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\begin{document}

\title[Stark  points and  $p$-adic iterated integrals]
{Stark  points and  $p$-adic iterated integrals \\
attached to modular forms of  weight one}



\author{Henri Darmon, Alan Lauder and Victor Rotger}

\dedicatory{To our families}

\begin{abstract}
Let $E$ be an elliptic curve over $\Q$   and let $\varrho_\flat$ and $\varrho_\sharp$ be   odd
  two-dimensional Artin representations for which 
$\varrho_\flat \otimes\varrho_\sharp$ is self-dual.
The progress on modularity achieved in the last decades ensures the existence of normalised eigenforms
$f$, $g$ and $h$  of respective weights $2$, $1$ and $1$,
 giving rise to $E$, $\varrho_\flat$, and $\varrho_\sharp$ via the constructions of Eichler-Shimura 
 and Deligne-Serre. 
This article  examines  certain   {\em $p$-adic iterated integrals} attached to  
the triple $(f,g,h)$, which are $p$-adic avatars of the leading term
of the Hasse-Weil-Artin $L$-series  $L(E,\varrho_\flat \otimes\varrho_\sharp,s)$ 
when it has a double zero at the center. A 
 formula is proposed for these iterated integrals, involving  
 the formal group logarithms of   
 global points on $E$---referred to as 
 {\em Stark points}---which are defined over the number field cut out by $\varrho_\flat\otimes \varrho_\sharp$.
This formula can   be viewed as an elliptic curve analogue of 
Stark's conjecture on units attached to weight one forms.
It is proved when $g$ and $h$ are binary theta series attached to a common
imaginary quadratic field in which $p$ splits, by relating the arithmetic quantities
that arise in it to elliptic units and Heegner points.
Fast algorithms for computing $p$-adic iterated integrals 
based on Katz expansions of overconvergent modular forms are
then  exploited
to gather numerical evidence   in more exotic scenarios,
encompassing
Mordell-Weil groups over  cyclotomic fields, ring class fields of real quadratic fields (a setting which
may shed light on the theory of Stark-Heegner points attached to
Shintani-type  cycles
 on $\cH_p\times\cH$),   and
extensions of $\Q$ with Galois group a central extension of
the dihedral group
$D_{2n}$   or  of one of the exceptional subgroups $A_4$, $S_4$,  and $A_5$
of $\PGL_2(\C)$.
 \end{abstract}

\address{H. D.: McGill University, Montreal, Canada}
\email{darmon@math.mcgill.ca}
\address{A. L.: University of Oxford, U. K.}
\email{lauder@maths.ox.ac.uk}
\address{V. R.: Universitat Polit\`{e}cnica de Catalunya, Barcelona, Spain}
\email{victor.rotger@upc.edu}

%\subjclass{11G18, 14G35}


\maketitle

\tableofcontents

\section*{Introduction}

The purpose of this article is to formulate an analogue of the Stark conjecture
on units in number fields in the setting of global points on elliptic curves. 
 We prove this ``elliptic Stark conjecture'' in some cases 
and provide numerical evidence in its support in many others. 
It is offered as a more constructive alternative to the 
 Birch and Swinnerton-Dyer (BSD)
conjecture, since it 
often 
allows the efficient analytic computation of  $p$-adic logarithms of global points.  
%in rank one situations. 
In contrast, the BSD conjecture 
(and its $p$-adic variant formulated by
 Mazur, Tate and Teitelbaum)   gives 
the  ($p$-adic) heights of such points, from which 
 it appears computationally   more difficult  to recover
  the points themselves \cite{silverman}. 
 Our conjecture gives a  unifying perspective on the two currently known constructions of
   global points on elliptic curves over $\Q$---Heegner
points and the still conjectural  Stark-Heegner points attached to real quadratic cycles on
 $\cH_p\times \cH$---and suggests a 
possible global underpinning for the latter.  
It  also yields conjectural constructions and explicit formulae,
in situations of rank one and two,
for global points  over 
cyclotomic fields,  abelian extensions of quadratic fields
which are not necessarily  anticyclotomic, 
and  extensions of $\Q$ with Galois group a central extension of
 $A_4$, $S_4$ or $A_5$. 
\medskip

Let $E$ be an elliptic curve over $\Q$  with  
Galois representation $V_p(E) := H^1_{\et}(E_{\bar\Q},\Q_p(1))$,
 and let
\begin{equation}\nonumber
%% \label{rho}
\varrho:  G_{\Q} \lra \Aut(V_\varrho) \simeq \GL_n(L), \qquad \dim_{L}(V_\varrho)  = n \geq 1
\end{equation}
be an Artin representation---i.e., a continuous  Galois representation factoring through a finite quotient $\Gal(H/\Q)$ of $G_\Q$---with coefficients in some
 finite extension $L\subset \C$ of $\Q$. 
The {\em Hasse-Weil-Artin $L$-series}  $L(E,\varrho,s)$ of  $E$ twisted by $\varrho$ is the $L$-function of the 
compatible system   $V_p(E)\otimes V_\varrho$ of $p$-adic representations of $G_\Q$.
It is defined on the 
right half-plane $\mathrm{Re}(s)>3/2$  by an absolutely convergent Euler product of degree $2n$ and
   is expected to admit an analytic continuation to the whole complex plane. 
Under this assumption, the {\em analytic rank} of the pair $(E,\varrho)$ is defined to be
\begin{equation}\nonumber
%% \label{r-an}
r_{\mathrm{an}}(E,\varrho) := \mathrm{ord}_{s=1} L(E,\varrho,s).
\end{equation}
A  natural Galois equivariant refinement of the  Birch and Swinnerton-Dyer conjecture predicts that
\begin{equation}
\label{BSD}
r_{\mathrm{an}}(E,\varrho)   \stackrel{?}=     r(E,\varrho),
\end{equation}
where $r(E,\varrho)$ is the {\em multiplicity}  of $V_\varrho$ as a subrepresentation of  $E(H)\otimes L$
when $\varrho$ is irreducible, 
 defined for   general $\varrho$ by 
 \begin{equation}\nonumber
%% \label{r}
r(E,\varrho) := \dim_{L}   \Hom_{G_\Q}(V_\varrho, E(H)\otimes L).
\end{equation}
Equality \eqref{BSD} has been proved when $r_{\an}(E,\varrho) =0$ and 
(in roughly chronological order) 
\begin{itemize}
\item[I.]
$\varrho$ is the odd  self-dual
 two-dimensional Galois representation induced from a ring class character of 
 an imaginary quadratic field, by a series of works building on the breakthroughs of 
 Gross-Zagier and Kolyvagin;
 \item[II.]
$\varrho$ is a Dirichlet character, by the work of Kato \cite{kato};
\item[III.]
$\varrho$ is an odd irreducible two-dimensional Galois 
representation satisfying some mild restrictions, by  \cite{BDR2};
\item[IV.]
 $\varrho$ is an irreducible constituent of the tensor product
 $ \varrho_\flat \otimes\varrho_\sharp$ of two odd
irreducible two-dimensional Galois representations 
which is self-dual and satisfies some other mild restrictions, by
\cite{DR2}.
\end{itemize}
The setting where  $r_{\mathrm{an}}(E,\varrho)>0$
raises the challenge of constructing  non-zero elements 
in 
$$ 
E(H)_L^{\varrho} := \sum_{\varphi} \varphi(V_{\varrho}), \qquad \mbox{where $\varphi$ runs over a basis of $
 \Hom_{G_\Q}(V_\varrho, E(H)\otimes L)$}.
$$
Devising strategies for constructing such global points is arguably the most fundamental 
issue surrounding the Birch and Swinnerton-Dyer conjecture. 
Yet very little is known in this direction beyond the setting 
alluded to  
 in item I above, where 
  the methods of Gross, Zagier and Kolyvagin  can also be exploited to establish  
\eqref{BSD}   when $r_{\an}(E,\varrho)=1$, by using Heegner points
(as recalled in Ch.~\ref{sec:heegner} below) to construct the predicted
non-trivial
elements of $E(H)_L^{\varrho}$. 

As explained  in the survey \cite{BCDDPR}, 
   the techniques used  to analyze
 settings II, III and IV can be organised under the common rubric of ``Euler systems of Garrett-Rankin-Selberg type".
The present work considers the same settings,  but in the
 less well-understood scenario of
``analytic rank $\le 2$."
More precisely, it will be   assumed that
$\varrho$ is an irreducible constituent of the tensor product  $\varrho_\flat \otimes \varrho_\sharp$, 
where $\varrho_\flat$ and $\varrho_\sharp$ are odd, two-dimensional, 
but not necessarily irreducible, 
Artin representations satisfying the self-duality condition
\begin{equation}\nonumber
%% \label{eqn:self-duality-condition}
\chi := \det(\varrho_\flat)^{-1}  = \det(\varrho_\sharp), 
\end{equation}
and that
 $r_{\an}(E,\varrho)=1$ or $2$. The primary
 goal of this paper is  to propose
 %  in favorable settings where  $r(E,\varrho)=1$,
  a conjectural 
$p$-adic analytic formula  relating  global points in $E(H)_L^{\varrho}$  
to``$p$-adic iterated integrals" 
attached to an appropriate triple of modular forms,
and to prove it  in some cases.




The following  general notations and background are  required to formulate
the ``elliptic Stark conjecture" (Conjecture  $\shi$) below. 
For any positive integer $N$, any Dirichlet character $\eta: (\Z/N\Z)^\times \ra \bar\Q^\times$ and any field  $L$, 
write $S_k(N,\eta)_L\subseteq M_k(N,\eta)_L$  for  the spaces of classical cusp forms and 
modular forms of weight $k$, level 
$N$ and character $\eta$ with Fourier coefficients in $F$. (When $\eta$ is suppressed from the notation, it is to be understood
 that $\eta$ is the trivial character.)
 
Thanks to Wiles' proof of the Shimura-Taniyama conjecture, the
 elliptic curve $E$ is known to be {\em modular}\hspace{0.05cm}:
 there is a normalised weight two newform 
 $f\in S_2(N_f)_\Q,$
 where $N_f$ is the conductor of $E$,
satisfying 
$$ L(f,s) = L(E,s).$$
The  work of  Khare and Wintenberger \cite{KW} on Serre's modularity conjecture
 likewise implies that any odd, two-dimensional Artin representation
  arises from a classical (cuspidal or Eisenstein) normalized newform of weight one. 
  More precisely, denoting by $N_g$ and $N_h$ the Artin conductors of
  $\varrho_\flat$ and $\varrho_\sharp$, there are weight one newforms
$g \in M_1(N_g,\chi^{-1})_L$ and $ h\in M_1(N_h,\chi)_L$ 
satisfying 
$$   L(g,s) = L(\varrho_\flat,s), \qquad   L(h,s) = L(\varrho_\sharp,s).$$
Henceforth,  it will be convenient to use  $\varrho_g$ and $\varrho_h$  to designate 
$\varrho_\flat$ and $\varrho_\sharp$, to  write
$$ \varrho_{gh} := \varrho_g\otimes\varrho_h: \Gal(H/\Q) \lra \SL_4(L),$$
and to denote by $V_g$, $V_h$ and  $V_{gh}$  the   $L$-vector spaces underlying 
$\varrho_g$, $\varrho_h$ and $\varrho_{gh}$, respectively. 


When either $g$ or $h$ is an Eisenstein series, the Rankin-Selberg method yields the
analytic continuation and functional equation for the $L$-function
$$L(E,\varrho_{gh},s)= L(f\otimes g\otimes h,s)$$  relating its values at $s$ and $2-s$. 
This has been extended to the general case where both $g$ and $h$ are cusp forms,
by a result of Garrett.   The self-duality condition on $\varrho_{gh}$ implies that the 
root number  $\varepsilon(E,\varrho_{gh})$ that
appears in this functional equation is $\pm 1$, 
and determines the parity of $r_{\an}(E,\varrho_{gh})$.
Furthermore, 
it can be written as a product of local
root numbers 
$$\varepsilon(E,\varrho_{gh}) = \prod_{v|\Nfgh\cdot\infty} \varepsilon_v(E,\varrho_{gh}), 
\qquad \mbox{(where }  \Nfgh:= \lcm(N_f,N_g,N_h) \mbox{)},$$ 
and the local sign $\varepsilon_\infty(E,\varrho_{gh})$ is always equal to $1$.
The following  hypothesis on the local root numbers 
$\varepsilon_v(E,\varrho_{gh})$  at the non-archimedean places,
which is discussed in further detail in Ch.~\ref{tri},
 shall be made throughout this article:

\medskip
\noindent
{\bf Hypothesis A}. (Local sign hypothesis)
For all finite places $v |\Nfgh$, the 
 local signs  $\varepsilon_v(E,\varrho_{gh})$
are equal to $+1$.
In particular, the same is true of the global root number, and hence
$L(E,\varrho_{gh},s)$ vanishes to {\em even order} at $s=1$.

\medskip\medskip
\noindent
This hypothesis on the local sign is often satisfied: for instance,  
whenever $\gcd(N_f, N_g,N_h)=1$.
Under Hypothesis A, the method of Garrett-Rankin-Selberg relates the 
central critical value $L(f\otimes g\otimes h,1)$ to the values of the trilinear form
$$I: S_2(\Nfgh)_\C \times M_1(\Nfgh,\chi^{-1})_\C \times M_1(\Nfgh,\chi)_\C \, \lra \, \C$$
defined by
$$ I(\testf, \testg,\testh) := \langle \testf, \testg\testh\rangle,$$
where $\langle \ , \ \rangle$ denotes the Petersson scalar product on 
$S_2(\Nfgh)_\C \times M_2(\Nfgh)_\C$.
More precisely, for any integer $N$, let
 $\mathbb T_N$ denote the Hecke algebra generated by  the good Hecke operators $T_n$ with 
 $n \nmid N$. If $M$ is any $\mathbb T_N$-module and $\phi\in M$ is a simultaneous eigenvector for
  the action of $\mathbb T_N$, satisfing $T_\ell\phi = a_\ell(\phi) \phi$ for all $\ell\nmid N$,
   let 
  $I_\phi  \subset \mathbb T_N $ be the ideal 
  associated to this system of Hecke eigenvalues and denote by 
\begin{equation}
\label{eqn:hecke-eigenspace}
M[\phi]= \mathrm{Ker}(I_\phi) =   \{ m\in M: T_\ell m = a_\ell(\phi) m \mbox{ for all } \ell \nmid N \}
\end{equation} 
the corresponding eigenspace.
 By the work \cite{HaKu} of Harris-Kudla,  
the restriction 
\begin{equation}
\label{eqn:trilinear-form-L}
 I_{fgh}: S_2(\Nfgh)[f] \times M_1(\Nfgh,\chi^{-1})[g] \times M_1(\Nfgh,\chi)[h] \, \lra\,  \C
\end{equation}
of $I$  to the product of isotypic components for $f$, $g$ and $h$  is identically zero
  if and only if the central critical value
$L(f\otimes g\otimes h,1)$ vanishes. 
% See Ch.~\ref{tri} for a more detailed discussion of this, and  the relation of the $p$-adic avatars \eqref{eqn:iterated-integral}  and \eqref{eqn:trilinear-form-iterated-integral} of the  trilinear  forms introduced below to the Garrett-Hida  triple product $p$-adic $L$-function constructed originally by Hida and Harris-Tilouine.
Since this article is solely interested  in the setting where $\varrho_{gh}$ 
admits irreducible constituents $\varrho$ for which 
$L(E,\varrho,1)=0$, the following hypothesis is also made throughout:

\medskip\medskip
\noindent
{\bf Hypothesis B}. (Global vanishing hypothesis). The $L$-function
$L(E,\varrho_{gh},s)$ vanishes at $s=1$, and hence the 
trilinear form $I_{fgh}$ 
in 
\eqref{eqn:trilinear-form-L} is identically zero.

