testf,\testg^*,\testh), \qquad \Lp^{h_\alpha}(\testf,\testg,\testh^*), \qquad \Lp^{h_\beta}(\testf,\testg,\testh^*).
 \end{equation}
They depend on the choice of certain test vectors 
$$(\testf,\testg,\testh)\in S_2(N; L)\times M_1(N,\chi;L) \times M_1(N,\chi^{-1};L)$$
 with the same system of Hecke eigenvalues as $f$, $g$ and $h$ respectively, 
and with fourier coefficients in $L$, and on the choice of dual test vectors 
$$(\testg^*,\testh^*) \in \Hom(M_1(N,\chi^{-1};L),L)\times \Hom(M_1(N,\chi;L),L)$$
 with the same system of Hecke eigenvalues as $g$ and $h$. 
 We refer to the introduction of 
\cite{DR2} for more details on their definition,    contenting ourselves with remark that the $p$-adic $L$-value $\Lp^{g_\alpha}(\testf,\testg^*,\testh)$ is  
 defined essentially as the $p$-adic limit of central critical values
$$
\Lp^{g_\alpha}(\testf,\testg^*,\testh) := \lim_{\ell \ra 1} \cE(f,g_\ell,h) \times C(\testf,\testg^*,\testh) \times \frac{L(V_f\otimes V_{g_\ell}\otimes V_h, (\ell+1)/2)}{\langle g_\ell,  g_\ell \rangle},
$$
 as  $g_\ell$ ranges over the specialisations of (odd) weight $\ell \ge 3$ of the Hida family $\hg$ specialising to 
 $g_\alpha$ in weight one. Here  $\cE(f,g_\ell,h)$ is a $p$-adic multiplier arising from a recipe of Panciskin, whose presence allows
 the $p$-adic interpolation of the special values above, and $C(\testf,\testg^*,\testh)$ is a  product 
 over the primes dividing
  $N\cdot \infty$ of local terms 
 which depend  in a simple way on the choice of   test vectors.


 Choose a basis of $V_{gh}$ (over $\Q_p$, for now) which is compatible with the decomposition 
\eqref{eqn:frob-decomp}, i.e., choose non-zero vectors
\begin{equation}
\label{eqn:basis-of-vs}
 v_{gh}^{\alpha\alpha} \in V_{gh}^{\alpha\alpha}, \quad v_{gh}^{\alpha\beta} \in V_{gh}^{\alpha\beta}, \quad
v_{gh}^{\beta\alpha} \in V_{gh}^{\beta\alpha}, \quad v_{gh}^{\beta\beta} \in V_{gh}^{\beta\beta}.
\end{equation}

Write
\begin{equation}
\label{eqn:reskappa-R}
\kappa_p(f,g_\alpha,h_\alpha) =  R_{\alpha\alpha} \otimes v_{gh}^{\beta\beta} + R_{\alpha\beta} \otimes v_{gh}^{\beta\alpha} + 
R_{\beta\alpha} \otimes v_{gh}^{\alpha\beta} + R_{\beta\beta} \otimes v_{gh}^{\alpha\alpha}.
\end{equation}
The coordinate
$R_{\xi} $ belongs to $E(H_p)_{\Q_p}^{\Frob_p = \xi}$, where
$\xi$ ranges over the index set
 $$\{\alpha \alpha=\alpha_g\alpha_h,  \ \ \ \alpha \beta=\alpha_g \beta_h, \ \ \ \beta\alpha=\beta_g \alpha_h, \ \ \  \beta \beta = \beta_g \beta_h \}.$$
  Note that $R_{\xi}$ is even the image of a global point in $E(H)_{\Q_p}$,
assuming the finiteness of the Shafarevich-Tate group of $E$ over $H$.
Let 
\begin{equation}
\label{eqn:E-log}
 \log_p: E(H_p)_{\Q_p} \lra H_p
 \end{equation} 
denote the formal group logarithm attached to an invariant differential  on $E/\Q$. 
The following theorem is
 stated in Section 6.4  of \cite{DR2}:
\begin{theorem}
\label{thm:DR2-D}
When $L(E,V_{gh},1) = 0$, there exists a choice of $\pi$ in 
\eqref{eqn:choice-pi} and
of test vectors for $f$, $g$ and $h$
 such that the coordinates in \eqref{eqn:reskappa-R} satisfy
\begin{equation}
\label{eqn:thm-D}
\log_p (R_{\alpha\beta}) \sim \Lp^{g_\alpha}(\testf,\testg^*,\testh),  \qquad 
\log_p (R_{\beta\alpha}) \sim \Lp^{h_\alpha}(\testf,\testg,\testh^*), \qquad 
\log_p(R_{\beta\beta}) = 0,
\end{equation}
where $\sim$ denotes equality up to a non-zero $p$-adic period in $H_p^\times$.
\end{theorem}
\begin{remark}
This theorem says nothing about the quantity $\log_p(R_{\alpha\alpha})$, which does not bear any direct relationship with $p$-adic $L$-values introduced above. We expect that $\log_p(R_{\alpha\alpha})$ may rather be connected with the first derivative of a putative  
refinement  of $\Lp^{f}(f,g_\alpha,h_\alpha)$  in which all three modular forms would be made to vary in a Hida family. 
\end{remark}
As explained in the introduction and in Section 6.3.~of 
\cite{DR2}, Theorem \ref{thm:DR2-D} has the following corollary which can be 
viewed as a $p$-adic Gross-Zagier formula  in ``analytic rank two":

\begin{corollary}
\label{cor:DR2}
If $L(E,V_{gh},1) =  0$ and $\Lp^{g_\alpha}(\testf,\testg^*,\testh)\ne 0$ for a suitable choice
$(\testf,\testg^*,\testh)$ of 
test vectors, then
 the two global classes
$$\kappa(f,g_\alpha,h_\alpha),  \qquad  \kappa(f,g_\alpha,h_\beta)$$
 are linearly independent in 
the Selmer group $H^1_{\fin}(\Q, V_{fgh})$ attached to  $E$ and  $V_{gh}$,
for a suitable choice of $\pi$ in \eqref{eqn:choice-pi}.
\end{corollary}
Theorem \ref{thm:DR2-D} and its corollary
 motivated the experimental study
undertaken in \cite{DLR}
 of the special values of $p$-adic $L$-functions
appearing in \eqref{eqn:thm-D}.
This led to a precise conjecture for these values up to a factor of $L^\times$ rather than
$\Q_p^\times$. 

To formulate this conjecture, recall that the
 class $\kappa(f,g_\alpha,h_\alpha)$ is expected to be trivial
when $\ord_{s=1} L(E,V_{gh},s)>2$. Assume that this $L$-function has a {\em double} zero at the center, which implies, by
Conjecture $\BSD(E,V_{gh})$, that
 $(E(H)_L\otimes V_{12})^{G_\Q}$ is a two-dimensional $L$-vector space.

Fix vectors $v_{gh}^{\alpha\alpha} ,\ldots,v_{gh}^{\beta\beta}$ chosen as in \eqref{eqn:basis-of-vs}, 
 with the difference  that they  belong to   $L$-vector space $V_{12}$ rather than the $\Q_p$-vector space $V_{gh}$.
Choose  a basis $(P, Q)$  for this $L$-vector space, and write
\begin{eqnarray*}
P &=& P_{\alpha\alpha} \otimes v_{gh}^{\beta\beta}  + P_{\alpha\beta} \otimes v_{gh}^{\beta\alpha} + P_{\beta\alpha} \otimes v_{gh}^{\alpha\beta} 
+ P_{\beta\beta} \otimes v_{gh}^{\alpha\alpha}, \\
Q &=& Q_{\alpha\alpha} \otimes v_{gh}^{\beta\beta}  + Q_{\alpha\beta} \otimes v_{gh}^{\beta\alpha} + Q_{\beta\alpha} \otimes v_{gh}^{\alpha\beta} 
+ Q_{\beta\beta}  \otimes v_{gh}^{\alpha\alpha},
\end{eqnarray*}
where  $P_{\xi}, Q_{\xi}$ are points in $E(H)_L^{\Frob_p = \xi}$  for every $\xi\in \{ \alpha \alpha=\alpha_g\alpha_h, \alpha \beta=\alpha_g \beta_h, \beta\alpha=\beta_g \alpha_h, \beta \beta = \beta_g \beta_h \}.$ 


These points can be used to 
define a   regulator attached to $g_\alpha$,  whose entries are the $p$-adic
formal group logarithms of  the coordinates attached to the vectors
$v_{gh}^{\alpha\alpha}$ and $v_{gh}^{\alpha\beta}$ (and similarly for  $h_\alpha$):
\begin{definition} The  regulators attached to $E$ and $V_{12}$ are 
\begin{equation*}
 \Reg_{g_\alpha}(E,V_{12}) = \det \left( \begin{array}{cc}
\log_p P_{\beta\beta}  & \log_p P_{\beta\alpha} \\
\log_p Q_{\beta\beta} & \log_p Q_{\beta\alpha} \end{array} \right) 
= \log_p P_{\beta\beta} \cdot \log_p Q_{\beta\alpha} -  \log_p Q_{\beta\beta} \cdot \log_p P_{\beta\alpha},
\end{equation*}

\begin{equation*}
 \Reg_{h_\alpha}(E,V_{12}) = \det \left( \begin{array}{cc}
\log_p P_{\beta\beta}  & \log_p P_{\alpha\beta} \\
\log_p Q_{\beta\beta} & \log_p Q_{\alpha\beta} 
\end{array} \right) = \log_p P_{\beta\beta} \cdot \log_p Q_{\alpha\beta} -  \log_p Q_{\beta\beta} \cdot\log_p P_{\alpha\beta}.
\end{equation*}
\end{definition}