\medskip  \medskip
Under Hypotheses A and B, the $L$-function $L(E,\varrho_{gh},s)$
vanishes to order at least $2$ at $s=1$. 
When $L''(E,\varrho_{gh},1)\ne 0$, the
 elliptic Stark conjecture of this article 
expresses the values of   certain
  {\em $p$-adic iterated integrals} attached to the triple $(f,g,h)$
in terms of the formal group logarithms of
   global points in the $\varrho_{gh}$-isotypic 
 subspace $E(H)_L^{\varrho_{gh}}$ 
 of  $E(H)\otimes L$.  
 
 \medskip
The definition of these $p$-adic iterated integrals is based on the theory of 
$p$-adic and 
overconvergent modular forms.
Let $p$ be a prime number and fix an embedding of $\bar\Q$ in $\bar\Q_p$. Let $\C_p$ denote the completion of $\bar\Q_p$, and let $H_p$ and $L_p$ denote the $p$-adic closure of $H$ and $L$ in $\C_p$ with respect to the fixed embedding. Assume that $p$ is chosen in such a way that
\begin{equation} \label{Hyp-p}
\ord_p(N_f)\leq 1 \, \mbox{ and } \, p\nmid N_gN_h, \, \mbox{ and set } \, N  := \mbox{ the prime-to-$p$ part of }  \Nfgh.
\end{equation}
(This hypothesis is made primarily for ease of exposition, and would eventually be
interesting to relax.)


 
The symbols $S_k^{(p)}(N,\chi)\subset M_k^{(p)}(N,\chi)$ refer to 
 the spaces of $p$-adic cusp forms (resp. modular forms) of weight $k$, level $N$ 
 and character $\chi$
with coefficients in $\C_p$,  while $S_k^{\oc}(N,\chi)\subset M_k^{\oc}(N,\chi)$ 
stand for the corresponding subspaces of overconvergent modular forms. 
The relations between these various spaces are summarised in the following sequence of natural inclusions of $\C_p$-vector spaces
\begin{equation}\nonumber
%% \label{eqn:modular-form-spaces}
 \xymatrix{
 S_k(Np,\chi)_{\C_p}  \ar@{^{(}->}[r]    \ar@{^{(}->}[d]   & S_k^{\oc}(N,\chi)   \ar@{^{(}->}[r]  \ar@{^{(}->}[d] &S_k^{(p)}(N,\chi)  \ar@{^{(}->}[d]  \\
  M_k(Np,\chi)_{\C_p}   \ar@{^{(}->}[r] &  M_k^{\oc}(N,\chi)   \ar@{^{(}->}[r] &  M_k^{(p)}(N,\chi).
 }
\end{equation}
These spaces are all endowed with an action of the Hecke algebra $\TT_N^{(p)}$ 
generated by the Hecke operators $T_\ell$ with $\ell\nmid Np$ and by the Hecke operator
 $U_p$. 
 Let   $e_{\ord}:= \lim_n U_p^{n!}$ be  
  Hida's ordinary idempotent and write $M^{\ord}=e_{\ord}\cdot M$, for 
  any $L_p[U_p]$-module $M$. 
  While the spaces $S_k^{\oc}(N,\chi)$ and $M_k^{\oc}(N,\chi)$ are infinite-dimensional over
  $\C_p$, their ordinary parts 
 $S_k^{\oc,\ord}(N,\chi)$ and 
  $M_k^{\oc,\ord}(N,\chi)$  are known to be finite-dimensional.
   In fact, Hida's classicality theorem  asserts that, for all $k\ge 2$, 
  the natural inclusion
\begin{equation}\nonumber
%% \label{Coleman-classicality}
M_k^{\ord}(Np,\chi)_{\C_p} \hookrightarrow M_k^{\oc,\ord}(N,\chi)
\end{equation}
is an isomorphism.

Serre's differential operator  
$$d= q \frac{d}{dq}:M_k^{(p)}(N,\chi) \lra M_{k+2}^{(p)}(N,\chi) $$
on $p$-adic modular forms 
sends overconvergent forms of weight $0$ to overconvergent forms of weight $2$.
For any $\testf\in S_2^{\oc}(N)$, the overconvergent modular function (or modular form of weight $0$)
$$\testF : = d^{-1} \testf  := \lim_{t\rightarrow -1} d^t \testf \in S_0^{\oc}(N) $$ is called the {\em overconvergent primitive} of $\testf$. Here the limit is taken over positive integers $t$ tending to $-1$ in weight space $\Z/(p-1)\Z\times \Z_p$.
The Fourier expansions of $\testf$  and $\testF$ are related by
$$  \testf(q) = \sum_{n\ge 1} a_n(\testf) q^n, \qquad \testF(q) = \sum_{p\nmid n} \frac{a_n(\testf)}{n} q^n.$$


For any $k\geq 2$, the
{\em $p$-adic iterated integral} attached to the data 
$$(\testf,\testgamma,\testh) \in S_2(N p)_L \times M_k^{\ord}(Np,\chi)_L^\vee \times M_k(Np,\chi)_L$$
(in which the superscript $\vee$ denotes the $L$-linear dual)
is defined to be
\begin{equation}
\label{eqn:iterated-integral}
 \int_{\testgamma} \! \testf \cdot \testh :=  \testgamma(e_{\ord}(\testF \cdot \testh)) \in \C_p.
 \end{equation}
Expressions of this sort with $k\ge 2$ 
play an important role in \cite{DR1} and \cite{DR2} where they are related (when $\testf$,  $\testgamma$ and $\testh$ are cuspidal eigenvectors) to the special 
values of Garrett-Hida triple product  $p$-adic  $L$-functions  and to the  images of certain
{\em generalised Gross-Schoen diagonal 
cycles} under the $p$-adic Abel-Jacobi map.  When $\testgamma$ and/or $\testh$ arise from
weight two  Eisenstein series,
these expressions are  also related to $p$-adic regulators of {\em Beilinson elements } 
in the $K$-groups
$K_2(X_1(N))$ or
in $K_1(X_1(N)^2)$ (cf.\,\cite{BD1} and \cite{BDR1}, \cite{BDR2} respectively). 


On the computational side, using a method for computing with overconvergent modular forms via Katz expansions
developed in \cite{lauder-alg}, the article \cite{lauder} describes an algorithm for the  efficient numerical evaluation of these
 $p$-adic iterated integrals, and uses it
 to  calculate  certain {\em Chow-Heegner points} on the elliptic curve $E$ which were first
defined and studied by Shouwu Zhang. 
 These global points,  which arise when $k\ge 2$ and 
$\testgamma$ and $\testh$ are    eigenvectors for $\TT_N$ with
 the same   eigenvalues, have a well-understood geometric provenance,
 and can also be calculated by complex analytic means following the strategy described
 in \cite{DRS} and \cite{arizona}. 




 The setting $k=1$ which is germane to the 
   present work is significantly  more
 mysterious: the $p$-adic iterated integrals (and indeed, the modular forms $g$ and $h$
 themselves)
cease to enjoy geometric interpretations in terms of the cohomology of Kuga-Sato varieties.
 On a more technical level,  the space $M_1^{\oc,\ord}(N,\chi)$,
 while still finite-dimensional, may contain the space $M_1^{\ord}(Np,\chi)_{\C_p}$ as a
 proper subspace. 
 Fix a root $\alpha_g$ of the Hecke polynomial 
 $$x^2-a_p(g) x + \chi^{-1}(p) = (x-\alpha_g)(x-\beta_g) $$
  attached to
 $g$ at $p$, and let $g_\alpha\in M_1(Np,\chi^{-1})_L$ denote the corresponding 
 $p$-stabilisation. It is an eigenform for the Hecke algebra $\TT_N^{(p)}$ satisfying
 $$U_p g_\alpha = \alpha_g\cdot g_\alpha.$$
 The weight one form $g$ is said to be   {\em regular at $p$} if  $\alpha_g \ne \beta_g$, i.e., if
 $g$ admits two distinct $p$-stabilisations.
 Let $g^*_\alpha = g_\alpha \otimes \chi$ denote the newform of level $N_g$ corresponding to the twist of $g_\alpha$ by $\chi$, the inverse of its nebentype character. 
 
If $\phi$ is a simultaneous eigenvector for either the Hecke algebra $\TT_N^{(p)}$ 
or $\TT_{Np}$, then, extending the definitions in  \eqref{eqn:hecke-eigenspace}, 
let
$$ M[\phi]:= \mathrm{Ker}(I_\phi), \qquad M[[\phi]] = \cup_{n\geq 1} \mathrm{Ker}(I_\phi^n)$$ 
denote the   eigenspace and generalized eigenspace attached to its system of
eigenvalues. 

 The following  assumption is  a crucial substitute for the
 classicality theorem when $k=1$:

 
\medskip\medskip
\noindent
{\bf Hypothesis C} (Classicality property for $g_\alpha$). \hspace{0.1cm}
The (overconvergent) cuspidal generalised \linebreak eigenspace $S_1^{\oc,\ord}(N,\chi)_{\C_p}[[g_\alpha^*]]$  is non-trivial and consists entirely of classical forms.

\medskip
\noindent
When $g$ is a cusp form,
Proposition \ref{BeDi-criterion} of Section \ref{sec:smooth} 
 establishes, as an immediate 
consequence of the   work  
 of Cho-Vatsal \cite{cho-vatsal}
and 
 Bella\"iche-Dimitrov \cite{BeDi}, 
that Hypothesis C holds---i.e., that the
 natural inclusion 
\begin{equation} 
 \label{eqn:inclusion-g}
i_{g_\alpha}: S_1(Np,\chi)_{\C_p}[[g^*_\alpha]] 
\hookrightarrow  S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]]
 \end{equation}
is an isomorphism of $\C_p$-vector spaces---whenever
 $g$  satisfies the explicit condition (1) of Hypothesis C' below. 
 (As discussed in Remark \ref{rem:hypC-Cprime} of
 Section \ref{sec:smooth}, this condition is expected to be necessary as well as sufficient.) 

When $g$  is an Eisenstein series, 
Proposition \ref{prop:hypothesis-D-eisenstein} shows that 
the cuspidal generalised  eigenspace  $S_1^{\oc,\ord}(N,\chi)[[g_\alpha^*]]$ is non-zero, 
i.e., 
$g_\alpha$ is a $p$-adic {\em cusp} form, if and only if $g$ satisfies  condition (2) of Hypothesis C' below. While the larger space $M_1^{\oc,\ord}(N,\chi)[[g_\alpha^*]]$  always contains non-classical
modular forms in this setting (whose Fourier expansions can be written down explicitly), it   nonetheless seems plausible that $S_1^{\oc,\ord}(N,\chi)[[g_\alpha^*]]$ should  consist  solely of classical forms, i.e., that the natural map
\begin{equation} 
 \label{eqn:inclusion-g-eis}
i_{g_\alpha}: M_1(Np,\chi)_{\C_p}[[g^*_\alpha]]  \cap S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]]
\hookrightarrow  S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]]
 \end{equation}
should be an isomorphism.  
However, the analogue of the theorem of Bellaiche-Dimitrov for irregular weight one Eisenstein points does not appear to be available in the literature, and therefore Hypothesis C could conceivably be stronger than 
condition (2) of Hypothesis C' in the Eisenstein setting.
 
Motivated by this discussion,  the following
Hypothesis C', which has the virtue of being entirely explicit, 
shall be  assumed throughout  along with Hypothesis C. (It is worth keeping in mind that
  both hypotheses
are quite likely  to be {\em equivalent}.)

\medskip\medskip
\noindent
{\bf Hypothesis C'} (Classicality property for $g_\alpha$, bis).  The modular form $g$  satisfies one of the following explicit  conditions:
\begin{itemize}
\item[(1)]
 It is a cusp form which is regular at  the prime $p$ (i.e., $\varrho_g$ is  irreducible and the frobenius element at $p$ acts on it with two distinct eigenvalues) 
  and it is 
 not the theta 
 series of a character of a real quadratic field in which  $p$ splits. 
\item[(2)]
 It is  an Eisenstein series which is irregular at $p$. (In other words, $\varrho_g$ is the direct sum of two   characters $\chi_1$ and $\chi_2$ for which $\chi_1(p)=\chi_2(p)$.)
\end{itemize}

\medskip\medskip
 Thanks to Hypothesis C and its implication that the inclusions in 
 \eqref{eqn:inclusion-g} and 
 \eqref{eqn:inclusion-g-eis} 
 are isomorphisms,
it becomes  possible to adapt the definition of the
  $p$-adic iterated integrals to
 define a $\C_p$-valued trilinear form
\begin{equation}
\label{eqn:trilinear-form-iterated-integral}
I_p': S_2(Np)_L[f]  \times M_1(Np,\chi)^\vee_L[g_\alpha] \times M_1(Np,\chi)_L[h] \,\lra\, \C_p
\end{equation}
 by the simple rule
 $$I_p'(\testf,\testgamma,\testh) =  \int_{\testgamma} \! \testf \cdot \testh \ \ 
 := \ \  \testgamma( e_{g^*_\alpha} e_{\ord}(\testF \cdot \testh)),$$
 where 
\begin{equation}
 \label{eqn:proj-g-alpha}
 e_{g^*_\alpha}: S_1^{\oc,\ord}(N,\chi) \lra S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]] = M_1(Np,\chi)_{\C_p}[[g^*_\alpha]]
 \end{equation}
 is the projector onto the generalized eigenspace attached to $g^*_\alpha$ for the action of the Hecke
 algebra $\TT_N^{(p)}$. 
 Note the crucial role played by Hypothesis C in the definition of this trilinear form via the equality in
 \eqref{eqn:proj-g-alpha}.
 
 \medskip
 We are now almost ready to state Conjecture $\shi$ relating the trilinear form in 
 \eqref{eqn:trilinear-form-iterated-integral} to arithmetic data associated to $(E,\varrho_g,\varrho_h)$. 
 Since the prime $p$ is unramified in $H$, there is an embedding $H\lra \Q_p^{\ur}$ 
 into the maximal unramified extension of $\Q_p$. 
 Fix such an embedding  and denote by 
  $\sigma_p \in \Gal(H/\Q)$ the associated {\color{blue} geometric}\footnote{In the published version we were not explicit about the precise normalization of the frobenius element we adopt throughout the article, which led to potential confusions in the interpretation of our statements and also caused a minor mistake in the proof of the main theorem of \S \ref{sec:Hee}, that we have corrected here. For the reader's convenience, we have marked in blue the modifications with respect to the published version.}  frobenius 
element at  $p$, {\color{blue} inducing the automorphism $x \mapsto x^{1/p}$ on the residue field.}
 If $F$ is any finite extension of 
$\Q_p$, with ring of integers $\cO_F$,
let $$  \log_p: \cO_F^\times  \longrightarrow F   $$
be the usual $p$-adic logarithm,  and let
$$
\log_{E,p}: E(F)   {\longrightarrow} F 
$$ 
denote
the canonical extension to $E(F)$ of the $p$-adic 
 formal group logarithm associated to a N\'eron differential on $E$.
Via the chosen   embedding  of $H$ into $\C_p$ the maps 
 $\log_p$ and $\log_{E,p}$ give rise to homomorphisms
  $$ \log_p: (\cO_H)^\times_L \lra  H_p \otimes L,   \qquad 
   \log_{E,p}: E(H)_L \lra H_p  \otimes L.$$
   
 Hypotheses $A$ and $B$  together imply that $r_{\an}(E,\varrho_{gh})\ge 2$, and  
 of course the 
 same   is expected of the algebraic rank. 
When $r(E,\varrho_{gh})=2$, let $(\Phi_1,\Phi_2)$ be an $L$-vector space basis
 for $\Hom_{G_\Q}(V_{gh},E(H)_L)$. 
 