The main conjecture of \cite{DLR} is the following\footnote{We warn the reader that here in this note we have chosen to state the main conjecture of \cite{DLR} in terms of the {\em arithmetic} frobenius $\Frob_p$ at $p$, while in \cite{DLR} we rather employ the {\em geometric} frobenius $\sigma_p=\Frob_p^{-1}$. It is for this reason that the roles of $\alpha$ and $\beta$  are swapped in both formulations.}, assuming $\frac{\alpha_g}{\beta_g} \ne \pm 1$ (resp.\,$\frac{\alpha_h}{\beta_h} \ne \pm 1$) so that the Stark unit $u_{g_\alpha}$ (resp.\,$u_{h_\alpha}$) is well-defined:
\begin{conjecture}
\label{conj:DLR}
Assume that $L(E,V_{gh},s) $ vanishes to order $2$ at $s=1$. 
Then there exists a choice of test vectors
$(\testf,\testg^*,\testh)$ and  $(\testf,\testg,\testh^*)$
 such that
$$
  \Lp^{g_\alpha}(\testf,\testg^*,\testh) =  \frac{\Reg_{g_\alpha}(E, V_{12})}{\log_p u_{g_\alpha}}, \qquad 
    \Lp^{h_\alpha}(\testf,\testg,\testh^*) =  \frac{\Reg_{h_\alpha}(E, V_{12})}{\log_p u_{h_\alpha}} \pmod{L^\times}.
$$
\end{conjecture}
\begin{remark}
\label{rem:gkc-computation}
 Conjecture \ref{conj:DLR} lends itself to 
 numerical verification and has been extensively tested in \cite{DLR}. This is because the
 $p$-adic $L$-values $\Lp^{g_\alpha}(\testf,\testg^*,\testh)$ and
  $\Lp^{h_\alpha}(\testf,\testg,\testh^*)$ can be expressed in terms of the  rather concrete
 {\em $p$-adic iterated integrals} of loc.cit., which can be computed efficiently using
 Alan Lauder's \cite{La} fast ordinary projection algorithms on the space of overconvergent
 modular forms. In contrast, the generalised Kato classes themselves 
 (like many objects constructed in \'etale cohomology) seem difficult to compute in practice, even though
  their theoretical usefulness is amply illustrated in \cite{BDR2} and \cite{DR2}.
  \end{remark}
 
  

 \subsection{Enhanced regulators}
 \label{sec:regulators}
 The goal of this article is to combine the  insights arising from Theorem \ref{thm:DR2-D}
and    Conjecture \ref{conj:DLR} 
   to formulate a conjecture on the position of the generalised Kato
 classes themselves in $(E(H)\otimes V_{gh})^{G_\Q}$, specifying this 
 position up to an
 ambiguity of $L^\times$ rather  than the   less precise
 $\Q_p^\times$ ambiguity of   Theorem  \ref{thm:DR2-D}.
 
 The most important ingredients in the formulation of this conjecture are the  so-called
 {\em enhanced regulators} 
 \begin{eqnarray*}
 \eReg(E,V_{12}) & \in & (E(H)_L\otimes V_{12})^{G_\Q}  \otimes  (E(H)_L\otimes V_{12})^{G_\Q}, \\
   \eReg_{\alpha\alpha}(E,V_{12}) & \in & (H_p)^{\Frob_p = \beta_g\beta_h} \otimes (E(H)_L\otimes V_{12})^{G_\Q}, \\
& & \\
  \eReg(E,V_{gh}) & \in & (E(H)_L\otimes V_{gh})^{G_\Q}  \otimes  (E(H)_L\otimes V_{gh})^{G_\Q}, \\
   \eReg_{\alpha\alpha}(E, V_{gh}) & \in & D(V_{gh}^{\alpha\alpha})   \otimes  (E(H)_L\otimes V_{gh})^{G_\Q},
    \end{eqnarray*}
 whose definition is somewhat in the spirit 
 of the regulator $R_S$ defined in equation (2) of \cite{darmon-thesis}, and which 
 we now proceed to describe. As in \eqref{Ohta-w}, here $D(V_{gh}^{\alpha\alpha})  := (\Q_p^{\nr} \otimes V_{gh}^{\alpha \alpha})^{G_{\Q_p}} = (H_p \otimes V_{gh}^{\alpha \alpha})^{G_{\Q_p}}$.

 \begin{definition} Choose an $L$-basis $(P,Q)$ of the two-dimensional
 vector space $(E(H)\otimes V_{12})^{G_\Q}$, 
 and set
   \begin{equation}
   \label{eqn:enhanced-regulator-formal}
 \eReg(E,V_{12}) := \det \left( \begin{array}{cc}
 P   & P \\
Q  & Q \end{array} \right) :=   P\otimes Q -  Q\otimes   P.
\end{equation}
It does not depend on the choice of basis that was made to define it, up to multiplication
by $L^\times$.



 The function 
 $\log_{\alpha\alpha}: (E(H)_L \otimes V_{12})^{G_\Q} \lra (H_p)^{\Frob_p=\beta_g\beta_h}$ defined
 by 
 $$ \log_{\alpha\alpha}(P) := \log_p(P_{\beta\beta})$$
  induces a linear map
 $$ \log_{\alpha\alpha} \otimes 1: 
 (E(H)_L\otimes V_{12})^{G_\Q} \otimes (E(H)_L\otimes V_{12})^{G_\Q} \lra 
 (H_p)^{\Frob_p = \beta_g\beta_h} \otimes (E(H)_L\otimes V_{12})^{G_\Q},$$
 and we  set
     \begin{equation}
  \eReg_{\alpha\alpha}(E,V_{12}) :=  (\log_{\alpha\alpha}\otimes 1)(\eReg(E,V_{12})) =  \log_{p}(P_{\beta\beta}) \otimes Q - \log_{p}(Q_{\beta\beta}) \otimes P.
   \end{equation}
\end{definition}

Recall the embedding $j_{gh}:V_{12} \lra V_{gh}^L \subset V_{gh}$ of
 \eqref{eqn:non-canonical-isom}. Although this embedding is completely non-canonical and only defined up to scaling by $\Q_p^\times$, there is a canonical
 way of embedding $V_{12}^{\otimes 2}$ into $V_{gh}^{\otimes 2}$.
 This is done by exploiting the
  canonical dualities
 on $V_g$ and $V_h$ described in
 Section \ref{sec:periods}, 
 which  gives rise to perfect pairings
 $$ V_g\times V_g \lra \Q_p(\chi), \qquad V_h\times V_h \lra \Q_p(\chi^{-1}),
 \qquad V_{gh}\times V_{gh} \lra \Q_p.$$
 These pairings allow us to define $L$-rational structures
 $V_{g}^{L*}$, $V_{h}^{L*}$ and $V_{gh}^{L*}$ which are dual
 to $V_{g}^L$, $V_h^L$ and $V_{gh}^L$ respectively, by letting $V_g^{L*}$ be the $L$-dual of $V_{g}^L$ in $V_g$, and likewise for $V_{h}^{L*}$ and $V_{gh}^{L*}$. 
We may then choose  $G_\Q$-equivariant embeddings 
 $$ j_g^*: V_1 \lra V_g^{L*}, \qquad j_h^*: V_2 \lra  V_{h}^{L*}, \qquad
 j_{gh}^*:= j_g^* \otimes j_h^*: V_{12} \lra V_{gh}^{L*},$$
 which are well-defined up to scaling by $L^\times$. Replacing $j_{gh}$ by 
 $\mu \cdot j_{gh}$, for any $\mu\in\Q_p^\times$, has the effect of replacing
 $j_{gh}^*$ by $\mu^{-1} \cdot j_{gh}^*$. Hence the map
 $$ j_{gh} \otimes j_{gh}^*: V_{12}\otimes V_{12} \lra V_{gh} \otimes V_{gh}$$
 is well defined up to scaling by $L^\times$.
 
\begin{definition}
The enhanced regulator $\eReg(E,V_{gh})$ associated to $E$ and $V_{gh}$ is
 \begin{equation}
 \label{eqn:enhanced-regulator-gh}
  \eReg(E,V_{gh}) := (j_{gh} \otimes j_{gh}^*)(\eReg(E,V_{12})) \ \in \ 
 (E(H) \otimes V_{gh})^{G_\Q} \otimes  (E(H)\otimes V_{gh})^{G_\Q}.
 \end{equation}
 \end{definition}
 
  Finally, let 
    $$ \Log_p: (E(H) \otimes V_{gh})^{G_\Q} \lra (H_p \otimes V_{gh})^{G_{\Q_p}} = 
    D(V_{gh})$$
    be the canonical $p$-adic logarithm map induced from the $p$-adic
    logarithm of \eqref{eqn:E-log} via
     the fixed  embedding $H\subset H_p$, and let 
     $$ \Log_{\alpha\alpha}: (E(H)\otimes V_{gh})^{G_\Q} \lra D(V_{gh}^{\alpha\alpha})$$
    be its composition with the functorial
     projection $D(V_{gh}) \lra D(V_{gh}^{\alpha\alpha})$. 
     This logarithm map is just the more canonical counterpart of
     the map $\log_{\alpha\alpha}$: the latter   depends on the choice of a basis vector
     $v_{gh}^{\alpha\alpha}$ for $V^{\alpha\alpha}$ and is related to 
     $\Log_{\alpha\alpha}$ by  the rule
     $$ \Log_{\alpha\alpha} := \log_{\alpha\alpha} \otimes v_{gh}^{\alpha\alpha}.$$
 We  set
  \begin{equation}
  \label{eqn:regulator-4}
  \eReg_{\alpha\alpha}(E,V_{gh}) := (\Log_{\alpha\alpha}\otimes 1)(\eReg(E,V_{gh}))   =    \Log_{\alpha\alpha}(P) \otimes Q - \Log_{\alpha\alpha}(Q) \otimes P.
   \end{equation}
    It is worth noting that the enhanced regulator
     $\eReg_{\alpha\alpha}(E,V_{gh})$ is a canonical invariant associated to 
     $E$ and $V_{gh}$, i.e., it is well-defined up to multiplication by $L^\times$,
     while the less canonical $\eReg_{\alpha\alpha}(E,V_{12})$   
     depends on the 
     choice of a basis $v_{gh}^{\alpha\alpha}$ for $V_{gh}^{\alpha\alpha}$.
     The two regulators are related by  
     \begin{equation}
     \label{eqn:abstract-to-concrete}
      \eReg_{\alpha\alpha}(E,V_{gh}) = \eReg_{\alpha\alpha}(E,V_{12}) \otimes v_{gh}^{\alpha\alpha}.
      \end{equation}     
     
     
   
 
 


\subsection{The conjecture}
\label{sec:conjecture}

Recall the periods
$$ \omega_{g_\alpha} \in D(V_g^\alpha), \qquad \omega_{h_\alpha} \in D(V_h^\alpha)$$
constructed in \eqref{Ohta-w}. The main conjecture of this note is:
\begin{conjecture}
\label{conj:main}
Assume that $r(E,V_{gh})=2$. 
The    generalised Kato class $\kappa(f,g_\alpha,h_\alpha)$ belongs to $(E(H)\otimes V_{gh})^{G_\Q}$ and satisfies the relation
$$  \omega_{g_\alpha} \omega_{h_\alpha} \otimes \kappa(f,g_\alpha,h_\alpha) \sim_L
 \eReg_{\alpha\alpha}(E, V_{gh}) $$
 in $D(V_{gh}^{\alpha\alpha}) \otimes (E(H)\otimes V_{gh})^{G_\Q}$, 
 where $\sim_L$ denotes an equality 
 up to scaling by  a factor in   $L$ which is non-zero for a
 suitable choice of $\pi$ in \eqref{eqn:choice-pi}.
\end{conjecture}

 
  The following  proposition  shows that, under Conjecture \ref{conj:stark} (relating the 
  canonical period attached to $g$ to the Stark unit $u_{g_\alpha}$) and Conjecture \ref{conj:non-van} (a mild 
  strengthening of $\BSD(E,\varrho_{gh})$),  
  Conjecture \ref{conj:main} implies the main
  conjecture  of
  \cite{DLR}. 
 Before dismissing this proposition   as  mere  conjectural relations between conjectures, the reader is reminded   that Conjecture \ref{conj:DLR}   lends itself to experiment  and has been extensively tested numerically  in \cite{DLR},    while the strengthening described in Conjecture \ref{conj:main}     lies
 for the moment  beyond the range of explicit calculations (cf.~Remark  \ref{rem:gkc-computation}). 
 %Any relation between   Conjecture \ref{conj:DLR} and \ref{conj:main} provides indirect  experimental support for the latter.
 \begin{proposition}
Assume Conjectures   \ref{conj:stark} and \ref{conj:non-van}.  Then  Conjecture
 \ref{conj:main}  implies Conjecture  \ref{conj:DLR}.
 \end{proposition}
 \begin{proof}
 Consider the product of periods
 $$\eta_{g_\alpha} \omega_{h_\alpha}  = (\Xi_{g_\alpha}  \otimes v_{g}^\beta ) \cdot
 (\Omega_{h_\alpha} \otimes v_h^\alpha) = 
  \Xi_{g_\alpha} \cdot
 \Omega_{h_\alpha} \otimes v_{gh}^{\beta\alpha} \in D(V_{gh}^{\beta \alpha})
 $$
 defined in Section \ref{sec:periods}.

The pairing introduced in \eqref{eqn:pairing-g} gives rise to a pairing
$$
 \langle \,, \rangle: D(V_{gh}^{\alpha\beta}) \times D(V_{gh}^{\beta \alpha}) \lra D(\Q_p)=\Q_p.
$$
As   shown in the proof of \cite[Theorem 6.10 (ii)]{DR2},
 \begin{equation}
 \label{eqn:lhs-etaomega}
 \left \langle  \Log_{\alpha\beta}\kappa(f,g_\alpha,h_\alpha), \eta_{g_\alpha} \omega_{h_\alpha} \right\rangle 
 = \Lp^{g_\alpha}(f,g,h) \pmod{L^\times}.
 \end{equation}
 On the other hand, by the definition of the enhanced regulator,
  \begin{eqnarray*}
  \Log_{\alpha\beta} \eReg_{\alpha\alpha}(E,V_{gh}) 
&=&   (\log_p P_{\beta\beta} \log_p Q_{\beta\alpha} - \log_p Q_{\beta\beta} \log_p P_{\beta\alpha})
\otimes v_{gh}^{\alpha\alpha} \otimes v_{gh}^{*\alpha\beta} \\
&=& \Reg_{g_\alpha}(E,V_{12}) 
\otimes v_{gh}^{\alpha\alpha} \otimes v_{gh}^{*\alpha\beta} \pmod{L^\times}.
\end{eqnarray*}
Hence the following equality holds  in $D(V_{gh}^{\alpha \alpha})$:
 \begin{equation}
 \label{eqn:rhs-etaomega}
\langle  \Log_{\alpha\beta} \eReg_{\alpha\alpha}(E,V_{gh}), \eta_{g_\alpha} \omega_{h_\alpha} \rangle
=  \Xi_{g_\alpha} \cdot
 \Omega_{h_\alpha} \cdot
 \Reg_{g_\alpha}(E,V_{gh})
\otimes v_{gh}^{\alpha\alpha} \pmod{L^\times}.
\end{equation}
By pairing the value of $\Log_{\alpha\beta}$ at both sides of the displayed identity in Conjecture \ref{conj:main} with the class 
$\eta_{g_\alpha} \omega_{h_\alpha}$ and invoking 
\eqref{eqn:lhs-etaomega} and 
 \eqref{eqn:rhs-etaomega},
 we obtain
$$  \omega_{g_\alpha} \omega_{h_\alpha} \otimes \Lp^{g_\alpha}(f,g,h) = 
\Xi_{g_\alpha} \cdot
 \Omega_{h_\alpha}\cdot
 \Reg_{g_\alpha}(E,V_{12})
\otimes v_{gh}^{\alpha\alpha} \in D(V_{gh}^{\alpha \alpha}) \pmod{L^\times}.$$
Since 
$$\omega_{g_\alpha} \omega_{h_\alpha} =  \Omega_{g_\alpha} \cdot \Omega_{h_\alpha}  \cdot v_{gh}^{\alpha\alpha} \pmod{L^\times},$$
it follows that
$$
 \Omega_{g_\alpha} \Lp^{g_\alpha}(f,g,h) = 
\Xi_{g_\alpha}
 \Reg_{g_\alpha}(E,V_{12})  \pmod{L^\times},$$
 and therefore that
 $$ \Lp^{g_\alpha}(f,g,h) = \frac{\Reg_{g_\alpha}(E,V_{12}) }{\mathcal{L}_{g_\alpha}} \pmod{L^\times}.$$
 Conjecture \ref{conj:DLR} now follows directly from this equality after invoking
 Conjecture \ref{conj:stark}.
\end{proof}
 
   \begin{remark}
 As explained in a number of the examples covered in Chapter \ref{sec:cases} below,
 it may happen that all four of the $p$-adic iterated integrals in 
 \eqref{eqn:Lvalues}  are  equal to zero even when 
some of the generalised Kato classes are non-trivial. 
 This suggests that Conjecture \ref{conj:main}
  is    a  genuine strengthening of
  Conjecture \ref{conj:DLR}. 
 \end{remark}
 
 
 

   






 


  


 


\section{Special cases}
\label{sec:cases}
This section examines Conjecture \ref{conj:main}, and   the special forms taken by
the enhanced regulators
$$\eReg_{\alpha\alpha}(E; V_{12}), \qquad
\eReg_{\alpha\beta}(E; V_{12}),  \qquad
\eReg_{\beta\alpha}(E; V_{12}), \qquad
\eReg_{\beta\beta}(E; V_{12}), $$
 in the  arithmetically interesting  cases  where
$V_{gh}$ is {\em reducible}. According to \cite[\S 2]{DLR2}, the following is a complete list of scenarios where this
occurs:
\begin{enumerate}
\item The original   {\em Beilinson-Kato setting} where $V_g$ and $V_h$ are both reducible, i.e., where
$g$ and $h$ are both Eisenstein series of weight one;
\item the {\em Beilinson-Flach setting} where exactly one of $V_g$ or  $V_h$ is reducible, i.e., where exactly one of 
$g$ or $h$ is cuspidal;
\item the {\em complex multiplication} 
case where $V_g$ and $V_h$ are both induced from characters of a common imaginary quadratic field;
\item the {\em real multiplication} case where $V_g$ and $V_h$ are induced from characters of mixed signature of a common
real quadratic field;
\item the {\em adjoint case} where $h$ is (a twist of) the dual of $g$, so that $V_{gh}$ is the direct sum of  a one-dimensional representation and a twist of the adjoint of $V_g$. 
\end{enumerate}

The reader will notice that some of the above settings arise when $\varrho_g$ and/or $\varrho_h$ are {\em reducible}, while in \S \ref{sec:periods} and \S \ref{sec:Kato} these representations were  assumed to be irreducible. This assumption was imposed to a large extent for the sake of simplicity of the exposition, and the statement (and presumed validity) of Conjecture \ref{conj:main} does not rely on it. For completeness, we have therefore described the enhanced regulators that appear in
 Conjecture \ref{conj:main} in  all of the above cases.
 