Using Hypothesis C', we  make a choice of  a one dimensional $G_{\Q_p}$-stable 
subspace of $V_g$, denoted $V_{g}^{g_\alpha}$, according to the following recipe:
\begin{enumerate}
\item When condition (1) in Hypothesis C' is satisfied, 
 the geometric frobenius $\sigma_p$ acts  on the 
Artin representations  $V_g$  with two distinct eigenvalues and  thus determines a decomposition 
$V_g = V_g^{g_\alpha} \oplus V_g^{g_\beta}$ into one-dimensional 
$\sigma_p$-eigenspaces with eigenvalue  {\color{blue} $\alpha_g\chi(p)$ and  $\beta_g\chi(p)$} respectively.
\item
When condition (2) in Hypothesis C' is satisfied, we simply let $V_g^{g_\alpha}$ be 
any one-dimensional subspace of the (reducible) representation $V_g$ which is {\em not} stable under $G_\Q$. 
\end{enumerate}
The choice  of $V_g^{g_\alpha}$  then
   determines a canonical two-dimensional subspace of 
$V_{gh}$  by the rule
$$ V_{gh}^{g_\alpha} := V_g^{g_\alpha} \otimes V_h.$$
Fix an $L$-vector space basis $(v_1, v_2)$  for this space.

The bases $(\Phi_1,\Phi_2)$ and $(v_1,v_2)$   give rise to a  $2\times 2$ regulator matrix
$$ R_{g_\alpha}(E,\varrho_{gh}) :=  \left(
\begin{array}{cc}
\log_{E,p}(\Phi_1(v_1)) & \log_{E,p}(\Phi_1(v_2)) \\
\log_{E,p}(\Phi_2(v_1)) & \log_{E,p}(\Phi_2(v_2))
\end{array} \right),$$
which is independent of the choice of basis $(\Phi_1,\Phi_2)$ (resp., of the basis
$(v_1,v_2)$) up to left (resp., right)
 multiplication by a matrix in $\GL_2(L)$.
 The determinant of  $R_{g_\alpha}(E,\varrho_{gh})$
 is  therefore well defined, up to multiplication by $L^\times$. It can also be shown that
 this determinant does not depend on the choice of $V_{g}^{g_\alpha}$ when Condition (2) of Hypothesis C' is satisfied.
In both  cases (1) and (2), it    belongs to 
 $\Q_p^{\ur}\otimes L$, and $\sigma_p$ acts on it with the same eigenvalue
as on
 $\wedge^2 V_{gh}^{g_\alpha}$, namely, 
  $\alpha_g \alpha_h \alpha_g \beta_h = \alpha_g/\beta_g$. 
 
 

   Let $\mathrm{Ad}_g := \Hom^0(V_g,V_g)$ denote the three-dimensional {\em  adjoint representation} attached to   $\varrho_g$, on which the frobenius element $\sigma_p$ acts with eigenvalues
   $1$, $\alpha_g/\beta_g$, and $\beta_g/\alpha_g$. 
Section  \ref{sec:stark}  attaches to   $g_\alpha$  
a  Stark unit $u_{g_\alpha} \in (\cO_{H_g}[1/p]^\times)_L^{\mathrm{Ad}_g}$, 
where $H_g$ is the  number field cut out by $\mathrm{Ad}_g$. 
\begin{enumerate}
\item
When $g$  satisfies condition (1) in  Hypothesis C' and 
 $\alpha_g/\beta_g\ne - 1$,   this unit is determined up to scaling  by the twin requirements 
that 
$$ u_{g_\alpha} \mbox{ belongs to }  (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g}, \qquad \sigma_p(u_{g_\alpha}) = \frac{\alpha_g}{\beta_g} u_{g_\alpha}.$$
A more detailed discussion of $u_{g_\alpha}$  is given in Section \ref{sec:stark}.
The unit 
 $u_{g_\alpha}$ is  analogous to the Stark units of
 \cite{stark2}
 and \cite{Duke-Li} whose logarithms encode 
 the Petersson scalar product of $g_\alpha$ with itself. Section
 \ref{sec:stark} explains the striking connection between the existence of $u_{g_\alpha}$ and Hypothesis C'.  
 \item
 When $g$ satisfies condition (2) in Hypothesis C', then
 $$ u_{g_\alpha} \mbox{ belongs to }  (\cO_{H_g}[1/p]^\times)_L^{\mathrm{Ad}_g}.$$
 Note that in this case $H_g$ is the abelian extension  cut out by
the (odd) Dirichlet character $\chi_1/\chi_2$, in which the prime $p$ splits completely.
 \end{enumerate}

The main conjecture of the present work is

   

\medskip\medskip
\noindent
{\bf Conjecture $\shi$:} (Elliptic Stark  Conjecture). 
{\em 
Assume that hypotheses A,$\,$B and C-C' are satisfied, so that in particular
$r_{\an}(E,\varrho_{gh}) \ge 2$.
If $r(E,\varrho_{gh})>2$, then
the trilinear form $I_p'$  of \eqref{eqn:trilinear-form-iterated-integral}
is identically zero. If $r(E,\varrho_{gh})=2$, there exist ``test vectors"
$$(\testf,\testgamma,\testh) \in S_2(Np)_L[f] \times M_1(Np,\chi)^\vee_L[g_\alpha] \times M_1(Np,\chi)_L[h]$$
for which
\begin{equation}
\label{eqn:main-conj-intro}
 \int_{\testgamma} \! \testf \cdot \testh \ \  =  \ \ \frac{\det(R_{g_\alpha}(E,\varrho_{gh}))}{\log_p(u_{g_\alpha})}.
 \end{equation}  
     }
     
   
 \medskip\medskip  
Note that both the numerator and denominator  on 
 the right-hand side of \eqref{eqn:main-conj-intro} belong to the same eigenspace for $\sigma_p$, 
 with eigenvalue $\alpha_g/\beta_g$, 
 and hence that the ratio belongs 
 to $L_p\subset H_p\otimes L$, consistent with the fact that this is 
 clearly true of the left-hand side.

     

The condition   $r(E,\varrho_{gh})=2$  implies that either:
\begin{enumerate}
\item
  the Artin representation
$\varrho_{gh}$ admits a decomposition
\begin{equation}
\label{eqn:conj-I-decomp-20}
 \varrho_{gh} = \varrho_1 \oplus \varrho_2, \quad \mbox{ with } \quad
 r(E,\varrho_1) = 2, \quad r(E,\varrho_2) = 0,
 \end{equation}
 where $\varrho_1$ is an  irreducible constituent of $\varrho_{gh}$
 and $\varrho_2$ is  possibly trivial, or
 reducible;
 \item 
  the Artin representation
$\varrho_{gh}$ admits a decomposition
\begin{equation}
\label{eqn:conj-I-decomp-11}
 \varrho_{gh} = \varrho_1 \oplus \varrho_2 \oplus \varrho_3, \quad \mbox{ with } \quad
 r(E,\varrho_1) = r(E,\varrho_2) = 1, \quad r(E,\varrho_3)=0,
 \end{equation}
where
$\varrho_1$ and $\varrho_2$ are irreducible constituents of
$\varrho_{gh}$ and
$\varrho_3$ is possibly trivial or reducible.
\end{enumerate}
 These scenarios will be referred to  respectively as the {\em rank 2} and {\em rank (1,1) settings}
 for the elliptic Stark conjecture.
 The apparent rarity of double zeroes   for primitive $L$-functions  
 at central critical points 
 suggests that
 rank $2$ settings may only occur sporadically, while a systematic supply of 
 rank $(1,1)$ settings arises  when the constituents
  $\varrho_1$ and $\varrho_2$  in \eqref{eqn:conj-I-decomp-11} 
  are self-dual and the
 signs in the functional 
 equations for $L(E,\varrho_1,s)$ and $L(E,\varrho_2,s)$
 are both equal to $-1$.
 This explains why more  of the experiments reported on below 
 have tested Conjecture $\shi$ in the rank $(1,1)$ setting (although enough
 data has been gathered to  lend convincing evidence for Conjecture
 $\shi$ in the rank $2$ scenario as well). 
 
Another attractive feature of the rank  $(1,1)$ setting   is
that it  often leads to explicit formulae for the $p$-adic logarithms of
global points in $E(H)_L^{\varrho_1}$ and $E(H)_L^{\varrho_2}$. 
More precisely, the regulator  $R_{g_\alpha}(E,\varrho_{gh})$ is 
said to be {\em factorisable}
if, after eventually interchanging $\alpha_h$ and $\beta_h$ and/or
$\varrho_1$ and $\varrho_2$, the frobenius $\sigma_p$ acting  on 
$V_{\varrho_1}$ admits $\alpha_g\alpha_h$ but
not $\alpha_g\beta_h$ as an eigenvalue. 
Relative to a basis $(\Phi_1,\Phi_2) \in \Hom_{G_\Q}(V_{\varrho_1},E(H)_L) \times
\Hom_{G_\Q}(V_{\varrho_2}, E(H)_L)$ and to a 
 $\sigma_p$-eigenbasis $(v_1,v_2)$ for
$V_{gh}^{g_\alpha}$ 
satisfying
$$ \sigma_p v_1 = \alpha_g\alpha_h v_1, \qquad 
\sigma_p v_2 = \alpha_g \beta_h v_2,$$
the regulator  matrix
$R_{g_\alpha}$ takes on the simple form
$$ R_{g_\alpha}(E,\varrho_{gh})= 
\left( \begin{array}{cc}
\log_{E,p} P_1 & 0 \\
 \log_{E,p} Q & \log_{E,p} P_2 \end{array}
 \right),$$
 where the points  $P_j$ belong  to $E(H)_L^{\varrho_j}$ and are uniquely determined (up to $L^\times$)
  by the 
 requirement that
 \begin{equation}
  \label{eqn:P1-P2-admissible}
  \sigma_p P_1 = \alpha_g\alpha_h P_1, \qquad \sigma_p P_2 = \alpha_g \beta_h P_2.
  \end{equation} 
  Conjecture $\shi$ then
 admits the
  simpler formulation
  \begin{equation}\nonumber
%% \label{eqn:main-conj-intro-simpler}
 \int_{\testgamma} \! \testf \cdot \testh \ \  =  \ \ \frac{   \log_{E,p}(P_1) \cdot \log_{E,p}(P_2)}{\log_p(u_{g_\alpha})}.
 \end{equation}
The prototype for such a formula, discovered experimentally, is 
equation (\ref{formula-precise}) in Remark \ref{rem-formula-precise}.
 
 
 One of the important motivations for  studying Conjecture $\shi$ is the connection between the 
 $p$-adic Rankin $L$-values it considers
 and the  
 {\em generalised Kato classes}
 $$ \kappa(f,g_\alpha,h_\alpha), \quad \kappa(f,g_\alpha,h_\beta), \quad \kappa(f,g_\beta, h_\alpha), \quad
 \kappa(f,g_\beta,h_\beta)  \quad \in \ \  H^1(\Q, V_p(E) \otimes V_{gh})$$
 constructed in \cite{DR2} (resp. in \cite{BDR2}) as $p$-adic limits of \'etale Abel-Jacobi images of 
 Gross-Kudla-Schoen diagonal cycles (resp.~of \'etale regulators of Beilinson-Kato and/or Beilinson-Flach elements)
 when $g$ and $h$ are cusp forms (resp.~when $g$ and/or $h$ are Eisenstein series).  
 When $L(E,\varrho_{gh},1)=0$, these generalised Kato classes can be viewed as playing the role of Heegner points in  settings of (analytic) rank two; 
  the reader is  referred to Theorems B and D  and \S 6.3.~of \cite{DR2}~for  the
 relationship between their $p$-adic logarithms and the $p$-adic iterated integrals attached to  $(f,g,h)$.
The calculations performed in this work have also played a crucial role in the more recent preprint \cite{DR3}, which  conjectures a precise expression for
 the generalised Kato classes as linear combinations of points in the relevant
 Mordell-Weil group.
 
 
 
 
 
% It is worth insisting on the fact that, unlike the setting where $k\ge 2$,  the  iterated integrals that appear in Conjecture $\shi$ do not enjoy   direct geometric interpretations in terms of $p$-adic Abel-Jacobi images of cycles or   $p$-adic regulators of elements in $K$-theory, and have no obvious complex counterpart. Conjecture $\shi$ above suggests rather that they should be   intimately connected with  conjectural  points on   $E$ over the field cut out by  $\varrho_{gh}$. These points  can be viewed as  the direct analogues of the Stark units whose logarithms conjecturally encode the leading terms of the  non-abelian $L$-series attached to odd two-dimensional   Artin representations.  

\medskip\medskip
Turning to a more detailed overview of the contents of the paper, 
Chapter \ref{sec:inv-g}
discusses  Hypothesis 
C  and defines the Gross-Stark unit $u_{g_\alpha}$, showing that
Hypothesis C  bears an
intimate connection with the existence of this unit.


Building on the results of Harris-Kudla and Ichino on trilinear periods and Jacquet's conjecture, 
Chapter \ref{tri}  recasts the trilinear form $I_p'$ 
as the value of a Garrett-Hida triple product $p$-adic $L$-function
 at a point lying outside its region of
 classical  interpolation. This connection with $p$-adic $L$-functions
justifies  viewing Conjecture $\shi$ 
 as an elliptic curve analogue  of the ($p$-adic) Stark conjectures. 
It also  suggests that  Conjecture $\shi$ might be regarded as a $p$-adic limit of Jacquet's conjecture on trilinear periods, prompting a possible strategy for proving it in some cases 
and suggesting that it might be viewed as a deep manifestation of the ``Artin formalism" for certain 
$p$-adic $L$-series.

The rest of the paper is devoted to giving 
both  theoretical and experimental evidence  in support of Conjecture  $\shi$.

Chapters \ref{sec:heegner} and \ref{sec:stark-heegner} treat the case where $\varrho_g$ and $\varrho_h$ are induced from characters
$\psi_g$ and $\psi_h$ of the same quadratic field $K$, and with inverse central characters.
The representation 
$$V_{gh} = V_{\psi_1} \oplus V_{\psi_2}$$ 
then decomposes as the direct sum of a pair of representations induced from 
ring class characters $\psi_1$ and $\psi_2$ of $K$.