\subsection{Beilinson-Kato classes}
\label{sec:bk-case}
Assume that $g$ and $h$ are both Eisenstein series. After possibly twisting $g$ or $h$, there is no real loss of generality in assuming that there exist Dirichlet characters $\chi_1$, $\chi_2$ such that $g$ and $h$ are given by
$$ g = E_1(\chi_1, \chi_2), \qquad h=E_1(1, \chi_{12}^{-1}), \qquad \mbox{ where } \quad \chi_{12} =   \chi_1 \chi_2.$$

We refer to e.g. \cite[\S 2.1.2]{BDR1} for the definition of these weight one Eisenstein series in terms of their $q$-expansions. The Galois representations attached to $g$ and $h$ are reducible, namely,
\begin{equation}
\label{eqn:decomp-g-h}
 V_1 = L(\chi_1) \oplus L(\chi_2), \qquad V_2 = L \oplus L(\chi_{12}^{-1}),  \qquad V_{12}  = L(\chi_1) \oplus L(\chi_1^{-1}) \oplus L(\chi_2) \oplus L(\chi_2^{-1}),
 \end{equation}
 where the coefficient field $L$ is the cyclotomic field generated by the images of $\chi_1$ and $\chi_2$. These representations
 factor through the Galois group $\Gal(H/\Q)$ of an abelian extension $H$ of $\Q$.
 We may set 
 $$ \alpha_g = \chi_1(p), \quad \beta_g = \chi_2(p), \qquad \alpha_h = 1, \quad \beta_h = \chi_{12}^{-1}(p).$$
The regularity asumption  implies that $V_1$ and $V_2$ decompose uniquely as a direct sum of two $G_{\Q_p}$-stable lines, which are also stable under $G_\Q$.  More precisely,
$$ V_{12}^{\alpha\alpha} =  L \cdot v_{\chi_1}, \qquad V_{12}^{\beta\beta} =  L  \cdot v_{\bar\chi_1}, \qquad V_{12}^{\alpha\beta} 
= L  \cdot v_{\bar\chi_2}, \qquad
V_{12}^{\beta\alpha} = L \cdot  v_{\chi_2},$$
where $(v_{\chi_1}, v_{\bar\chi_1}, v_{\bar\chi_2}, v_{\chi_2})$ is a basis for $V_{12}$ 
on which $G_\Q$ acts
via the characters $\chi_1$, $\bar\chi_1$, $\bar\chi_2$, and $\chi_2$ respectively.


 
The class $\kappa(f,g_\alpha,h_\alpha) = \kappa_{\BK}(f,g_\alpha,h_\alpha)$ was  
constructed by Kato as a $p$-adic 
limit of Beilinson elements attached to
pairs of modular units whose logarithmic derivatives are weight two Eisenstein series. 
Theorem \ref{thm:DR2-C}
in this case boils down to Kato's reciprocity law, which asserts that
$ \kappa(f,g_\alpha,h_\alpha) $ belongs to the Selmer group of $E$  over $H$
 if and only if the $L$-function
$$ L(E,V_{gh},s) = L(E,\chi_1,s) L(E,\bar\chi_1,s) L(E,\chi_2,s)  L(E,\bar\chi_2,s) $$
vanishes at $s=1$.
In this case, it clearly vanishes to even order, and vanishes to order two if and only if (after eventually interchanging the characters 
$\chi_1$ and $\chi_2$) 
$$ \ord_{s=1}L(E,\chi_1,s) = \ord_{s=1} L(E,\bar\chi_1,s) = 1, \qquad L(E,\chi_2,1),  L(E,\bar\chi_2,1) \ne 0.$$
Assuming that this is the case,  Conjectures $\BSD(E,\chi_1)$ and $\BSD(E,\chi_2)$ predict that
$(E(H)_L \otimes V_{12})^{G_\Q}$ is two-dimensional over $L$ and that a basis for it can be chosen to be
$$ P := P_{\bar\chi_1} \otimes v_{\chi_1}, \qquad Q := Q_{\chi_1} \otimes v_{\bar\chi_1},$$
where $P_{\bar\chi_1}$ and $Q_{\chi_1}$ are global points in $E(H)_L$ generating   the 
$\bar\chi_1$ and $\chi_1$ eigenspaces respectively for the natural   action of $G_\Q$.
With these notations, we have 
$$ P_{\alpha\alpha} = P_{\alpha\beta} = P_{\beta\alpha} = 0, \qquad P_{\beta\beta} =  P_{\bar\chi_1},$$
$$ Q_{\alpha\beta} = Q_{\beta\alpha} = Q_{\beta\beta} = 0, \qquad Q_{\alpha\alpha} = Q_{\chi_1}.$$
This immediately implies that 
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_p(P_{\bar\chi_1}) \cdot  Q,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = 0, \\
 \eReg_{\beta\alpha}(E, V_{12}) = 0, &  \quad &
 \eReg_{\beta\beta}(E;V_{12}) = \log_p(Q_{\chi_1}) \cdot  P.
 \end{array}
 $$
 It follows that
\begin{equation*}
\label{eqn:regulators-eisenstein}
\Reg_{g_\alpha}(E; V_{12}) = \Reg_{g_\beta}(E;V_{12}) = \Reg_{h_\alpha}(E; V_{12}) = \Reg_{h_\beta}(E; V_{12}) = 0.
\end{equation*}
This
accounts for the fact that the $p$-adic iterated integrals
$$\Lp^{g_\alpha}(f,g,h),  \ \ \    \Lp^{g_\beta}(f,g,h), \ \ \ \Lp^{h_\alpha}(f,g,h),  \ \ \ \Lp^{h_\beta}(f,g,h)$$ 
systematically vanish\footnote{But see
the experiments described in \cite[\S 7]{DLR}
in the case where $g$ is {\em irregular} at $p$, which suggest that the irregular setting
of Conjecture \ref{conj:main}  would merit further investigation.}
 when $g$ and $h$ are Eisenstein series that
are regular at $p$. 
 Conjecture \ref{conj:main} makes the stronger prediction that the generalised Kato classes themselves 
are non-trivial, and is consistent with a Conjecture of Perrin-Riou, since it predicts that
$$ \log_{\beta\beta} (\kappa(f,g_\alpha,h_\alpha)) =  \log_{\alpha\alpha}(\kappa(f,g_\beta,h_\beta))  =
\log_p(P_{\bar\chi_1}) \log_p(Q_{\chi_1}) \pmod{L^\times}.$$ 

  
 

 
  
\subsection{Belinson-Flach classes}
\label{sec:bf-case}
In the Beilinson-Flach setting, it can be assumed without  loss of generality
 that $g$ 
is a weight one cusp form with nebentypus character $\chi$ and Galois representation
$V_g = V_1\otimes_L \Q_p$, and that
$h := E_1(1,\chi^{-1})$
is the weight one Eisenstein series attached to the pair $(1,\chi^{-1})$ of Dirichlet characters.
The relevant four-dimensional representations are then equal to 
$$ V_{gh} = V_g \oplus V_{\bar g}; \qquad V_{12} =  V_1 \oplus \bar V_{1},$$
and the Hasse-Weil-Artin $L$-series 
$$ L(E, V_{gh},s) = L(E,V_g,s) L(E, \bar V_{g} ,s)$$
has a double zero at  $s=1$ precisely when each of the primitive $L$-series 
$L(E,V_g,s)$ and $L(E,V_{\bar g},s)$ have a simple zero at $s=1$.
Conjecture $\BSD(E,V_g)$ then implies that each of the $L$-vector spaces on the right-hand side of
$$(E(H)_L\otimes V_{12})^{G_\Q}) = (E(H)_L\otimes V_1)^{G_\Q} \oplus (E(H)_L\otimes \bar V_{1})^{G_\Q},$$
 is one-dimensional.
Let $P$ be an $L$-basis for $(E(H)_L\otimes V_1)^{G_\Q}$ and let $\bar P$ be the associated
$L$-basis for $(E(H)_L\otimes \bar V_1)^{G_\Q}$, obtained by applying complex conjugation to the coefficients in $L$.


After fixing an ordering $\alpha_g, \beta_g\in L$ for the eigenvalues  of $\Frob_p$ on 
$V_1$,  and setting 
$$ \alpha_h = 1, \qquad  \beta_h = \chi^{-1}(p) = (\alpha_g\beta_g)^{-1},$$
we have
$$ V_{12}^{\alpha\alpha} = V_1^{\alpha_g}, \quad V_{12}^{\alpha\beta} = \bar V_1^{\beta_g^{-1}}, 
\quad V_{12}^{\beta\alpha} = V_1^{\beta_g}, \quad V_{12}^{\beta\beta} = \bar V_1^{\alpha_g^{-1}},$$
and hence
$$ 
\begin{array}{llll}
P_{\alpha\alpha} = P_{\alpha_g}, &  P_{\beta\alpha} = P_{\beta_g},  & P_{\alpha\beta} =  0 & P_{\beta\beta} =  0,\\
 \bar P_{\alpha\alpha} =  0 & \bar P_{\beta\alpha} = 0, & \bar P_{\alpha\beta} =  \bar P_{\beta_g^{-1}}, 
 & \bar P_{\beta\beta} = \bar P_{\alpha_g^{-1}}. 
 \end{array}
 $$
A direct calculation reveals that, up to multiplication by $L^\times$,
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_p(\bar P_{\alpha_g^{-1}}) \cdot  P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_p(P_{\beta_g}) \cdot  \bar P, \\
 \eReg_{\beta\alpha}(E, V_{12}) =    \log_p(\bar P_{\beta_g^{-1}}) \cdot    P, &  \quad &
 \eReg_{\beta\beta}(E;V_{12}) = \log_p(P_{\alpha_g}) \cdot  \bar P.
 \end{array}
 $$
It follows that
\begin{equation*}
\begin{array}{ll}
\Reg_{g_\alpha}(E, V_{12}) =  \log_p(\bar P_{\alpha_g^{-1}}) \cdot  \log_p(P_{\beta_g}), &
 \Reg_{g_\beta}(E,V_{12}) = \log_p(\bar P_{\beta_{g}^{-1}}) \cdot \log_p(P_{\alpha_g}), \\
 \Reg_{h_\alpha}(E, V_{12}) = 0, &
 \Reg_{h_\beta}(E,V_{12}) = 0. 
 \end{array}
 \end{equation*}
as described in \cite[\S 6]{DLR}.


\subsection{Complex multiplication classes and Heegner points}
\label{sec:cm-case}
 
 In this chapter we consider the setting where $g$ and $h$ are theta series attached to 
 characters $\psi_g$ and $\psi_h$ of the same imaginary quadratic field $K$, and with 
 inverse  nebentypus character. Given any character $\psi$ of $G_K$, let $\psi'$ denote the character obtained  by 
 conjugating it with the involution in $\Gal(K/\Q)$. 
Then
 $$ V_g = \Ind_K^\Q \psi_g = \Ind_K^\Q \psi_g', \qquad V_h = \Ind_K^\Q \psi_h = \Ind_K^\Q \psi_h', $$
 and therefore
$$ V_{gh} = \Ind_K^Q \psione \oplus \Ind_K^\Q \psitwo, \qquad \mbox{ where }
 \psione = \psi_g \psi_h, \qquad \psitwo = \psi_g \psi_h'.$$
 The self-duality assumption implies that $\psione$ and $\psitwo$ are
 are  {\em ring class characters}, i.e., they satisfy
 $$ \psione' = \psione^{-1}, \qquad \psitwo' = \psitwo^{-1}.$$
 Assume that the induced representations
 $$\Vone := \Ind_K^\Q \psione, \qquad \Vtwo := \Ind_K^\Q \psitwo$$
 appearing in the decomposition
 \begin{equation}
 \label{eqn:quadratic-decomp}
 V_{12} = \Vone \oplus \Vtwo
 \end{equation}
  (viewed as representations with coefficients in the number field $L$)
 are {\em irreducible}, which  is always the case unless $\psione$ or $\psitwo$ is a quadratic, i.e., a genus character.  (The more degenerate case where this arises can be subsumed 
 under the ``adjoint setting" considered in Section \ref{sec:adjoint-case}.)
 
 
The Hasse-Weil-Artin $L$-series 
$$ L(E, V_{gh},s) = L(E, \Vone,s) L(E, \Vtwo ,s) = L(E/K, \psione,s) L(E/K,\psitwo,s)$$
has a double zero at  $s=1$ in one of the following two cases:
\begin{enumerate}
\item 
  the primitive $L$-series 
$L(E,\Vone,s)$ and $L(E,\Vtwo,s)$ each have a simple zero at $s=1$.
This setting, which resembles more closely  the phenomena described in the 
previous two sections on Beilinson-Kato and Beilinson-Flach elements, will be referred to as the 
{\em rank (1,1) setting} of Conjecture \ref{conj:main}.
 \item
 Exactly one of the primitive $L$-series 
 $L(E,\Vone,s)$ or $L(E,\Vtwo,s)$ has a double zero at $s=1$, and the other is non-vanishing at the center.
 This case shall be referred to as the {\em rank (2,0) setting} of Conjecture \ref{conj:main}. The possible
 non-triviality of the generalised Kato classes in the presence of a ``genuine" double zero 
 of a primitive Hasse-Weil-Artin $L$-function
 represents a novel feature that did not arise in the setting of
  Beilinson-Kato or Beilinson-Flach elements.
  \end{enumerate}
 
 


\subsubsection{The rank $(1,1)$ setting}
\label{sec:quad-11-imag}

In this case, Conjectures $\BSD(E,\Vone)$ and $\BSD(E,\Vtwo)$ predict that the 
Mordell-Weil groups $(E(H)_L\otimes \Vone)^{G_\Q}$ and
$(E(H)_L \otimes \Vtwo)^{G_\Q}$ are both one-dimensional $L$-vector spaces, with
generators $\Pone$ and $\Ptwo$ respectively.
It is natural to write
\begin{equation}
\label{eqn:Ponetwo}
 \Pone = P_{\psione}  \otimes v_{\psione'}  + P_{\psione'} \otimes v_{\psione}, 
\qquad
\Ptwo = P_{\psitwo}  \otimes v_{\psitwo'}  + P_{\psitwo'} \otimes v_{\psitwo}, 
\end{equation}
where $P_{\psione}$, $P_{\psione'}$, $P_{\psitwo}$, and $P_{\psitwo'}$ are generators for the one-dimensional subspaces of $E(H)_L$ on which $G_K$ acts via the characters
$\psione$, $\psione'$, $\psitwo$, and $\psitwo'$ respectively.

\medskip
The description of the enhanced regulators attached to  $V_{12}$ and to
$(\Pone,\Ptwo)$ 
 can be further subdivided into two cases, with markedly different features: the case where the prime $p$ is split in $K$, and the case where it is inert in $K$.


\medskip\noindent {\em a) The case where $p$ is split in $K$}.   
In this case, let $p = \fp \fp'$ be the factorisation of $p$ into distinct primes of $K$.
We can then set
$$ \alpha_g =\psi_g(\fp), \quad \beta_g = \psi_g(\fp'), \qquad \alpha_h = \psi_h(\fp), \quad
\beta_h = \psi_h(\fp'),$$
so that
$$ \alpha_g\alpha_h  = \psione(\fp), \quad \alpha_g\beta_h = \psitwo(\fp), \quad
\beta_g\alpha_h = \psitwo(\fp'), \quad \beta_g\beta_h = \psione(\fp').$$
The decomposition of the $G_{K_{\fp}} = G_{\Q_p}$ representations attached to
 \eqref{eqn:quadratic-decomp} into $\Frob_p$-eigenspaces is also stable under the action of the global Galois group $G_K$, and is  described by:
 $$ V_{12}^{\alpha\alpha} = \Vone^{\psione}, \quad
 V_{12}^{\alpha\beta} = \Vtwo^{\psitwo}, \quad
 V_{12}^{\beta\alpha} = \Vtwo^{\psitwo'}, \quad
 V_{12}^{\beta\beta} = \Vone^{\psione'}.
 $$
 It follows  that, up to multiplication by $L^\times$,
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{\fp}(P_{\psione'}) \cdot  \Ptwo,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_{\fp}(P_{\psitwo'}) \cdot  \Pone, \\
 \eReg_{\beta\alpha}(E, V_{12}) =    \log_{\fp}(P_{\psitwo}) \cdot  \Pone, &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{\fp}(P_{\psione}) \cdot  \Ptwo,
 \end{array}
 $$
and therefore that
\begin{equation*}
\begin{array}{ll}
\Reg_{g_\alpha}(E, V_{12}) =  \log_{\fp}(P_{\psione'}) \cdot  \log_{\fp} (P_{\psitwo'}), 
& 
 \Reg_{g_\beta}(E,V_{12}) = \log_{\fp}(P_{\psione}) \cdot \log_{\fp}(P_{\psitwo}),
 \\
\Reg_{h_\alpha}(E, V_{12}) =  \log_{\fp}(P_{\psione'}) \cdot  \log_{\fp} (P_{\psitwo}), 
&
 \Reg_{h_\beta}(E,V_{12}) = \log_{\fp}(P_{\psione}) \cdot \log_{\fp}(P_{\psitwo'}).
 \end{array}
 \end{equation*}
The corresponding formulae 
for the $p$-adic iterated integrals $\Lp^{g_\alpha}(f,g,h)$, 
$\Lp^{g_\beta}(f,g,h)$, $\Lp^{h_\alpha}(f,g,h)$, and $\Lp^{h_\beta}(f,g,h)$
were proved in 
\cite[\S3]{DLR}, 
by using the $p$-adic Gross-Zagier formula of \cite{BDP1} to express these $L$-values in terms of products of $p$-adic logarithms of Heegner points.
Theorem 3.3 of loc.cit. is one of the   few pieces of theoretical evidence in support of
Conjecture \ref{conj:main}.