Chapter \ref{sec:heegner} focusses on the setting where $K$ is imaginary quadratic. Building on the results of Chapter \ref{tri}
 and the precise formulae of Watson \cite{Wat} and Woodbury \cite{Wood}, 
  Conjecture  $\shi$ is proved
 in Section \ref{sec:proof}   under the
 assumption that the prime $p$ splits in $K$ and a Heegner hypothesis is satisfied. The main ingredients in the proof are 
 the $p$-adic Gross-Zagier formula of \cite{BDP1} 
   expressing 
  Heegner points  in terms of values of 
  a Katz-style $p$-adic Rankin $L$-series, the relation
  established   in \cite{DR1} 
between $p$-adic iterated integrals
  and Garrett-Hida $p$-adic  $L$-series, and a subtle relation 
  between these two types of $L$-series. 
 It is in the non-trivial comparison between periods used
  in their definitions  
  that a value of Katz's two variable $p$-adic $L$-function 
  makes an appearance. %% (see \eqref{eqn-sigmak} and \eqref{sym-formula-Lp}). 
  The unit
   $u_{g_\alpha}$ can be expressed in terms
  of  elliptic units in this case, and emerges from Katz's 
  $p$-adic analogue of the Kronecker limit formula.

Katz's approach to  $p$-adic $L$-functions (as well as the closely related 
approach of \cite{BDP1})
 is based on the evaluation of 
$p$-adic modular forms at CM points when they lie  on the
 ordinary locus of the modular curve,
i.e., when $p$ is {\em split}  in $K$.  This approach breaks down entirely
 when $p$ is inert in the quadratic imaginary field.
Section  \ref{p-inert} gives  numerical evidence in support of Conjecture $\shi$ 
in the inert case, both in the rank $(1,1)$ setting---where the  theory of elliptic units and
Heegner points can still be used 
 to produce  the global objects that  should 
 enter into the right-hand side of \eqref{eqn:main-conj-intro}---and in the more mysterious
 rank $2$ setting.



%\medskip
Chapter \ref{sec:stark-heegner} attempts a parallel treatment 
 when $\varrho_g$ and $\varrho_h$ are induced from characters (of mixed signature)
 of the same {\em real} quadratic field $K$. In some cases, 
the global points $P_1$ and  $P_2$ 
  can then be   expressed, at least conjecturally, in 
  terms of the Stark-Heegner points
 of \cite{darmon-hpxh} and  \cite{Gre} attached to 
 ``real quadratic cycles on $\cH_p\times\cH$".
 The resulting  relation between the Stark points of the present work
 and these Stark-Heegner points  might
  be amenable to proof via the methods of
 Section \ref{sec:proof}.
 The prospect of tying
  the real quadratic cycles on $\cH_p\times\cH$
  to global (but not necessarily motivic)
  cohomology classes in the Selmer groups
of $E$ over ring class fields of real quadratic fields represents an
encouraging first step towards
better understanding their global nature.

Chapters \ref{sec:exotic-points}, \ref{sec:beilinson-flach-points}, and \ref{sec:abelian-extensions} 
 consider the remaining interesting cases, all of which appear to lie beyond the
reach of previously available  theory or conjectural construction. 

Chapter \ref{sec:exotic-points} reports on a number of experiments
in the setting where $h$ is an  exotic weight one form,
whose associated Galois representation  has  projective image 
$A_4$, $S_4$ or $A_5$, and $g = h^*$ is its dual. 
The representation $V_{gh}$ then breaks up as
$$V_{gh} = L \oplus \mathrm{Ad}_g.$$
In the rank $(1,1)$ setting,
 Conjecture $\shi$ leads  to a $p$-adic analytic construction of a global point
on $E$ over the field  cut out by $\mathrm{Ad}_g$.
The setting where the
Hasse-Weil $L$-series $L(E,s)$ has a double zero at the center 
and $L(E,\mathrm{Ad}_g,1)\ne 0$ is particularly interesting:   
Conjecture $\shi$ then predicts the vanishing of the $p$-adic iterated integrals, even though
the generalised Kato class $\kappa(f,g_\alpha,\overline{g}_{\bar \alpha})$ is  expected to be 
a canonical, non-trivial element of  the 
pro-$p$ Selmer group of $E/\Q$: we refer to \cite{DR3} for   a fuller discussion
of  this setting. 



Chapter \ref{sec:beilinson-flach-points}
takes up the setting where $g\in S_1(N,\bar\chi)$ is a cusp form 
and $h:= E_1(1,\chi)\in M_1(N,\chi)$ is the
weight one Eisenstein series attached to the pair $(1,\chi)$ of Dirichlet characters.
A novelty of this setting is that 
$$ V_{gh} = V_g \oplus V_{g^*} $$
decomposes into a direct sum of two representations which 
are not self-dual in general,
and  the regulator matrix $R_{g_\alpha}(E,\varrho_{gh})$
  involves points in the isotypic parts for these non self-dual 
representations.
 

Chapter \ref{sec:abelian-extensions} concludes with  the setting
in which $g := E_1(1,\chi^{-1})$
 is an (irregular)  Eisenstein series. 
The case where $h :=  E_1(\chi_1,\chi_2) $
 is also an Eisenstein series (with  $\chi_1\chi_2 = \chi$)
  is particularly appealing, since
 $$ V_{gh} = \chi_1 \oplus \bar\chi_1 \oplus \chi_2 \oplus\bar\chi_2$$
is then a direct sum of one-dimensional representations. 
In this setting---the most ``degenerate instance" of the one
 described in Ch.~\ref{sec:beilinson-flach-points}---the relevant
  ``generalised Kato classes"
coincide with the   classes constructed by Kato from $p$-adic families of 
Beilinson elements, whose behaviour has a bearing on the arithmetic of elliptic curves over
cyclotomic fields.
Chapter \ref{sec:abelian-extensions} reports on  experimental evidence  in support of
Conjecture $\shi$, where  the regulators
attached to $(f,g_\alpha,h_\alpha)$    involve 
 logarithms of  points on $E$ defined 
over the abelian extension of $\Q$ cut out by  either $\chi_1$ or $\chi_2$. 
The scenario in which   these Dirichlet characters are  non-quadratic
is particularly intriguing.
According to predictions of 
David, Fearnley and Kisilevsky \cite{DFK},
the  occurence of global points over cyclic extensions of degree greater than $2$  is 
quite sporadic;
 the $L$-series that control them have non-real coefficients, and their functional equation gives no
 indication as to the parity of their order of vanishing at the center. 
 Although ostensibly rare, global points over cyclic cubic extensions do occur in practice. Fairly
  extensive tables of pairs $(E,\chi)$ with $\chi$ cubic
   for which 
  $r_{\an}(E,\chi)>0$ have been 
 drawn up by the authors of \cite{DFK}, and the computationally
 more challenging task of verifying that $r(E,\chi)>0$
for such pairs is carried out---albeit in more
 limited ranges---in
 \cite{cowan}. 
 Section
\ref{sec:abelian-extensions-evidence} describes a few experimental verifications of
Conjecture $\shi$ involving the $p$-adic logarithms
of such cyclic cubic points.

The article concludes with  
 a table summarizing the main features of the numerical experiments 
described  in this work.

\medskip\noindent
{\small{\bf Acknowledgements.} The authors thank Kevin Buzzard for illuminating exchanges on weight one modular forms,
Mladen Dimitrov for helpful discussions around Section \ref{sec:smooth}, Neil Dummigan for his guidance on some classical formulae
relating adjoint $L$-functions and the Petersson norm, Haruzo Hida for a suggestion which helped shape the statement of the conjectures,
Mike Woodbury for sharing his expertise on trilinear periods and Daniele Casazza for his comments on Section \ref{sec:heegner}.
The experimental work in this paper was undertaken using the {\sc Magma} computer algebra package and 
the authors are very grateful and deeply indebted to Steve Donnelly for his help and advice on a number of computational problems.


The  authors  gratefully acknowledge the following institutions 
for their support and  the pleasant
 locales in which  their
collaboration on this project was pursued:  
the Mathematics Center Heidelberg (MATCH), 
McGill's Bellairs Research Station in Barbados, 
the CRM and Universit\`a Politecnica de Catalunya in Barcelona, 
the Mathematical Institute at the University
of Oxford,
 the Pohang Mathematics Institute in Pohang, South Korea,
and the Centro de Ciencias de Benasque Pedro Pascual in the Spanish Pyr\'en\'ees.

The first author was supported by an NSERC Discovery grant. The
work of the middle author was supported
 in part by a grant from the European Research Council (204083), and largely undertaken
during a sabbatical year at the University of Kyushu---he thanks Shin Hattori, Masanobu Kaneko, and Yuichiro Taguchi for their hospitality. The third author was supported by Grant MTM2012-34611.
}

\medskip
\medskip




  

\section{Overconvergent weight one forms and Gross-Stark units}
\label{sec:inv-g}

\def\hS{\underline{S}}
\def\TT{\mathbb{T}}
\def\hT{\underline{\TT}}



\subsection{Classical and overconvergent modular forms of weight one}
 \label{sec:smooth}

One of the main results of this section is the following concrete criterion,
following directly from a result of
 Cho-Vatsal and  Bellaiche-Dimitrov,
 which shows that
 Hypothesis C is frequently
 satisfied in practice.
\begin{proposition}
\label{BeDi-criterion}
Let $g\in S_1(N,\chi^{-1})$ be a  cusp form of weight one which is regular at $p$,
 and let $g_\alpha$ denote  one of its 
$p$-stabilisations. Then the natural inclusion
$$S_1(Np,\chi)_{\C_p}[g^*_\alpha] \hookrightarrow S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]]$$
is an isomorphism of $\C_p$-vector spaces  if and only if 
\begin{equation}
\label{psplit-red}
\varrho_g \mbox{  is not induced from a character of a 
 real quadratic field in which $p$ splits}.
 \end{equation}
\end{proposition}
\begin{proof}
Let 
$\TT_k(N)$ denote the $\Z_p$-algebra generated by the  Hecke operators 
$T_\ell$ (with $\ell\nmid N$) and $U_q$ (with $q|N$), together 
with the diamond operators, acting on the space
$M_k(N)$ of classical  modular forms on 
$\Gamma_1(N)$. When $k\ge 2$, there is a natural perfect duality
\begin{equation}
\label{eqn:classical-duality}
 \TT_k(N) \times M_k(N) \lra \Z_p
 \end{equation} 
defined by $\langle T, f \rangle := a_1(Tf)$. 

Hida's 
$\Lambda$-adic Hecke algebra is defined to be
$$ \hT(N) = \lim_{\leftarrow,t} (e_{\ord}\TT_2(Np^t)) \otimes_{\Z_p} \Q_p,$$
where  the inverse limit is taken with respect to the natural projection maps. 
The algebra $\hT(N)$ has a natural structure of a $\Lambda \otimes_{\Z_p} \Q_p$-module
 arising from the action of the
diamond operators, and it can also be viewed as a module over the $\Lambda\otimes\Q_p$-subalgebra
 $\hT^0(N)$ of $\hT(N)$ generated by the ``good" Hecke operators
$T_n$ with $(n,pN)=1$ and by $U_p$.
Let $\varphi_k:\Lambda \otimes \Q_p \lra \Q_p$ denote the ``weight $k$ specialisation map" sending 
a group-like element $\langle a \rangle \in 1+p\Z_p\subset \Lambda^\times$ to $a^{k-1}$,
and let $\wp_k$ denote its kernel. The pairing of 
\eqref{eqn:classical-duality} induces a perfect duality
$$ ((\hT(N)/\wp_k)\otimes_{\Q_p} \C_p)   \times M_k^{\oc,\ord}(N)  \lra \C_p.$$ 
In particular, the space $M_1^{\oc,\ord}(N)$ is naturally identified
with the vector space of $\C_p$-linear homomorphisms $\Hom(\hT(N)/\wp_1,\C_p)$. 
Under this identification, the classical weight one form  
$g_\alpha \in M_1^{\oc,\ord}(N)$,  which is a normalised eigenvector
 for the good Hecke operators $T_n$ with $(n,Np)=1$ and for $U_p$, 
corresponds to an algebra homomorphism
$$ \varphi_{g_\alpha} \in \Hom_{\alg}(\hT^0(N)/\wp_1, \C_{p}). $$
Let $\wp_{g_\alpha}\subset \hT^0(N)$ be the kernel of 
$\varphi_{g_\alpha}$. Its image 
  under the  ``projection map to weight space"
$$\Hom_{\alg}(\hT^0(N),\C_p)\lra \Hom_{\alg}(\Lambda,\C_p)$$
induced by restriction is equal to
 $\varphi_1$, i.e.,  $\wp_{g_\alpha}$ contains
the ideal
$\wp_1 \hT^0(N)$. 


The eigenspace $M_1^{\oc,\ord}(N)[g^*_\alpha]$ 
that arises in the statement of   Proposition \ref{BeDi-criterion}
can be identified with 
the dual of $\hT(N)/\wp_{g_\alpha}$:
$$ M_1^{\oc,\ord}(N)[g^*_\alpha]  = \Hom(\hT(N)/\wp_{g_\alpha}, \C_p).$$
Furthermore,
$$M_1^{\oc,\ord}(N)[[g^*_\alpha]]  = \Hom(\hT(N)_{\wp_{g_\alpha}}/\wp_1, \C_p),$$
where  
$$\hT^0(N)_{\wp_{g_\alpha}} :=  \lim_{\leftarrow,n} \hT^0(N)/\wp_{g_\alpha}^n, \qquad \hT(N)_{\wp_{g_\alpha}} := \lim_{\leftarrow,n} 
\hT(N)/\wp_{g_\alpha}^n$$
denote 
 the completions at the maximal ideal
  $\wp_{g_\alpha}$ of $\hT^0(N) \otimes \C_p$.

Hence the dual of  the natural inclusion 
$M_1^{\oc,\ord}(N)[g^*_\alpha]\hookrightarrow M_1^{\oc,\ord}(N)[[g^*_\alpha]]$
is  the natural surjection 
\begin{equation}
\label{eqn:bellaiche-dimitrov}
 \hT(N)_{\wp_{g_\alpha}}/\wp_1  \lra  \hT(N)/\wp_{g_\alpha}.
 \end{equation}
The ideal $\wp_1 \hT^0(N)_{\wp_{g_\alpha}}$ is   equal to a positive  power 
$\wp_{g_\alpha}^t$ (with $t\ge 1)$ of the maximal ideal $\wp_{g_\alpha}$ in the local ring $\hT^0(N)_{\wp_{g_\alpha}} \otimes\C_p$.
The  map 
\eqref{eqn:bellaiche-dimitrov}
is an isomorphism if and only if $t=1$, i.e., if and only if the point 
of $\Hom(\hT^0(N)_{\wp_g},\C_p)$ attached to $\varphi_{g_\alpha}$ is smooth and
\'etale over weight space. 
Under the regularity assumption on $g$ at $p$, Theorem 1.1. of 
\cite{BeDi} implies that this \'etaleness condition is satisfied precisely when  
\eqref{psplit-red}
holds. 
 The proposition follows.
\end{proof}
 \begin{remark}
 \label{rem:hypC-Cprime}
 When $g$ is the theta series of a real quadratic field in which $p$ splits, explicit Fourier expansions of non-classical forms in $S_1^{\oc,\ord}(N,\chi)[[g^*_\alpha]]$ are described
  in \cite{DLR3}.     Such non-classical forms are also expected to be present in all settings where
 $g$ is irregular at $p$: see \cite{DLR-irregular}  for a study of the non-classical forms in the  associated  generalised eigenspace in this case. 
 Hypothesis C and condition (1) of Hypothesis C' are thus expected to be equivalent when 
 $g$ is a cusp form. 
The extension of Conjecture   $\shi$ to     settings where $S_1^{\oc,\ord}(N,\chi)_{\C_p}[[g_\alpha^*]]$ contains non-classical forms,   which is far from routine and involves significant new features, is taken up in  \cite{DLR-irregular}. 
\end{remark}



We next turn  to the case where
 $g = E_1(\chi_1,\chi_2)$ is a weight one Eisenstein series with
  $\alpha_g:= \chi_1(p)$ and  $ \beta_g:=\chi_2(p)$,
and let 
 $$
g_\alpha(q)    := g(q) - \chi_2(p) g(q^p), \qquad
 g_{\beta}(q) := g(q) - \chi_1(p) g(q^p)
$$
denote its (not necessarily distinct) 
$p$-stabilisations.

\begin{proposition} 
\label{prop:hypothesis-D-eisenstein}
The cuspidal  generalised eigenspace $S_1^{\oc,\ord}(N,\chi)[[g_\alpha^*]]$ is non-trivial,
i.e., 
the eigenform $g_\alpha$  is a  $p$-adic cusp form, 
 if and only if $\chi_1(p)=\chi_2(p)$.
%, i.e., $S_1^{\oc,\ord}(N,\chi)_{\C_p}[[g_\alpha^*]]$ is non-zero.
\end{proposition}
 \begin{proof}
 Assume for simplicity that $\chi_1$ and $\chi_2$  are primitive with relatively prime
conductors $N_1$ and $N_2$, so that $\chi := \chi_1\chi_2$ is primitive
and $N = N_1 N_2$. 
%We  show that $S_1^{\oc}(N,\chi)[g_\alpha^*]\ne 0$ by proving that $g_\alpha\in
%M_1(Np,\chi)$ is  a $p$-adic cusp form, i.e, has zero 
We prove the proposition by analyzing the constant terms of $g_\alpha$ and $g_\beta$ at the cusps   in the ordinary locus
of the modular curve $X(N;p)= X_1(N) \times_{X(1)} X_0(p)$  associated to
 the congruence group $\Gamma_1(N)\cap \Gamma_0(p)$.
More precisely,
let  $X_1(N)_{\C_p}^{\ord}$ denote the complement of the supersingular 
 residue discs in  $X_1(N)(\C_p)$, regarded as an affinoid domain. 
 For a suitable wide open neighbourhood $\mathcal W$ of  $X_1(N)_{\C_p}^{\ord}$, let 
$$ s: \mathcal W \lra X(N;p)_{\C_p}$$ 
be the  rigid analytic section of the natural projection $X(N;p) \lra X_1(N)$,
defined by
sending the point attached to an ordinary  pair $(E, P_N)_{/\C_p} \in \mathcal W$ 
to the (moduli of the) triple  $(E,P_N,C_{\can})$, where $P_N$ is a point of order $N$ 
on $E$ and $C_{\can}$ denotes the canonical subgroup of
order $p$. 
The inclusion  $i_{g_\alpha}$ of   
 \eqref{eqn:inclusion-g-eis} in the Introduction
 is described geometrically  in terms of $s$, 
as follows: 
letting $\underline\omega$ denote the sheaf  over $X(N;p)$ with logarithmic poles
 at the cusps discussed in \cite[\S 1.1]{BDP1}, a classical weight one form $\phi$ 
 in the source of \eqref{eqn:inclusion-g-eis} may be viewed as a section of $\underline\omega$ over $X(N;p)$, 
 and the map of \eqref{eqn:inclusion-g-eis} sends it to its pull-back $s^*(\phi) \in H^0(\mathcal W,\underline{\omega})$, 
 naturally identified with an overconvergent modular form of level $N$. 

The non-zero constant terms  of the classical form  $g=E_1(\chi_1,\chi_2)$ are concentrated at the cusps of $X_1(N)$ 
attached to the  ``Katz test objects"
$$ c_{ab} := (\G_m/q^{\Z}, \zeta_{N_1}^a q^{b/N_2}), \qquad
d_{ab} := (\G_m/q^{\Z},  q^{a/N_1}\zeta_{N_2}^b ),$$
where $\G_m/q^\Z$ is the Tate  generalised  elliptic curve over ${\rm Spec} (\C_p[[q]])$, and 
  $\zeta_{N_1}$ and $\zeta_{N_2}$ are fixed primitive $N_1$ and $N_2$-th roots of unity; the indices
   $(a,b)$ range over $(\Z/N_1\Z)^\times \times (\Z/N_2\Z)^\times$. 
If $g$ is any weight one modular form on $\Gamma_0(N)$ with character $\chi$, then the modular form $g_p$ 
 on $X(N;p)$ whose $q$-expansion is
equal to $g(q^p)$ at the cusp $\infty = (\G_m/q^\Z,\zeta_N, \mu_p)$ 
is defined on test objects of the form
$(E,P_N,C_p)$,  with    $C_p\subset E$ of order $p$,  by the rule
\begin{equation}\nonumber
%% \label{eqn:rule-for-gp}
 g_p((E,P_N,C_p)) := \chi^{-1}(p) p^{-1} \varphi_{C_p}^\ast (g(E/C_p, P_N + C_p)), 
 \end{equation}
 where $\varphi_{C_p}:E \lra E/C_p$ is the natural quotient morphism.
This can be seen by checking this identity on  the test object $(\G_m/q^\Z, \zeta_N, \mu_p)$
and invoking the $q$-expansion principle. 
Let $g_{ab}(q)$ and $g'_{ab}(q) \in \C_p[[q^{1/N}]]$ denote the $q$-expansions of $g$ at the cusps
$c_{ab}$ and $d_{ab}$ respectively, and let 
$x_{ab}$ and $y_{ab}$ denote the associated leading terms. 
The $q$-expansion of $g_p$, viewed as a $p$-adic modular form, at the  cusps $c_{ab}$ and $d_{ab}$ is
then  given by the rule
\begin{eqnarray}
g_p(\G_m/ q^{\Z}, \zeta_{N_1}^a q^{b/N_2})  & = &  
\chi^{-1}(p) p^{-1} \varphi^*(g(\G_m/q^{p\Z}, \zeta_{N_1}^{pa} q^{pb/N_2})) \nonumber \\ 
&  =  &   \chi_2^{-1}(p)  g_{ab}(q^p) \frac{dt}{t}  \label{eqn:q-exp-at-cab}  \\
g_p(\G_m/q^{\Z}, q^{a/N_1} \zeta_{N_2}^b)  & = &
\chi^{-1}(p) p^{-1}  \varphi^*(g(\G_m/q^{p\Z},  q^{pa/N_1} \zeta_{N_2}^{pb})) \nonumber \\ 
& =   & \chi_1^{-1}(p)  g_{ab}'(q^p) \frac{dt}{t}. \label{eqn:q-exp-at-dab}
\end{eqnarray}
Here we have used the fact the canonical subgroup 
 $\mu_p\subset \G_m/q^\Z$
 of the Tate elliptic curve is 
  the kernel of the map $\varphi: \G_m/q^\Z \lra \G_m/q^{p\Z}$ sending $t$ to $t^p$. 
Equations \eqref{eqn:q-exp-at-cab}
and \eqref{eqn:q-exp-at-dab} 
yield the  constant terms of the forms $g$, $g_p$, $g_\alpha$, 
and $g_\beta$
at the cusps $c_{ab}$ and $d_{ab}$, as   summarised in the table below:
$$
\begin{array}{c|c|c} 
  & c_{ab} & d_{ab}  \\ \hline
g & x_{ab} & y_{ab} \\
g_p & \chi_2^{-1}(p) x_{ab} & \chi_1^{-1}(p) y_{ab} \\
g_\alpha & 0 & (1-\chi_2/\chi_1(p)) y_{ab} \\
g_\beta & (1-\chi_1/\chi_2(p)) x_{ab}  & 0 
\end{array}
$$
The proposition follows.
% from the fact that $\chi_1(p)= \chi_2(p)$.
\end{proof}
\begin{remark}
If $g= E_1(\chi_1,\chi_2)$ is  regular, 
then the fact that
$S_1^{\oc,\ord}(N,\chi)[[g_\alpha^*]]=0$
shows that
the $p$-adic iterated integrals attached to $(f,g_\alpha,h_\alpha)$ vanish
identically, since the overconvergent weight one
forms $e_{\ord}(d^{-1} \testf \times \testh)$ belong to 
$S_1^{\oc,\ord}(N,\chi)$.  It is not hard to check that the regulator term
$\det(R_{g_\alpha}(E,\varrho_{gh}))$ attached to the subspace $V_{g}^{g_\alpha} = \C_p(\chi_1)$ vanishes as well, 
 in  harmony with    Conjecture 
$\shi$, which can be seen as amounting to the statement
$0=0$
 when $g$ is a regular Eisenstein series.
  \end{remark}

 
 
  
\subsection{The Gross-Stark unit attached to a weight one form}
\label{sec:stark}

The aim of this section is to associate to a $p$-stabilised eigenform
$g_\alpha$  satisfying Hypothesis C' 
a global
 element $u_{g_\alpha}\in H^\times\otimes L$ (often well-defined up to scaling
 by $L^\times$), 
 whose $p$-adic logarithm arises in Conjecture $\shi$.
This element belongs to the $\mathrm{Ad}_g$-isotypic subspace of
$H^\times\otimes L$, where 
$$ \mathrm{Ad}_g :={\rm End}_0(V_g) $$ 
is  the {\em adjoint representation }
attached to $V_g$, the $3$-dimensional
 Artin representation consisting of trace zero $L$-linear endomorphisms of 
$V_g$ on which $G_\Q$ acts  by conjugation  via $\varrho_g$.
 
 In defining $u_{g_\alpha}$,
  it shall  be convenient to 
  replace the field $H$ of the introduction  by the smaller field $H_g$ cut out by $\mathrm{Ad}_g$, 
  and $L$ by any field over which the representation $\mathrm{Ad}_g$ can be defined.
To give a rough feeling for the nature of   $u_{g_\alpha}$, let us mention
 at the outset
that  
\begin{equation}
\label{eqn:nature-of-ug}
L \cdot u_{g_\alpha} \subset \left\{
\begin{array}{ll} 
 (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} & \mbox{ if $g$ is cuspidal,} \\
(\cO_{H_g}[1/p]^\times)_L^{\mathrm{Ad}_g} & \mbox{ if   $g$ is Eisenstein},
 \end{array}\right.
 \end{equation}
and 
\begin{equation} \nonumber
%% \label{eqn:nature-of-ug2}
\mbox{the geometric frobenius element } \sigma_{p} \mbox{ acts on } u_{g_\alpha} \mbox{ with eigenvalue } \alpha_g/\beta_g. 
\end{equation}
As explained below, 
these requirements often characterize $u_{g_\alpha}$ uniquely up to scaling by $L^\times$. 

Write $\sigma_\infty \in G:=   \Gal(H_g/\Q)$ for the involution given by complex conjugation 
induced by the fixed choice of complex embedding of $H_g\subset \bar\Q$.
The representation $\mathrm{Ad}_g$ decomposes as a direct sum of  $+$ and $-$ eigenspaces for the action of 
$\sigma_\infty$, denoted $\mathrm{Ad}_g^+$ and $\mathrm{Ad}_g^-$. These eigenspaces are  of dimensions $1$ and $2$ 
respectively:
\begin{equation}
\label{eqn:dimWplusminus}
 \dim_L \mathrm{Ad}_g^+ = 1, \qquad \dim_L \mathrm{Ad}_g^-=2.
 \end{equation} 
As suggested by \eqref{eqn:nature-of-ug}, the nature of the invariant $u_{g_\alpha}$ depends on whether $\varrho_g$ is irreducible or not, and each case will be dealt with in turn.


\subsubsection{The case where $\varrho_g$ is irreducible}
If  $\varrho_g$ is irreducible, then
 $g$ is a {\em cuspidal} eigenform of weight one. The definition of $u_{g_\alpha}$ rests   on the following proposition:
\begin{proposition}
\label{Stark-rank1}
\begin{enumerate}
\item The vector space $\Hom_{G_\Q}(\mathrm{Ad}_g, \cO_{H_g}^\times \otimes L)$ is $1$-dimensional.
\item The Artin $L$-function 
$L(\mathrm{Ad}_g,s)$ has a simple zero at $s=0$.
\end{enumerate}
\end{proposition}
\begin{proof}  
 The finite group $G=\Gal(H_g/\Q)$
  acts faithfully on $\mathrm{Ad}_g$ and is identified with the image of  the projective representation attached to $V_g$.
Note that $G_\infty = \langle \sigma_\infty\rangle$ is of order two since
$\mathrm{Ad}_g$ is not totally even.
Dirichlet's unit theorem determines the structure of $\cO_{H_g}^\times\otimes L$ as
an  $L[G]$-module:
$$ \cO_{H_g}^\times \otimes L = (\Ind_{G_\infty}^G L) -  L,$$
where the two occurrences of $L$ on the right refer to trivial representations (of $G_\infty$ and $G$ respectively).
 The irreducibility of $V_g$ implies that the representation $\mathrm{Ad}_g$ contains no $G$-invariant vectors, by Schur's lemma, and hence
 $$ \Hom_G(\mathrm{Ad}_g,\cO_{H_g}^\times\otimes L) = \Hom_G(\mathrm{Ad}_g, \Ind_{G_\infty}^G L).$$
 By Frobenius reciprocity,
 $$ \Hom_G(\mathrm{Ad}_g, \Ind_{G_\infty}^G L) = \Hom_{G_\infty}(\mathrm{Ad}_g,L) = \Hom(\mathrm{Ad}_g^+,L).$$
 The first part of the proposition now follows from \eqref{eqn:dimWplusminus}.
   The second statement  follows from 
   the shape of the $\Gamma$-factors in the functional equation of $L(\mathrm{Ad}_g,s)$ determined by
   \eqref{eqn:dimWplusminus}.
\end{proof}
  Choose a basis $\varphi$ for the one-dimensional vector space 
 $\Hom_G(\mathrm{Ad}_g,\cO_{H_g}^\times \otimes L)$, and let
 $$  (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} := \varphi(\mathrm{Ad}_g) \subset \cO_{H_g}^\times \otimes L.$$
 The $L[G]$-module $ (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g}$ is of dimension $\le 3$ over $L$ and is abstractly isomorphic to
 the unique irreducible subrepresentation of $\mathrm{Ad}_g$ containing $\mathrm{Ad}_g^+$.  