\medskip\medskip\noindent
{\em b) The case where $p$ is inert in $K$}.
In this case, the eigenvalues of the Frobenius automorphism $\Frob_p$  acting on 
$V_g$  and $V_h$ are
of the form
$$ \alpha_g, \qquad \beta_g = -\alpha_g, \qquad  \alpha_h = \alpha_g^{-1}, \qquad \beta_h = - \alpha_g^{-1}.$$
Let $(v_{\psi_g}, v_{\psi_g'})$ be a eigenbasis  of $V_g$  for the action of $G_K$ 
relative to the  distinct characters $\psi_g$ and $\psi_g'$, and let 
$(v_{\psi_h}, v_{\psi_h'})$ be a similar basis for $V_h$.
These vectors can be scaled so that $\Frob_p$ acts on them as
$$ 
\Frob_p  (v_{\psi_g}) =  \alpha_g \cdot v_{\psi_g'}, \qquad 
\Frob_p (v_{\psi_g'}) =  \alpha_g  \cdot v_{\psi_g}, \qquad 
\Frob_p (v_{\psi_h} )=  \alpha_g^{-1} \cdot v_{\psi_h'}, \qquad 
\Frob_p (v_{\psi_h'}) =  \alpha_g^{-1} \cdot v_{\psi_h},$$
and therefore we may set
$$ V_g^{\alpha} = L\cdot(v_{\psi_g}+ v_{\psi_g'}), \qquad V_g^{\beta} = L\cdot(v_{\psi_g} - v_{\psi_g'}), \qquad
V_h^{\alpha} = L\cdot(v_{\psi_h}+v_{\psi_h'}), \qquad V_h^\beta = L\cdot(v_{\psi_h} - v_{\psi_h'}).$$
After setting
$$ v_{\psione} := v_{\psi_g}\otimes v_{\psi_h}, \qquad
v_{\psione'} := v_{\psi_g'}\otimes v_{\psi_h'}, \qquad
v_{\psitwo} := v_{\psi_g} \otimes v_{\psi_h'}, \qquad
v_{\psitwo'} := v_{\psi_g'} \otimes v_{\psi_h},$$
and letting
$$\vone^+ := v_{\psione} + v_{\psione'},  \qquad
\vone^- := v_{\psione} - v_{\psione'}, \qquad
\vtwo^+ :=  v_{\psitwo} + v_{\psitwo'},  \qquad
\vone^- := v_{\psitwo} - v_{\psitwo'}, $$
it is easy to see that 
 \begin{equation}
 \label{eqn:alpha-beta-inert-decomp} 
 \begin{array}{ll}
  V_{12}^{\alpha\alpha} =  L \cdot (\vone^+ + \vtwo^+),  &  \qquad
 V_{12}^{\alpha\beta} = L \cdot (\vone^- + \vtwo^-), \\
 V_{12}^{\beta\alpha} =  L \cdot (\vone^- - \vtwo^-), & \qquad
 V_{12}^{\beta\beta} =  L \cdot (\vone^+ - \vtwo^+).
 \end{array}
\end{equation}
 Note that  $\Frob_p$ acts  on both $V_{12}^{\alpha\alpha}$ and $V_{12}^{\beta\beta}$ with the
   eigenvalue  $1$, and  on $V_{12}^{\alpha\beta}$ and $V_{12}^{\beta\alpha}$ with the eigenvalue
   $-1$.
  In particular, while
$V_g$ and $V_h$ are {\em always} regular at $p$, the tensor product $V_{gh} \simeq V_{12}$ never 
enjoys this property, even though the vector spaces described in  
\eqref{eqn:alpha-beta-inert-decomp} are one dimensional.


Keeping the same notations as in \eqref{eqn:Ponetwo}, let
 $$ 
 \Pone^+ := P_{\psione}+ P_{\psione'}, \qquad \Pone^- := P_{\psione}- P_{\psione'}, \qquad 
  \Ptwo^+ := P_{\psitwo}+ P_{\psitwo'}, \qquad \Ptwo^- := P_{\psitwo}- P_{\psitwo'}.$$
 With these notations, 
 the enhanced regulators describing the associated generalised Kato classes  are given by 
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(\Pone^+) \cdot  \Ptwo - \log_{p}(\Ptwo^+) \cdot \Pone,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_{p}(\Pone^-) \cdot  \Ptwo - 
\log_p(\Ptwo^-) \cdot \Pone, \\
 \eReg_{\beta\alpha}(E,V_{12}) = \log_{p}(\Pone^-) \cdot  \Ptwo + \log_{p}(\Ptwo^-) \cdot \Pone,  &  \quad &  \eReg_{\beta\beta}(E,V_{12}) = \log_{p}(\Pone^+) \cdot  \Ptwo + 
\log_p(\Ptwo^+) \cdot \Pone. \\
 \end{array}
 $$
The four regulators $ \Reg_{g_\alpha}(E,V_{12})$, 
$ \Reg_{g_\beta}(E,V_{12}) $, $ \Reg_{h_\alpha}(E,V_{12})$, 
and $ \Reg_{h_\beta}(E,V_{12})$ are all seen to be simple $L^\times$-multiples of the expression
 \begin{equation}
 \label{eqn:inert-DLR-regulators}
\log_{p}(\Pone^+) \cdot  \log_p(\Ptwo^-) - \log_{p}(\Ptwo^+) \cdot \log_p(\Pone^-).
\end{equation}
  The resulting formula 
  for $ \Lp^{g_{\alpha}}(f,g,h)$  predicted by Conjecture \ref{conj:DLR}
  has been  extensively tested numerically in  \cite[\S 3.3]{DLR}. 
 
  
  \begin{remark}
  Even though the points $\Pone^+$, $\Pone^-$, $\Ptwo^+$, and $\Ptwo^-$
  that figure in the  generalised Kato classes are in principle expressed as 
 linear combinations of Heegner points, the methods used to prove Conjecture \ref{conj:DLR} when $p$ is split in $K$, which are based on the $p$-adic Gross-Zagier formula of \cite{BDP1} and on properties of the Katz $p$-adic $L$-function attached to $K$,  seem to break down completely when $p$ is inert in $K$. A theoretical understanding of the $p$-adic iterated integrals of
 \cite{DLR} in this setting would seem to require a  new idea.
 \end{remark}
 
 \begin{remark}
 It is worth contrasting the    expressions arising in 
 \eqref{eqn:inert-DLR-regulators} with the simpler formulae
 \begin{eqnarray*}
 \log_{\alpha\alpha} \eReg_{\beta\beta}(E,V_{12}) = \log_{\beta\beta} \eReg_{\alpha\alpha}(E, V_{12}) & = &  \log_p(\Pone^+) \cdot 
 \log_p(\Ptwo^+),  \\
  \log_{\alpha\beta} \eReg_{\beta\alpha}(E,V_{12}) = \log_{\beta\alpha} \eReg_{\alpha\beta}(E, V_{12}) & = &  \log_p(\Pone^-) \cdot 
 \log_p(\Ptwo^-). 
 \end{eqnarray*}
 In certain very special settings--notably, when the elliptic curve $E$ has multiplicative 
 reduction at $p$--these expressions arise as the leading terms of the $p$-adic $L$-series 
 $$\Lp^{f}(f,g_\alpha,h_\alpha),  \quad \ldots, 
\quad \Lp^{f}(f, g_\beta,h_\beta).$$
Methods based on the Cerednik-Drinfeld theory of $p$-adic uniformisation of 
Shimura curves make it possible to relate these leading terms to the $p$-adic 
logarithms   of Heegner points, leading to some indirect theoretical evidence
 for Conjecture \ref{conj:main} in the setting where $p$ is inert in $K$.
 See  \cite{DR3} for a   description of this approach.
 \end{remark}
 



\subsubsection{The rank $(2,0)$ setting}
\label{sec:quad-20-imag}

Assume, after possibly interchanging $\Vone$ and $\Vtwo$, that
$L(E,\Vone,s)$ has a double zero at $s=1$ and that $L(E,\Vtwo,1) \ne 0$. 
In this case, Conjectures $\BSD(E,\Vone)$ and $\BSD(E,\Vtwo)$ predict that  
$$(E(H)_L\otimes V_{12})^{G_\Q} = (E(H)_L \otimes \Vone)^{G_\Q} $$ is
 two-dimensional over $L$. Choose a basis $(P,Q)$ for this vector space.
It is natural to write
$$ P  = P_{\psione}  \otimes v_{\psione'}  + P_{\psione'} \otimes v_{\psione}, 
\qquad
Q = Q_{\psione}  \otimes v_{\psione'}  + Q_{\psione'} \otimes v_{\psione}, 
$$
where $(P_{\psione},Q_{\psione})$ and 
 $(P_{\psione'}, Q_{\psione'})$ are bases  for the two-dimensional subspaces of $E(H)_L$ on which $G_K$ acts via the characters
$\psione$ and $\psione'$ respectively.

As in the rank $(1,1)$ setting, the shape of the
 enhanced regulators attached to  $V_{12}$ and to the basis
$(P,Q)$ 
depend very much on whether the prime $p$ is split or inert  in $K$.


\medskip\noindent
{\em a) The case where $p$ is split in $K$}.
In this case, letting $p = \fp \fp'$ be the factorisation of $p$ into distinct primes of $K$, we can
 set
$$ \alpha_g =\psi_g(\fp), \quad \beta_g = \psi_g(\fp'), \qquad \alpha_h = \psi_h(\fp), \quad
\beta_h = \psi_h(\fp'),$$
so that
$$ \alpha_g\alpha_h  = \psione(\fp), \quad \alpha_g\beta_h = \psitwo(\fp), \quad
\beta_g\alpha_h = \psitwo(\fp'), \quad \beta_g\beta_h = \psione(\fp').$$
The decomposition of the $G_{K_{\fp}} = G_{\Q_p}$ representations attached to
 \eqref{eqn:quadratic-decomp} into $\Frob_p$-eigenspaces is also stable under the action of the global Galois group $G_K$, and is  described by:
 $$ V_{12}^{\alpha\alpha} = \Vone^{\psione}, \quad
 V_{12}^{\alpha\beta} = \Vtwo^{\psitwo}, \quad
 V_{12}^{\beta\alpha} = \Vtwo^{\psitwo'}, \quad
 V_{12}^{\beta\beta} = \Vone^{\psione'}.
 $$
 It follows  that, up to multiplication by $L^\times$,
\begin{equation}
\label{eqn:20imag-split}
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{\fp}(P_{\psione'}) \cdot  Q - \log_{\fp}(Q_{\psione'}) \cdot P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) =0 , \\
 \eReg_{\beta\alpha}(E, V_{12}) =   0, &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{\fp}(P_{\psione}) \cdot Q - \log_{\fp}(Q_{\psione}) \cdot P.
 \end{array}
 \end{equation}
This suggests that the generalised Kato classes 
$\kappa(f,g_\alpha,h_\alpha)$ and $\kappa(f,g_\beta,h_\beta)$ give non-trivial
elements of  the two-dimensional vector space $(E(H)_L\otimes \Vone)^{G_\Q}$,
while the generalised Kato classes 
$\kappa(f,g_\alpha,h_\beta)$ and $\kappa(f,g_\beta,h_\alpha)$ should vanish.
Furthermore, a direct calculation shows that 
\begin{equation*}
\Reg_{g_\alpha}(E, V_{12}) =   \Reg_{g_\beta}(E,V_{12}) = \Reg_{h_\alpha}(E, V_{12}) =   \Reg_{h_\beta}(E,V_{12})  = 0,
 \end{equation*}
 which is consistent with the fact, proved in \cite[\S 3.2]{DLR}, that all the $p$-adic iterated integrals attached to $(f,g,h)$ vanish in the rank $(2,0)$ setting when $p$ is split in $K$.
In this case the generalised Kato classes carry more arithmetic information that the
$p$-adic iterated integrals which describe (certain of) their $p$-adic logarithms. This represents yet another setting where
 Conjecture \ref{conj:main} is a genuine strengthening of 
Conjecture \ref{conj:DLR} of \cite{DLR}.