Recall the prime number $p$ chosen in the introduction. The frobenius element $\sigma_p$
 acts on $\mathrm{Ad}_g$ with the (not necessarily different) eigenvalues $1$,
  $\alpha_g/\beta_g$ and $\beta_g/\alpha_g$. Define
 $$ U_{g_\alpha} := \left\{ u\in  (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} \mbox{ such that } \sigma_p(u) = \frac{\alpha_g}{\beta_g} u
\right\}.$$


  
\begin{lemma}
\label{le} 
Assume that $V_g$ is regular.
 If $\varrho_g$ is induced from a character of a real quadratic field in which $p$ splits, 
then $U_{g_\alpha}=0$. 
Otherwise,
$ \dim_L (U_{g_\alpha}) \ge  1$, with equality if either
\begin{enumerate}
\item  $\alpha_g\ne -\beta_g$, i.e.,  $\mathrm{Ad}_g$ is regular,  or,
\item 
 $\varrho_g$ is induced from a character of a quadratic field in which 
$p$ is inert.
\end{enumerate}
\end{lemma}
 \begin{proof}
 This lemma is proved by a case by case analysis, according to the following  
 possibilities for $V_g$:
 
 a) $V_g$ has imaginary dihedral projective image;
 
 b) $V_g$ has real dihedral projective image;
 
 c) $g$ is  an exotic weight one form with projective image 
 $A_4$, $S_4$ or $A_5$.
 
 
 \medskip
 \noindent
 {\em a) Imaginary dihedral forms}. Assume that the representation $V_g$ is induced from a character
 $\psi_g$ of an imaginary quadratic field $K$. Then
 \begin{equation}
 \label{eqn:adjoint-of-induced}
  \mathrm{Ad}_g = L(\chi_K) \oplus  V_\psi, 
  \end{equation}
 where $\chi_K$ is the quadratic Dirichlet character associated to $K$
 and 
$ V_\psi:= \Ind_K^\Q \psi$ is the two-dimensional 
representation induced from the ring class  character 
 $ \psi := \psi_g/\psi_g'$ of $K$.
Since $\sigma_\infty$ acts with eigenvalue $-1$ on 
 the first  factor $L(\chi_K)$, it follows that $ (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g}$ 
 is a subrepresentation of $V_{\psi}$.

If   $V_\psi$ is irreducible, then
$$ (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} \simeq V_{\psi}. $$ 
The eigenvalues of $\sigma_p$ on $V_{\psi}$ are 
 $\alpha_g/\beta_g$ and $\beta_g/\alpha_g$  if
 $\chi_K(p)=1$, and $1$ and $-1 = \alpha_g/\beta_g$ 
 if  $\chi_K(p)=-1$. In either case,
  $\dim_L (U_{g_\alpha}) \ge  1$, with equality if either 
  $\alpha_g/\beta_g\ne - 1$ or if $p$ is inert in $K$. 
  
 If $V_\psi$ is reducible, the 
character 
 $\psi$ is necessarily quadratic, and $V_\psi$  further breaks up
 as a sum of one-dimensional representations attached to  odd and
   even Dirichlet quadratic characters $\chi^-$ and $\chi^+$, respectively, 
   satisfying $\chi^+\chi^- = \chi_K$. 
Because of this,
$$ (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} = (\cO_{H_g}^\times)_L^{\chi^+} \simeq L(\chi^+)$$ 
is isomorphic to
the one-dimensional representation associated to $\chi^+$. 
The space $U_{g_\alpha}\subset  (\cO_{H_g}^\times)_L^{\chi^+}$ is therefore
of dimension 
at most one, with equality if and only if 
$\alpha_g/\beta_g = \chi^+(p)$. 
This equality always holds when $p=\wp \bar\wp$ splits in $K$,
since in that case
$$ \alpha_g/\beta_g = \psi(\wp) = \psi(\bar\wp) = \chi^+(p)=\chi^-(p).$$
 (Note that the
regularity assumption $\alpha_g\ne \beta_g$  that is made 
on $g$ forces all these expressions to be equal to $-1$ rather than $1$.)
If $p$ is inert in $K$, then $\sigma_p$ acts on  $V_{\psi}$ 
with eigenvalues $1$ and $-1$, 
and hence $\alpha_g/\beta_g = \beta_g/\alpha_g =-1$.
In that case, either $\chi^+(p)=-1$ and $\dim(U_{g_\alpha})=1$,
or 
$p$ splits in the real quadratic field $K^+$ cut out by $\chi^+$, and $U_{g_\alpha}=0$.
In this last scenario, the representations
$V_{\psi}$ and $V_g$ are   also induced from  a character of 
the real quadratic field $K^+$, in which $p$ splits. 

 

 
   
 \medskip \noindent
 {\em b) Real dihedral forms}. If the  representation $V_g$ is induced from a
  character $\psi_g$ of signature $(+,-)$ 
 of a real quadratic field $K$, then the 
  character $\psi = \psi_g/\psi_g'$ is a totally odd ring class
   character of $K$, and cuts out an abelian extension of $K$ of dihedral type.
    The decomposition
 \eqref{eqn:adjoint-of-induced} continues to hold, 
 but now, since $\sigma_\infty$ acts with eigenvalue $1$ on $L(\chi_K)$, it follows
 that 
\begin{equation}
  \label{eqn:gs-unit-real-K}
 (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} \simeq L(\chi_K)
 \end{equation}
 in all cases.  If $p$ splits in $K$, then 
 $\chi_K(p)= 1 \ne \alpha_g/\beta_g$, and hence
 $U_{g_\alpha}=0$. 
    If $p$ is inert in $K$, then   $\chi_K(p) = \alpha_g/\beta_g=-1$,  and hence
$U_{g_\alpha}$ is also one-dimensional,
and generated over $L$ by  a fundamental unit of the real quadratic field $K$.
 
 
 
 \medskip\noindent
 {\em c) Exotic forms}. Assume to conclude that the  projective image $G$ of the representation $V_g$ is 
 isomorphic to $A_4$, $S_4$ or $A_5$.
 Since these groups admit no faithful irreducible representations of dimension $<3$,
 it follows that $\mathrm{Ad}_g$ is irreducible, and hence
 $$  (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g} \simeq \mathrm{Ad}_g.$$
 The lemma follows directly in this case, since $\sigma_p$ acts on $\mathrm{Ad}_g$
  with eigenvalues
 $1$, $\alpha_g/\beta_g$, and $\beta_g/\alpha_g$. 
 \end{proof}
 
\begin{remark}  
Note  that condition \eqref{psplit-red} which
  was shown in Ch.~\ref{sec:smooth}
   to imply Hypothesis C for $g_\alpha$ 
  is precisely what guarantees the non-triviality of the group 
  $U_{g_\alpha}$. 
  \end{remark}
 \begin{remark}
  The scenarios considered in $(a)$ and $(b)$ in the 
  above proof are not mutually exclusive, and can arise simultaneously in certain cases where $\psi_g$ is quadratic and
  $\varrho_g$ has  image isomorphic to the dihedral group of order $8$: some of the numerical examples below 
  focus on such settings.
 \end{remark}
 
  \begin{definition}
  A non-zero element $u_{g_\alpha}$ of $U_{g_\alpha}$ is called a
 {\em  Stark unit attached to $g_\alpha$}.
 \end{definition}
 \begin{remark}
Lemma \ref{le} furnishes us with many concrete instances where
 the unit $u_{g_\alpha}$  is  well-defined up to multiplication by $L^\times$.
This happens, for instance, when 
 $\varrho_g$ is induced from a character of a real
  quadratic field $K$ in which  $p$
 is inert, when
  the vector space $U_{g_\alpha}$ is always
  generated by
 the fundamental unit of $K$.
 The case where 
 $p$   splits in $K$ is more mysterious, since 
 $U_{g_\alpha} = 0$, and  a natural definition of $u_{g_\alpha}$ does not seem 
  to be available, consistent with the failure of  Hypothesis C.
  
More generally,  $ \dim_L (U_{g_\alpha}) > 1$ occurs precisely when $\alpha_g/\beta_g=\pm 1$.
 In the special case where $V_g$ is induced from a character of an imaginary quadratic field 
 in which $p$ splits,
there is  nonetheless 
a natural candidate for $u_{g_\alpha}$ which is well-defined up to scaling by
 $L^\times$: cf.~\S \ref{sec:proof} for more details.
 \end{remark}
  \begin{remark}
 Several results in the literature   relate the Petersson norm of
 $g$   to $L'(\mathrm{Ad}_g,0)$, which in turn is expected to admit
 an elementary expression as  the  logarithm  of 
 units in the subspace of $ (\cO_{H_g}^\times)_L^{\mathrm{Ad}_g}$ on which
 $\sigma_\infty=1$. See for example \S 6 of \cite{stark1}, 
 or Prop.~4.1 of \cite{Duke-Li}.
It is in this sense that the $p$-adic logarithm of $u_{g_\alpha}$ might
 be viewed   as a $p$-adic 
 avatar of $\langle g_\alpha,g_\alpha\rangle$.
 \end{remark}
 
\subsubsection{The case where $\varrho_g$ is reducible}
Consider now  the case where $g$ corresponds to the Eisenstein series $E_1(\chi_1,\chi_2) \in M_1(N,\chi)$, 
 where $\chi_1$ and $\chi_2$ are odd and even Dirichlet characters satisfying
 (as  motivated in Section \ref{sec:smooth}) the condition
  \begin{equation}
   \label{eqn:cond-p-chi1-chi2}
\chi_1(p) = \chi_2(p).
\end{equation}
  In that case, 
 $$ V_g = L(\chi_1) \oplus L(\chi_2), \qquad  \quad \mathrm{Ad}_g = L \oplus L(\eta) \oplus L(\eta^{-1}), \quad \mbox{ where } 
 \eta := \chi_1/\chi_2.$$
Condition 
 \eqref{eqn:cond-p-chi1-chi2} implies that $\eta(p)=1$ and hence 
  $p$ splits completely in the abelian extension $H_\eta/\Q$ cut out by $\eta$. 


To define $u_{g_\alpha}$, 
choose a prime ideal $\wp$ of $\cO_{H_\eta} $ over $p$ inducing the
 embedding $H_\eta \hookrightarrow \C_p$ chosen at the outset, and let 
 $u_\wp\in \cO_{H_\eta}[1/p]^\times $ be a generator of the principal ideal $\wp^h$, where $h$ is the class number of $H_\eta$:
$$ \wp^h = (u_\wp), \qquad u_\wp \in (\cO_{H_\eta}[1/\wp])^\times \pmod{\cO_{H_\eta}^\times}.$$

The unit $u_{g_\alpha}$ is defined as follows (in additive notation):
 \begin{definition} 
Let 
$$ e_\eta :=  \frac{1}{\#G}\sum_{\sigma\in G} \eta^{-1}(\sigma) \sigma, \qquad
 e_{\eta^{-1}} :=  \frac{1}{\#G}\sum_{\sigma\in G} \eta(\sigma) \sigma, $$
  be the usual idempotents in $L[G]$ giving the 
projection onto the $\eta$ and $\eta^{-1}$-isotypic component for the action of $G$. Set
\begin{equation}\nonumber
%% \label{def-uP}
u_{g_\alpha} = e_{\eta} u_\wp + e_{\eta^{-1}} u_\wp.
\end{equation}
 \end{definition}
 
 The unit $u_{g_\alpha}$ is a linear combination of {\em Gross-Stark units} attached to the characters
$\eta$ and $\eta^{-1}$, whose $p$-adic logarithms
$$
\log_\wp(u_{g_\alpha}) = \frac{1}{\#G} \sum_{\sigma \in \Gal(H_\eta/\Q)} (\eta(\sigma) + \eta^{-1}(\sigma)) \log_{\wp}(\sigma(u_\wp))
$$
account  for the derivative at $0$ of the Kubota-Leopoldt $p$-adic
$L$-function of $\eta$. 
 In this formula,  $\log_\wp$ denotes   Iwasawa's logarithm  satisfying $\log_\wp(p) = 0$, composed with the
 $p$-adic embedding of $H_\eta$ attached to $\wp$.
   It is not hard to see that
\begin{enumerate}
\item Although the unit $u_\wp$ is only well-defined modulo $\cO_{H_\eta}^\times$, there is no resulting ambiguity in the  
  unit $u_{g_\alpha}$, since $\eta$ and $\eta^{-1}$ are odd characters and hence the idempotents
  $e_{\eta}$ and $e_{\eta^{-1}}$ annihilate $\cO_{H_\eta}^\times\otimes L$.
  \item 
  The unit $u_{\wp}$, and hence also the unit $u_{g_\alpha}$, depends crucially 
  (and not just by scaling by $L^\times$!)
  on the choice of the ideal $\wp$ above $p$, but the expression 
  $\log_{\wp} u_{g_\alpha}$ is independent of this choice, up to multiplication by a root of unity.
  \end{enumerate}

\section{Iterated integrals and the Garrett-Hida $p$-adic $L$-function}
\label{tri}

The main goal of this chapter is to recast the $p$-adic iterated integral 
that arose in the introduction as the value of a $p$-adic $L$-function at
 a point which lies outside its region of classical interpolation. 
 This interpretation is crucial for the proof of Theorem \ref{main-theta} 
of Section  \ref{sec:proof}, 
 which provides the main piece of theoretical evidence for Conjecture  $\shi$ 
 in a setting where the $p$-adic trilinear form $I_p'$ does not vanish. 

We begin by briefly recalling a few key facts about Garrett's triple product classical $L$-function and the work of Harris-Kudla, Ichino, Watson and Woodbury on Jacquet's conjecture on trilinear periods. 

Let $(k,\ell,m)$ be a triple of positive integers and let 
$$g_k\in M_k(N_g,\chi_g), \quad f_\ell \in M_\ell(N_f,\chi_f), \quad h_m \in M_m(N_h,\chi_h)$$
 be three newforms (assumed to be cuspidal if the weight is at least $2$) satisfying 
 $\chi_g \chi_f \chi_h=1$. Note that this forces $k+\ell+m$ to be even. Set $N=\mathrm{lcm}(N_f,N_g,N_h)$ and write $\Q(g_k,f_\ell,h_m)$
  for the number field generated by the Fourier coefficients of the three forms.

Let $L(g_k \otimes f_\ell \otimes h_m,s)$ denote Garrett's triple-product $L$-function associated to $(g_k,f_\ell,h_m)$. It is a self-dual $L$-function
defined by an Euler product of degree $8$ and
 satisfying a functional equation with central critical point $c=\frac{k+\ell+m-2}{2}$ and root 
 number $$\varepsilon = \prod_{v\mid N \infty} \varepsilon_v(g_k \otimes f_\ell \otimes h_m), \qquad \mbox{ with } \qquad \varepsilon_v(g_k \otimes f_\ell \otimes h_m) \in \{\pm 1\}.$$ 


The archimedean sign $\varepsilon_\infty$ is equal to $1$
 if and only if the triple $(k,\ell,m)$ of weights is {\em unbalanced},
 i.e., one of the weights  is greater  or equal to the sum of the other two. If $v\mid  N$ is a finite place, the main result of \cite{Pr} asserts that $\varepsilon_v(g_k \otimes f_\ell \otimes h_m)=+1$ if and only if 
there exists a $\GL_2(\Q_v)$-invariant linear form in the tensor product
$\pi_{f,v}\otimes \pi_{g,v}\otimes \pi_{h,v}$ of local admissible representations of $\GL_2(\Q_v)$ associated to the three eigenforms. This is  the case if $v\nmid \mathrm{gcd}(N_f,N_g,N_h)$, or if one of $\pi_{f,v}$, $\pi_{g,v}$ or $\pi_{h,v}$ does not belong to the discrete series.


Motivated by the Ph.\,D thesis of D. Prasad, Jacquet formulated a conjecture suggesting a necessary and sufficient criterion for the non-vanishing of the central critical value  $L(g_k \otimes f_\ell \otimes h_m,c)$ in terms of certain {\em trilinear periods}. This conjecture was proved by Harris and Kudla in \cite{HaKu}, and several authors have subsequently refined this result by providing explicit formulae relating the $L$-value to the periods. 


In order to describe the 
 precise relationships between $L$-values and trilinear periods,
  we {\em assume for the rest of this chapter} that 
\begin{equation}\label{HypA}
\varepsilon_v(g_k \otimes f_\ell \otimes h_m) = +1 \quad \mbox{ for all } \quad v\mid N \infty.
\end{equation}
When $(k,\ell,m)=(1,2,1)$ and $(g_1,f_2,h_1) = (g,f,h)$
are the modular forms considered
 in the introduction, then $L(g_k \otimes f_\ell \otimes h_m,s)= L(E/\Q,\varrho_{gh},s)$ is the Hasse-Weil-Artin $L$-function considered there, and 
\eqref{HypA} is just Hypothesis A. The above discussion yields sufficient conditions for Hypothesis A to hold, and shows that this is not a very stringent assumption, in practice.
Condition \eqref{HypA} at  $v=\infty$ implies that the triple $(k,\ell,m)$ is unbalanced;
assume for the sake of definiteness that
 $k\geq \ell + m$ from now on. 



As in the introduction, let 
$$S_k(N,\chi_g)[g_k], \quad S_\ell(N,\chi_f)[f_\ell], \quad S_m(N,\chi_h)[h_m]$$
 denote the eigenspaces corresponding to the three forms in level $N$ 
 with respect to the Hecke operators $\{ T_n, n\nmid N\}$. 

For any  $w\geq 1$ and any Dirichlet character $\chi$ of conductor $N$, 
let $S_w^{\nh}(N,\chi)$ denote the space of nearly holomorphic modular forms of 
weight $w$, level $N$ and nebentype $\chi$, and let 
$$\delta  =\frac{1}{2\pi i}\left(\frac{d}{dz} + \frac{w}{z-\bar z}\right): S_w^{\nh}(N,\chi) \lra S_{w+2}^{\nh}(N,\chi) $$
 denote the Shimura-Maass operator,  as described for instance in \cite[Ch.\,X]{hida-book}. Let also
\begin{equation} \nonumber
%% \label{Petersson-first}
\langle \,,   \rangle: S_w^{\nh}(N,\chi)\times S_w^{\nh}(N,\chi)\lra \C,
\qquad \langle f_1,f_2\rangle =
\int_{\Gamma_1(N)\backslash \uhp} \overline{f_1(z)} \, f_2(z)\, y^{w} \frac{dx dy}{y^2},
\end{equation}
denote the Petersson scalar product on these spaces.


Set $t=(k-\ell-m)/2 \geq 0$. A {\em test vector} for the triple $(g_k,f_\ell,h_m)$ is a triplet 
$$(\testg_k,\testf_\ell,\testh_m) \in S_k(N,\chi_g)_L[g_k] \times S_\ell(N,\chi_f)_L[f_\ell]\times S_m(N,\chi_h)_L[h_m].$$
 For such a test vector, recall that we denote $g_k^*=g_k\otimes \chi_g^{-1}$, and  define the quantities
$$
I(\testg_k,\testf_\ell,\testh_m) := \langle \testg^*_k, \delta^t(\testf_\ell)\cdot \testh_m\rangle,
$$
as well as 
\begin{eqnarray}
J(\testg_k,\testf_\ell,\testh_m) & := & I(\testg_k,\testf_\ell,\testh_m)/\langle g_k^*,g_k^*\rangle, \nonumber \\
\omega(\testg_k,\testf_\ell,\testh_m)  &:=  &I(\testg^*_k,\testf^*_\ell,\testh^*_m)/I(\testg_k,\testf_\ell,\testh_m). \label{J-omega}
\end{eqnarray}


The following result is a direct consequence of the fundamental work of Ichino \cite{Ichino}. The precise formulation given here is only valid under assumption \eqref{HypA} and follows by combining loc.\,cit.\,with a formula due to Hida for a special value of the $L$-function of the adjoint of a modular form and the precise formulae of  Watson \cite{Wat}  and Woodbury \cite{Wood}, \cite{Wood2}.

\begin{proposition}
\label{propHK} 
{\bf a)} For any test vector  $(\testg_k,\testf_\ell,\testh_m)$,
\begin{equation}\label{formulaHK}
|I(\testg_k,\testf_\ell,\testh_m) |^2 =  \mathfrak{a}(k,\ell,m) \cdot \mathfrak{f}(k,\ell,m) \cdot \prod_{v\mid N\infty} C_v(\testg_k,\testf_\ell,\testh_m) \cdot L(g_{k} \otimes f_\ell \otimes h_m,c),
\end{equation}
where
\begin{enumerate}
\item[i)]  $\mathfrak{a}(k,\ell,m)= (\frac{k+\ell+m-4}{2})! \cdot (\frac{k+\ell-m-2}{2})! \cdot (\frac{k-\ell+m-2}{2})!\cdot (\frac{k-\ell-m}{2})! \cdot \pi^{-2k}$,
\vspace{0.1cm}
\item[ii)]  $\mathfrak{f}(k,\ell,m)  =  A\cdot 2^B $ where $A\in \Q^\times$ depends only 
on the levels of the eigenforms and of their characters, $B=B(k,\ell,m)\in \Z$ is a linear function on the weights $k$,$\ell$,$m$,
\vspace{0.1cm}
\item[iii)]   for all $v\mid N \infty$, the local term $C_v(\testg_k,\testf_\ell,\testh_m) \in L$   depends only on the admissible representations of $\GL_2(\Q_v)$ associated to the three cuspforms and on the local components at $v$ of the choice of  $(\testg_k,\testf_\ell,\testh_m)$. 
\end{enumerate}

 \medskip\noindent
{\bf b)} There exists a test vector  $(\testg_k,\testf_\ell,\testh_m)$ for which $C_v(\testg_k,\testf_\ell,\testh_m)\ne 0$ for all $v\mid N \infty$.

\medskip\noindent
{\bf c)} Assume $N$ is square-free. Then there exists a particular test vector  $(\testg_k,\testf_\ell,\testh_m)$ such that $W_v := C_v(\testg_k,\testf_\ell,\testh_m) \in \Q^\times$ is a rational constant for all $v\mid N$.
% and only depends on whether $v$ divides or not each of $N_g$, $N_f$ and $N_h$.
\end{proposition}
\begin{proof}
Let $\Gamma_{\R}(s):=\pi^{-s/2}\Gamma(s/2)$ and $\Gamma_{\C}(s):=2(2\pi)^{-s} \Gamma(s)$,
where
 $\Gamma(s)$ is the usual gamma function. 
 For any eigenform $\varphi \in S_w(N,\chi)$, let 
$$L(\mathrm{Ad}(\varphi),s) = L(\mathrm{sym}^2(\varphi), \chi^{-1}, s)$$ 
denote the $L$-function of the adjoint square of $\varphi$ and let 
$$L^*(\mathrm{Ad}(\varphi),s) = L_\infty(\mathrm{Ad}(\varphi),s) L(\mathrm{Ad}(\varphi),s)  $$
denote the completed $L$-function,
with 
$$ L_\infty(\mathrm{Ad}(\varphi),s) := \Gamma_{\R}(s-w+2) \Gamma_{\C}(s)$$ 
the idoneous factor at $\infty$.

The Rankin-Selberg method   relates the special value of 
$ L(\mathrm{Ad}(\varphi),s)$ at $s=w$ to the Petersson  norm of $\varphi$. 
A completely explicit formula is given in \cite[Theorem 5.1]{Hi81}, which,
combined with the value of $L_\infty(\mathrm{Ad}(\varphi),s)$ at $s=w$, yields
\begin{equation}\label{Ad-ff}
L^*(\mathrm{Ad}(\varphi),w) = 2^{w}A_\varphi\times \langle \varphi,\varphi\rangle
\end{equation}
where $A_\varphi\in \Q^\times$ is a constant which only depends on the level of $\varphi$ and conductor of $\chi$.

Likewise, let 
$$L^*(g_{k} \otimes f_\ell \otimes h_m,s) := L_\infty(g_{k} \otimes f_\ell \otimes h_m, s)  L(g_{k} \otimes f_\ell \otimes h_m,s)$$ 
denote the completed Garrett $L$-function, where
\begin{equation}\label{Loo}
L_\infty(g_{k} \otimes f_\ell \otimes h_m,s) :=\Gamma_{\C}(s) \Gamma_{\C}(s+1-m)  \Gamma_{\C}(s+1-\ell)  \Gamma_{\C}(s+2-\ell-m).
\end{equation} 

The main result of \cite{Ichino} (see also \cite[Theorem 2.1]{Wood} for a reformulation of the same statement) asserts that for {\em any} test vector  $(\testg_k,\testf_\ell,\testh_m)$,
\begin{equation}\label{Ichino-formula}
\frac{|I(\testg_k,\testf_\ell,\testh_m) |^2 }{\langle g_k,g_k\rangle \cdot \langle f_\ell,f_\ell\rangle \cdot \langle h_m,h_m\rangle} = C\cdot  \frac{\prod_{v\mid N\infty} C_v(\testg_k,\testf_\ell,\testh_m) \cdot L^*\left(g_{k} \otimes f_\ell \otimes h_m,c\right)}{L^*(\mathrm{Ad}(g_k),k) L^*(\mathrm{Ad}(f_\ell),\ell) L^*(\mathrm{Ad}(h_m),m)},
\end{equation}
where
\begin{itemize}
\item $C\in \Q^\times$ is an absolute constant which only depends on the normalisation taken for the Petersson scalar product;
\item for any $v\mid N\infty$, $C_v(\testg_k,\testf_\ell,\testh_m)\in \Q(g_k,f_\ell,h_m)$ is as described in iii) above.
\end{itemize}
Combining \eqref{Ichino-formula} with \eqref{Ad-ff}, and evaluating \eqref{Loo} at $s=c$ it follows that 
\begin{equation} \nonumber
%% \label{Ichino-formula2}
|I(\testg_k,\testf_\ell,\testh_m) |^2   =  \mathfrak{a}(k,\ell,m) \cdot \mathfrak{f}(k,\ell,m)  \cdot \prod_{v\mid N\infty} C_v\cdot L\left(g_{k} \otimes f_\ell \otimes h_m,c\right),
\end{equation}
where we have set $\mathfrak{f}(k,\ell,m) = 2^{-3k-\ell-m+4}\cdot C \cdot (A_{g_k} A_{f_\ell} A_{h_m})^{-1}$. This proves {\bf a)}.
Statement {\bf b)} was originally known as Jacquet's conjecture, and was proved by Harris and Kudla in \cite{HaKu}. As for {\bf c)}, we refer to \cite[Theorem 5.1]{Wood}. 
\end{proof}
\begin{remark}
\label{WW-classic}
The choice of test vector in c)  is a completely explicit one, given by  Watson \cite{Wat} in the case $N_g=N_f=N_h$ and more recently by Woodbury \cite{Wood} for arbitrary square-free levels. We shall refer to it as the {\em Watson-Woodbury test vector}. The resulting value of $W_v\in \Q^\times$ for $v\mid N$ is also explicit (see \cite[Theorem 3]{Wat} 
and \cite[Theorem 5.1]{Wood} for the recipes), and we shall refer to it as 
the {\em Watson-Woodbury local constant at $v$}. 
Note we say nothing about $C_\infty(\testg_k,\testf_\ell,\testh_m)$, although 
having a tight control of it would be desirable. See \cite{Wood2} for the latest 
developments, which assert that in favorable scenarios (such as the one considered
 in \S \ref{sec:proof} below) the Watson-Woodbury test vector yields $W_\infty=1$. 
\end{remark}

Up to this point we have summarized the basic material we need about Garrett triple-product classical $L$-functions. Fix as in the introduction a prime number $p\nmid N$ and let us discuss now the $p$-adic interpolation properties of their central critical values. The vehicle that allows to do so is the notion of Hida families, which are families of $p$-adic overconvergent ordinary modular forms
parametrized by a rigid analytic space: we refer to \cite[\S 2.6]{DR1} and \cite[\S 1.8]{DR2} for a brief introduction to this subject, presented in a language suitable for our purposes. Let 
\begin{equation}
\label{Omega}
\Omega:= \Hom(\Z_p^\times,\Z_p^\times) \simeq \Z/(p-1)\Z \times \Z_p
\end{equation}
denote the weight space, which contains the integers as a dense subset by means of the identification $k \leftrightarrow \varphi_k$, where $\varphi_k\in \Omega$ is the homomorphism defined by the rule $\varphi_k(n)=n^{k-1}$. 

Let $\hg$, $\hf$, and $\hh$ be three Hida families of new tame level $N_g$, $N_f$, and $N_h$ and tame nebentype characters $\chi_g$, $\chi_f$ and $\chi_h$, respectively. Assume $\chi_g \chi_f \chi_h = 1$ as above.




 
  % Assume that the weight $1\in \Omega$ belongs to the image of both $w_{\hg}$ and $w_{\hh}$, and that $2\in \Omega$ lies in the image of $w_{\ff}$
%When the weight map is \'etale at these points,  
%one can choose   suitably small neighborhoods of these points, denoted 
%, which 
%can be  identified with subsets of $\Omega$ (and are even contained in a single residue class modulo $p-1$). 
%This identification will be made from now on in order to lighten the discussion. More precisely, we assume that $2\in \mathcal U_f$, while weight $1$ belongs both to $\mathcal U_g$ and $\mathcal U_h$. In %addition, we assume that the specialisations $g_{1,\alpha}$ and $h_{1,\alpha}$ of $\hg$ and $\hh$ at weight $1$ are both classical eigenforms.

Let $\mathcal U_g$ denote the parameter space of the Hida family $\hg$, and let $k_{\hg}: \mathcal U_g \ra \Omega$ denote the finite weight map associated to it. If confusion is not likely to arise, in order to simplify the notation we may freely identify a point $x\in \cU_{\hg}$ with its image $k=k_{\hg}(x)\in \Omega$.

For any point $x\in \mathcal U_g$, let $x(\hg)$ denote the specialisation of $\hg$ at $x$. In general $x(\hg)$ is an overconvergent ordinary modular form. Define $\mathcal U_g^{\cl} \subseteq \mathcal U_g$ to be the subset of points $x\in \mathcal U_g$ such that $x(\hg)$ is a {\em classical} ordinary eigenform. By the work of Hida, $\mathcal U_g^{\cl}$ contains all points of weight $k_{\hg}(x)=k\in \Z_{\geq 2}$. Points of weight $1$ may belong to $\mathcal U_g^{\cl}$ or not.
Define further $\mathcal U_g^{\circ}$ to be the subset of classical points  $x\in\mathcal U_g^{\cl}$ such that the eigenform $x(\hg)$ is old at $p$. For such points, there exists a classical newform that we denote $g_k \in M_k(N_g,\chi_g)$ whose ordinary $p$-stabilisation is the specialisation of $\hg$ at $x$. Phrased in terms of $q$-expansions, this amounts to saying that
$$
x(\hg)(q) = g_k(q)-\beta g_k(q^p) \in M_k(N_g p,\chi_g), 
$$
where $\beta\in \C_p$ is a root of the $p$-th Hecke polynomial $x^2-a_p(g_k)x+\chi_g(p)p^{k-1}$ whose $p$-adic valuation is $k-1$. Adopt throughout similar notations as above for $\hf$ and $\hh$. 