\medskip\noindent
{\em b) The case where $p$ is inert in $K$}.
After scaling the points $P_{\psione}$, and $P_{\psione'}$, $Q_{\psione}$ and $Q_{\psione'}$ in such a way that
$$ \Frob_p(P_{\psione}) = P_{\psione'}, \qquad \Frob_p(P_{\psione'}) = P_{\psione}, \qquad
\Frob_p(Q_{\psione})  = Q_{\psione'}, \qquad \Frob_p(Q_{\psione'}) = Q_{\psione}$$
and letting
$$ P^\pm := P_{\psione} \pm P_{\psione'}, \qquad Q^{\pm}:= Q_{\psione} \pm Q_{\psione'},$$
 the enhanced regulators
  are  given by 
\begin{equation}
\label{eqn:20imag-inert}
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(P^+) \cdot  Q - \log_{p}(Q^+) \cdot  P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_{p}(P^-) \cdot  Q - 
\log_p(Q^-) \cdot P, \\
 \eReg_{\beta\alpha}(E,V_{12}) = \log_{p}(P^-) \cdot  Q - \log_{p}(Q^-) \cdot P,  &  \quad &  \eReg_{\beta\beta}(E,V_{12}) = \log_{p}(P^+) \cdot  Q - 
\log_p(Q^+) \cdot P. 
 \end{array}
\end{equation}
 In this case,  the four regulators  $\Reg_{g_\alpha}(E,V_{12}), \ldots, \Reg_{h_\beta}(E,V_{12})$
 attached to $(f,g,h)$ 
 are  explicit multiples of the expression
 \begin{equation}
 \label{eqn:ii-pinert-quad}
  \log_{p}(P^+) \cdot \log_p(Q^-) - \log_{p}(Q^+) \cdot  \log_p(P^-).
  \end{equation}
See \cite[Ex.~3.14]{DLR} for some numerical verifications of the agreement between this value and the 
$p$-adic iterated integrals attached to $(f,g,h)$. 

Equations
 \eqref{eqn:20imag-split}
 and
 \eqref{eqn:20imag-inert} combined with Conjecture \ref{conj:main}
 suggest that the generalised Kato classes   always
 generate the Mordell-Weil group $(E(H)_L\otimes \Vone)^{G_\Q}$  (tensored over $L$ with $\Q_p$)
 in the rank $(2,0)$ setting.   Since the irreducible representation $\Vone$ occurs with multiplicity
 two in $E(H)_L$, none of the $\Vone$-isotypic part of the  Mordell-Weil group is expected to be accounted for by
  Heegner points, as discussed in the introduction.
 
 
 
 
 


\subsection{Real multiplication classes and Stark-Heegner points}
\label{sec:rm-case}

 In this chapter we consider the setting where $g$ and $h$ are theta series attached to 
 characters $\psi_g$ and $\psi_h$ of mixed signature of the same {\em real}
  quadratic field $K$.
 In that case, we have, exactly as in Section \ref{sec:cm-case},
 $$ V_g = \Ind_K^\Q \psi_g, \quad V_h = \Ind_K^\Q \psi_h, $$
 and
 $$
 V_{gh} = \Vone \oplus \Vtwo :=  \Ind_K^\Q \psione \oplus \Ind_K^\Q \psitwo, \qquad \mbox{ where }
 \psione = \psi_g \psi_h, \qquad \psitwo = \psi_g \psi_h'.$$
 The characters  $\psione$ and $\psitwo$ are also ring class characters of $K$, with one  totally even, and the other
  totally odd. Once again, it is convenient  to assume  that $\Vone$ and $\Vtwo$ are both irreducible, i.e., that neither $\psione$ nor
 $\psitwo$ is a genus character of $K$.

 As in the case where $K$ is imaginary,  the study of the generalised Kato classes divides naturally into the rank $(1,1)$ 
and rank $(2,0)$ settings, depending on the orders of vanishing of $L(E/K,\psione,s)$ and $L(E/K,\psitwo,s)$
(or, alternately, on the dimensions of $(E(H)_L\otimes \Vone)^{G_\Q}$ and $(E(H)_L\otimes \Vtwo)^{G_\Q}$), and 
continue to depend in a crucial way on whether $p$ is split or inert in $K$. 
In all four cases,  the formulae for the enhanced regulators are identical to those obtained in 
Section \ref{sec:cm-case},  so it is unecessary to reproduce them here,  contenting ourselves with
 the following comments in connection with the rank $(1,1)$ setting.


\medskip\noindent
{\em a) The case where $p$ is split in $K$}.
This setting,   where the greatest amount of theoretical evidence
was available when $K$ is imaginary quadratic, thanks to the theory of Heegner points, is a lot more mysterious when $K$
is real quadratic.  With notations being the same as in Section \ref{sec:quad-11-imag}, we have
\begin{equation}
\label{eqn:stark-heegner}
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{\fp}(P_{\psione'}) \cdot  \Ptwo,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_{\fp}(P_{\psitwo'}) \cdot  \Pone, \\
 \eReg_{\beta\alpha}(E, V_{12}) =    \log_{\fp}(P_{\psitwo}) \cdot  \Pone, &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{\fp}(P_{\psione}) \cdot  \Ptwo,
 \end{array}
\end{equation}
 and 
\begin{equation}
\label{eqn:stark-heegner-bis}
\begin{array}{ll}
\Reg_{g_\alpha}(E, V_{12}) =  \log_{\fp}(P_{\psione'}) \cdot  \log_{\fp} (P_{\psitwo'}),  & 
 \Reg_{g_\beta}(E,V_{12}) = \log_{\fp}(P_{\psione}) \cdot \log_{\fp}(P_{\psitwo}), \\
\Reg_{h_\alpha}(E, V_{12}) =  \log_{\fp}(P_{\psione'}) \cdot  \log_{\fp} (P_{\psitwo}),  &
 \Reg_{h_\beta}(E,V_{12}) = \log_{\fp}(P_{\psione}) \cdot \log_{\fp}(P_{\psitwo'}).
 \end{array}
 \end{equation}
 This setting  has special appeal in connection with an eventual 
(for now, highly conjectural, and not even clearly fomulated) theory of 
{\em Stark-Heegner points} over ring class fields of real quadratic fields.
 It would be of great interest to relate (conjecturally, at least)  the  regulators  in 
 \eqref{eqn:stark-heegner} and in \eqref{eqn:stark-heegner-bis} to 
 generalised Kato classes and $p$-adic iterated integrals,  respectively.
 
The obstruction to doing this is that the modular forms $g$ and $h$ (more precisely, their stabilisations)
 fail to obey Hypothesis IV in Section \ref{sec:periods}.
  When $g$ is a modular form of RM type which is regular at a prime $p$ which splits in $K$,
 the Stark unit $u_{g_\alpha}$ is also unavailable, and an analogue of 
 Conjecture \ref{conj:DLR} has yet to be  formulated precisely in this setting.
Because of the tantalising connection with  Stark-Heegner points defined over ring class fields of $K$,
it would be of great interest to extend the Conjectures of \cite{DLR}, as well as Conjecture 
 \ref{conj:main}, to the real quadratic context.  
 A first step has been made in \cite{DLR3} towards understanding the periods of \S \ref{sec:periods}
 in this setting.



\medskip\noindent
{\em b) The case where $p$ is inert in $K$}.
The formulae for the enhanced regulators  are identical to those in Part b) of Section \ref{sec:quad-11-imag}, namely:
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(\Pone^+) \cdot  \Ptwo - \log_{p}(\Ptwo^+) \cdot \Pone,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) = \log_{p}(\Pone^-) \cdot  \Ptwo - 
\log_p(\Ptwo^-) \cdot \Pone, \\
 \eReg_{\beta\alpha}(E,V_{12}) = \log_{p}(\Pone^-) \cdot  \Ptwo + \log_{p}(\Ptwo^-) \cdot \Pone,  &  \quad &  \eReg_{\beta\beta}(E,V_{12}) = \log_{p}(\Pone^+) \cdot  \Ptwo + 
\log_p(\Ptwo^+) \cdot \Pone. \\
 \end{array}
 $$
 The $p$-adic logarithms of these enhanced regulators ought to involve  linear combinations of 
 products of logarithms of so-called {\em Stark-Heegner points}. This prediction   has been extensively tested numerically in  \cite[\S 4.2]{DLR}. 
 
 \begin{remark}
The logarithms of the generalised Kato classes that are not amenable to expressions in terms of 
$p$-adic iterated integrals are expected to admit particularly simple expressions, as suggested by 
the formulae
  \begin{eqnarray*}
 \log_{\alpha\alpha} \eReg_{\beta\beta}(E,V_{12}) = \log_{\beta\beta} \eReg_{\alpha\alpha}(E, V_{12}) & = &  \log_p(\Pone^+) \cdot 
 \log_p(\Ptwo^+),  \\
  \log_{\alpha\beta} \eReg_{\beta\alpha}(E,V_{12}) = \log_{\beta\alpha} \eReg_{\alpha\beta}(E, V_{12}) & = &  \log_p(\Pone^-) \cdot 
 \log_p(\Ptwo^-). 
 \end{eqnarray*}
 In the special case where $E$ has {\em multiplicative} reduction at $p$, 
 the article \cite{DR3} in progress proves a  formula of the shape
 $$ \log_{\alpha\alpha} \kappa(f,g_\alpha,h_\alpha) =  \log_p(\Pone^{?\pm}) \cdot 
 \log_p(\Ptwo^{?\pm}),  $$
 where $\Pone^{?\pm}$ and $\Ptwo^{?\pm}$ are the Stark-Heegner points 
 of \cite{darmon_hpxh}, and the sign that arises  depends on whether $E$ has split or non-split 
 multiplicative reduction at $p$. 
 \end{remark}
 

\subsection{Adjoint classes} 
\label{sec:adjoint-case}
The case where $h$ is dual to $g$ is of considerable arithmetic interest, since in that case
the representation 
$$V_{gh} = \Q_p \oplus  M_g, \qquad (\mbox{where } M_g := {\rm Ad}^0(V_g)),$$
admits the trivial representation as a constituent.
 The generalised Kato classes attached to $g$ and $h$ may then, in
  appropriate circumstances, contribute to the Mordell-Weil group
$E(\Q)$, and it is interesting to understand when this occurs.

The Hasse-Weil-Artin $L$-series 
$$ L(E, V_{gh},s) = L(E, s) L(E, M_g ,s)$$
has a double zero at  $s=1$ in one of the following three cases:
\begin{enumerate}
\item the rank $(0,2)$ setting where $L(E,1)\ne 0$ and $L(E,M_g,s)$ has a double zero at
$s=1$;
\item 
the rank $(1,1)$ case where $L(E,s)$ and $L(E,M_g,s)$ each vanish to order $1$ at
$s=1$;
 \item
 the rank $(2,0)$ setting where $L(E,s)$ has a double zero at the center and $L(E,M_g,1)\ne 0$.
 This case is particularly intriguing for its direct connection with
  the arithmetic of elliptic curves of rank
 two  over $\Q$. 
 \end{enumerate}
 

In all the examples that will be treated below, we always set
$$ \alpha_h = \alpha_g^{-1}, \qquad \beta_h = \beta_g^{-1},$$
so that
$$ \alpha_g \alpha_h = 1, \quad \alpha_g\beta_h = \alpha_g/\beta_g, \quad
\beta_g\alpha_h = \frac{\beta_g}{\alpha_g}, \quad \beta_g\beta_h = 1.$$


\subsubsection{The rank $(0,2)$ setting}
\label{sec:adjoint-02}
Let $(P,Q)$ be an $L$-basis for $(E(H)_L\otimes V_{12})^{G_\Q} = (E(H)_L \otimes M_g)^{G_\Q}$, 
and write
\begin{eqnarray*}
 P &=&  P_{\frac{\alpha}{\beta}} \otimes v_{\beta/\alpha} + P_1 \otimes v_1 + P_{\frac{\beta}{\alpha}} \otimes v_{\alpha/\beta}, \\
 Q & = & Q_{\frac{\alpha}{\beta}} \otimes v_{\beta/\alpha} + Q_1 \otimes v_1 + Q_{\frac{\beta}{\alpha}} \otimes v_{\alpha/\beta}, 
\end{eqnarray*}
where $v_{\beta/\alpha}$, $v_1$, and $v_{\alpha/\beta}$ are bases for the $\Frob_p$-eigenspaces of  $W_g$ attached to the eigenvalues $\frac{\beta_g}{\alpha_g}$, $1$, and
$\alpha_g/\beta_g$ respectively.
Then the four enhanced regulators are given, up to $L^\times$, as follows:
$$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(P_{1}) \cdot  Q - \log_{p}(Q_{1}) \cdot P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) =  \log_{p}(P_{\frac{\beta}{\alpha}}) \cdot  Q - \log_{p}(Q_{\frac{\beta}{\alpha}}) \cdot P,   \\
 \eReg_{\beta\alpha}(E, V_{12}) =  \log_{p}(P_{\frac{\alpha}{\beta}}) \cdot  Q - \log_{p}(Q_{\frac{\alpha}{\beta}}) \cdot P,     &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{p}(P_{1}) \cdot Q - \log_{p}(Q_{1}) \cdot P.
 \end{array}
 $$
Conjecture \ref{conj:main} suggests
 in this case that the generalised Kato class $\kappa(f,g_\alpha, h_\beta)$ generates the kernel of the map 
$\log_{\alpha\beta}$ in the two dimensional $\Q_p$-vector space $(E(H)_L\otimes M_g)^{G_\Q} \otimes_L \Q_p$.

We refer the reader to \cite[Example 5.4]{DLR} for the numerical verification of Conjecture \ref{conj:DLR} for two different instances in this setting.

 


\subsubsection{The rank $(1,1)$ setting}
\label{sec:adjoint-11}
Let $P$ be a generator of $E(\Q)_L$ and let $Q$ be a generator of $(E(H)_L\otimes M_g)^{G_\Q}$. With
 the same notational conventions as before, we find:
\begin{equation}
\label{eqn:ad-reg-enhanced} 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(P) \cdot  Q - \log_{p}(Q_{1}) \cdot P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) =   \log_{p}(Q_{\frac{\beta}{\alpha}}) \cdot P,   \\
 \eReg_{\beta\alpha}(E, V_{12}) =   \log_{p}(Q_{\frac{\alpha}{\beta}}) \cdot P,     &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{p}(P) \cdot Q - \log_{p}(Q_{1}) \cdot P.
 \end{array}
\end{equation}
 In contrast, we have
\begin{equation}
\label{eqn:ad-reg}
\begin{array}{lll}
\Reg_{g_\alpha}(E,V_{12}) = \log_{p}(P) \cdot  \log_p(Q_{\frac{\beta}{\alpha}})   &  \quad & 
 \Reg_{g_\beta}(E,V_{12}) =   \log_p(P) \cdot \log_{p}(Q_{\frac{\alpha}{\beta}}),   \\
 \Reg_{h_\alpha}(E, V_{12}) =  \log_p(P) \log_{p}(Q_{\frac{\alpha}{\beta}}),     &  \quad &
 \Reg_{h_\beta}(E, V_{12}) = \log_{p}(P) \cdot \log_{p}(Q_{\frac{\beta}{\alpha}}).
 \end{array}
\end{equation}
 Many numerical examples where the $p$-adic iterated integrals atttached to $(f,g,h)$ are
 seen to agree with these 
 regulators  are described in   in \cite[\S 5]{DLR}. It is worth noting that the expression
$\log_p(Q_1)$ that appears in the enhanced regulators  of 
\eqref{eqn:ad-reg-enhanced} disappears from the 
regulators  \eqref{eqn:ad-reg} that arose in \cite{DLR}. 


 
 


\subsubsection{The rank $(2,0)$ setting}
\label{sec:adjoint-20}

Let $(P,Q)$ be an $L$-basis of the two-dimensional $L$-vector space
 $E(\Q)_L$. With
 the same notational conventions as before, we find:
 $$ 
\begin{array}{lll}
\eReg_{\alpha\alpha}(E,V_{12}) = \log_{p}(P) \cdot  Q - \log_{p}(Q) \cdot P,  &  \quad &  \eReg_{\alpha\beta}(E,V_{12}) =  0,   \\
 \eReg_{\beta\alpha}(E, V_{12}) =  0,     &  \quad &
 \eReg_{\beta\beta}(E, V_{12}) = \log_{p}(P) \cdot Q - \log_{p}(Q) \cdot P.
 \end{array}
 $$
 In other words, the generalised Kato classes $\kappa(f,g_\alpha,h_\alpha)$ and 
 $\kappa(f,g_\beta,h_\beta)$ give (essentially, the same, up to $L^\times$-multiples)   canonical element 
 of $E(\Q)_{\Q_p}$, which is expected to be non-trivial precisely when
 $$ L''(E,1) \ne 0, \qquad L(E,M_g,1) \ne 0.$$
 Note that on the other hand
  $$ 
\Reg_{g_\alpha}(E,V_{12}) =   \Reg_{g_\beta}(E,V_{12}) =    \Reg_{h_\alpha}(E, V_{12}) =   \Reg_{h_\beta}(E, V_{12}) = 0.$$
 This last example gives yet another instance where Conjecture \ref{conj:main} represents
  a genuine strengthening of the elliptic Stark conjectures of \cite{DLR}.
  It predicts that  that generalised Kato classes 
 of the form $\kappa(f,g_\alpha, \bar g_{1/\alpha})$ 
 ought to give non-trivial elements in the pro-$p$-Selmer groups of elliptic curves of rank two over $\Q$, when the auxiliary $L$-value $L(E,M_g,1)$ is non-zero.
 Testing this prediction  experimentally seems to
present an  interesting  
 challenge.


{\small
 \thanks{{\bf Acknowledgements.} The first author was supported by an NSERC Discovery Grant and
 the second author was supported by Grant MTM2012-34611.
}}


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\end{document}

 