Note that for any triple of classical points $(x,y,z)\in \mathcal U_g^{\circ} \times \mathcal U_f^{\circ} \times \mathcal U_h^{\circ}$ of weights $(k,\ell,m)$,
 the subfield $\Q(g_k,f_\ell,h_m)$  of $\C_p$ generated by the Fourier coefficients of $g_k$, $f_\ell$ and $h_m$  is a finite extension of $\Q$, whose degree 
 varies in general as a function of the weights $(k,\ell,m)$.
 

Let $\htg$, $\htf$, $\hth$ be $\Lambda$-adic test vectors of tame level $N$ associated to $\hg$, $\hf$ and $\hh$ as described in \cite[Definition 2.16]{DR1}. Such test vectors may be written down explicitly as  $\htg = \sum_{d\mid N/N_g} \lambda_d \hg(q^d)$, where $d$ ranges over the positive divisors of $N/N_g$ and $\lambda_d\in \Lambda_g = \mathcal O(\mathcal U_g)$ are Iwasawa functions on the domain $\mathcal U_g$ (and similarly for $\htf$ and $\hth$).

Assume that $\mathrm{gcd}(N_f,N_g,N_h)=1$ and that the three Hida families are cuspidal, meaning that all their specialisations at classical points of weight greater than $1$ (though not necessarily at points of weight $1$) are {\em cuspidal} eigenforms. The first assumption implies that the local signs hypothesis \eqref{HypA} holds for all triples $(g_k,f_\ell,h_m)$ at points $(k,\ell,m)\in \mathcal U_g^{\circ} \times \mathcal U_f^{\circ} \times \mathcal U_h^{\circ}$.

Define a dense subset  $\mathcal U^{\circ}_{gfh}\subset \mathcal U_g \times \mathcal U_f \times \mathcal U_h$ by
$$
\mathcal U^{\circ}_{gfh} = \{ (k,\ell,m)\in \mathcal U_g^{\circ} \times \mathcal U_f^{\circ} \times \mathcal U_h^{\circ} \, \mbox{ such that }  k\geq \ell + m\}.
$$ 
Under the running assumptions, a three-variable $p$-adic $L$-function 
\begin{equation}\label{padicL-Garrett-def}
\Lp^g(\htg,\htf,\hth): \mathcal U_g \times \mathcal U_f \times \mathcal U_h \, \lra \,\C_p
\end{equation}
 was constructed  in \cite[\S 4.2]{DR1}, which is 
characterized by the following interpolation property at triplets $(k,\ell,m)\in \mathcal U^{\circ}_{fgh}$:
\begin{equation}\label{int-Garrett}
\Lp^g(\htg,\htf,\hth)(k,\ell,m) = \mathfrak{e}(k,\ell,m) \cdot J(\testg_k,\testf_\ell,\testh_m),
\end{equation}
where 
\begin{equation}\nonumber
%% \label{euler}
\mathfrak{e}(k,\ell,m) =\cE(g_{k},f_\ell,h_m)/\cE_0(g_k) \cE_1(g_k) =\cE(g^*_{k},f^*_\ell,h^*_m)/\cE_0(g^*_k) \cE_1(g^*_k)
\end{equation} denotes the Euler factor introduced in \cite[Theorem 1.3]{DR1}. 
%(A glance to the formulas in loc.\,cit.\,reveals that $\mathfrak{e}(k,\ell,m) =\cE(g^*_{k},f^*_\ell,h^*_m)/\cE_0(g^*_k) \cE_1(g^*_k)$. 

The next proposition spells out the interpolation property  tying
this  $p$-adic $L$-function to central critical $L$-values.
\begin{proposition}
\label{padicL-Garrett} For any $\Lambda$-adic test vectors $\htg$, $\htf$, $\hth$ and every $(k,\ell,m)\in \cU^{\circ}_{fgh}$, the following formula is valid:
\begin{equation}
\label{ggg}
\begin{split}
   \Lp^g(\htg,\htf,\hth) &  \cdot \Lp^g(\htg^*,\htf^*,\hth^*)(k,\ell,m)      \\ 
    &  = \mathfrak{e}(k,\ell,m)^2 \cdot   \mathfrak{a}(k,\ell,m) \cdot   \mathfrak{f}(k,\ell,m)  \cdot \prod_{v\mid N \infty} C_v(\testg_k,\testf_\ell,\testh_m) \\
 & \quad \times   \frac{L(g_k \otimes f_\ell \otimes h_m,\frac{k+\ell+m-2}{2})}{ \langle g^*_{k},g^*_{k}\rangle^{2}}. 
\end{split}
\end{equation}

\end{proposition}
\begin{proof} 
Note that
$$|I(\testg_k,\testf_\ell,\testh_m)|^2 = I(\testg_k,\testf_\ell,\testh_m)\cdot \overline{I(\testg_k,\testf_\ell,\testh_m)} = I(\testg_k,\testf_\ell,\testh_m)\cdot I(\testg^*_k,\testf^*_\ell,\testh^*_m), $$
and that $$\langle g_{k},g_{k}\rangle=\langle g^*_{k},g^*_{k}\rangle.$$
It thus follows from \eqref{int-Garrett} and \eqref{formulaHK} that 
\begin{equation*}
\begin{split}
  \Lp^g(\htg,\htf,\hth) & \cdot \Lp^g(\htg^*,\htf^*,\hth^*)(k,\ell,m) = 
 \mathfrak{e}(k,\ell,m)^2 \frac{|I(\testg_k,\testf_\ell,\testh_m)|^2}{\langle g^*_{k},g^*_{k}\rangle^2}   \\
& =  \mathfrak{e}(k,\ell,m)^2  \cdot   \mathfrak{a}(k,\ell,m) \cdot \mathfrak{f}(k,\ell,m) 
 \cdot  \prod_{v\mid N \infty} C_v(\testg_k,\testf_\ell,\testh_m) \\ 
 & \qquad \times  \frac{ L(g_k \otimes f_\ell \otimes h_m,\frac{k+\ell+m-2}{2})}{\langle g^*_{k},g^*_{k}\rangle^2}.
\end{split}
\end{equation*}
%Since $\frac{\Lp^g(\htg^*,\htf^*,\hth^*)(k,\ell,m)}{\Lp^g(\htg,\htf,\hth)(k,\ell,m)} = \frac{\mathfrak{e}^*(k,\ell,m) \cdot J(\testg^*_k,\testf^*_\ell,\testh^*_m)}{\mathfrak{e}(k,\ell,m) \cdot J(\testg_k,\testf_\ell,%\testh_m)} = \frac{\mathfrak{e}^*(k,\ell,m)}{\mathfrak{e}(k,\ell,m)} (\testg_k,\testf_\ell,\testh_m)$
\end{proof}
\begin{remark}
\label{WW-adic}
An inspection of the definition of the Watson-Woodbury test vector shows that, when we let it vary along the Hida families $\hg$, $\hf$ and $\hh$, it gives rise to a triplet $(\htg,\htf, \hth)$ of $\Lambda$-adic test vectors of tame level $N$, whose specialization at any $(k,\ell,m)\in  \mathcal U_g^{\circ} \times \mathcal U_f^{\circ} \times \mathcal U_h^{\circ}$ is an eigenform with coefficients in $\Q(g_k,f_\ell,h_m)$, a finite extension of $\Q$; and moreover, the constants $W_v = C_v(\testg_k,\testf_\ell,\testh_m) \in \Q^\times$
in Proposition \ref{propHK} {\bf c)} depend on $v$ and $N$ but not on the triplet $(k,l,m)$ of weights.
\end{remark}


For the remainder of this section (and throughout the rest of the paper, as we shall recall again in later sections) we impose the following hypothesis:

\begin{ass}
There exists a point $x\in \cU^\circ_{\hg}$ of weight $k_{\hg}(x)=1$ such that the specialisation $g_\alpha:=x(\hg)
\in M_1(N_g p,\chi_g)$ satisfies Hypothesis C.
\end{ass}

To make sure that the notations introduced in the Introduction and in this section are compatible, let us emphasize that $g_\alpha$ is an eigenform of weight $1$ and level $N_g p$ that is new at $N_g$ and old at $p$. The eigenform $g_\alpha$ is the $p$-stabilisation of a newform $g:=g_1\in M_1(N_g,\chi_g)$.

The next proposition relates this restriction of the Garrett-Hida $p$-adic $L$-function
 to  $p$-adic iterated integrals:
\begin{proposition}
\label{iter=Lp}
There exists a linear form
$$
\testgamma_\alpha: S_1(N p,\chi^{-1}_g)_L[g^*_\alpha] \,\lra \, L
$$
with $L=\Q(g_1,f_2,h_1)$ such that
$$
\Lp^g(\htg,\htf,\hth)(1,2,1)  = \int_{\testgamma_{\alpha}} \! \testf_2 \cdot \testh_1.
$$
\end{proposition}
\begin{proof}
Fix a finite flat extension $\Lambda^\dag$ of $\Lambda$, sufficiently large so that it contains the coefficients of all the $\Lambda$-adic modular forms considered below, and write $\uS^{\ord}(N;\Lambda^\dag)$ for the space of $\Lambda$-adic modular forms with coefficients
in $\Lambda^\dag$. The Hida family $\hg$ 
gives rise to a subspace
$$ \uS^{\ord}(N;\Lambda^\dag)[\hg]:= \left\{ 
\htg \in \uS^{\ord}(N;\Lambda^\dag)  \ \  \mbox{ such that }  \ \ 
T_n \htg = a_n(\hg) \htg,  \ \  \forall (n,N)=1  \right\}.$$
Letting $\mathcal L^\dag$ denote the fraction field of $\Lambda^\dag$, the vector space $\uS^{\ord}(N;\mathcal L^\dag)[\hg]$ is finite-dimensional
over $\mathcal L^\dag$ and has for basis the set  
\begin{equation}\label{basis}
\{ \hg(q^d) \}_{d|(N/N_g)}
\end{equation}
of  $\Lambda$-adic forms.
By \cite[Lemma 2.19]{DR1}, there exists a linear operator 
$$
J(\htg^*): \uS^{\ord}(N,\Lambda^\dag)\lra \mathcal L^\dag \otimes_\Lambda \Lambda^\dag, \quad \hphi \mapsto J(\htg^*,\hphi)
$$
that is characterized uniquely by the following interpolation property: at every point in $\mathcal U^\circ_g \cap \mathcal U^\circ_\phi$ of weight $k\geq 2$, the specialisation of $J(\htg^*)$ is regular (meaning that there is no pole at this point) and is described by the following rule:
$$
\nu_{k}\big(J(\htg^*)\big): S^{\ord}_k(N,\chi_g^{-1})_{\Q_p(g_k)} \lra \Q_p(g_k),   \quad \phi \mapsto \frac{\langle \testg^*_k, \phi\rangle}{\langle g^*_k,g^*_k\rangle}.
$$ 
%Note that, by \cite[Corollary 2.13]{DR1}, $\nu_k\big(J(\htg^*)\big)$ is in fact $\Q(g_k)$-rational. 

We turn now to analyze the specialisation of $J(\htg^*)$ in weight $1$. For any $d|(N/N_g)$ let $c_d: \uS^{\ord}(N,\mathcal L^\dag)\lra \mathcal L^\dag$ denote the functional which associates to $\hphi$ its coefficient in $\hg(q^d)$ with respect to the basis in \eqref{basis}.
It follows from the proofs of \cite[Lemmas 2.12 and 2.19]{DR1} that 
$$
J(\htg^*) = \sum_{d|(N/N_g)} \lambda_d\cdot c_d
$$
where $\lambda_d\in \mathcal L^\dag$ are elements which, viewed as functions of
 $\Z_{\geq 2}$ by means of the rule $k \mapsto \nu_k(\lambda_d)$, can be expressed as
  polynomials in the expressions
$q^k$, $a_q(g_k)$, $\frac{1}{q}$ and $\frac{1}{q+1}$ as $q$ ranges over the divisors of $N$. This implies that the specialisation of $J(\htg^*)$ in weight $1$ is also regular and hence gives rise to a linear operator 
$$
\testgamma_\alpha := \nu_1\big(J(\htg^*)\big): S^{\oc,\ord}_1(N,\chi_g^{-1})_{L_p} \lra L_p.
$$

It factors through $S^{\oc,\ord}_1(N,\chi_g^{-1})_{L_p}[[g^*_\alpha]]$, 
which by the classicality hypothesis imposed on $g^*_\alpha$ is isomorphic to the corresponding 
 space $S_1(Np,\chi_g^{-1})_{L_p}[g^*_\alpha]$ of classical modular forms. 

As explained in the introduction, Hypothesis C equips
 $S^{\oc,\ord}_1(N,\chi_g^{-1})_{L_p}[[g^*_\alpha]]$ with an
  $L$-rational structure, denoted $S^{\oc,\ord}_1(N,\chi_g^{-1})_{L}[[g^*_\alpha]]$. 
In view of the description given above for $\lambda_d$ as functions of $k$, it follows that $\nu_1(\lambda_d)\in L$ and hence $\breve\gamma_\alpha$ is $L$-rational, that is to say, it belongs to $S_1(Np,\chi_g^{-1})_{L}[g^*_\alpha]^\vee = S^{\oc,\ord}_1(N,\chi_g^{-1})_{L}[[g^*_\alpha]]^\vee$.

Let $e_{\ord}(d^{\bullet}\testf_2^{[p]} \times \testh_1)$ denote the $\Lambda$-adic modular form whose specialisation in weight $k$ is $e_{\ord}(d^{\frac{k-3}{2}} \testf_2^{[p]} \times \testh_1)$ for all $k\in \Z_{\geq 2}$.
By \cite[Definition 4.4]{DR1}, $\Lp^g(\htg,\testf_2,\testh_1) = J(\htg,e_{\ord}(d^{\bullet}\testf_2^{[p]} \times \testh_1))$ and, by construction,  
$$
\Lp^g(\htg,\testf_2,\testh_1)(1) = \testgamma_\alpha(e_{\ord}(d^{-1}\testf_2^{[p]} \times \testh_1)) = \int_{\testgamma_{\alpha}} \! \testf_2 \cdot \testh_1,
$$
as was  to be shown.
\end{proof}

Proposition
\ref{iter=Lp} recasts
the $p$-adic iterated integral of 
 Conjecture $\shi$ with $(\testgamma,\testf,\testh) = 
 (\testgamma_\alpha, \testf_2, \testh_1)$ 
  as a $p$-adic avatar of the classical
 special value $L(E,\varrho_{gh},1)$. 
 This suggests a possible strategy for proving Conjecture $\shi$ in the
 the rank  $(1,1)$ setting where
 $\varrho_{gh}$ admits a decomposition of the form $ \varrho_{gh} = \varrho_1 \oplus \varrho_2 \oplus \varrho_3$ as in \eqref{eqn:conj-I-decomp-11}. 
 Assuming the regulator  $R_{g_\alpha}(E,\varrho_{gh})$ is factorisable, Conjecture $\shi$ might then be proved by  constructing appropriate    ``$p$-adic Artin $L$-series" $\mathcal L_p(\mathrm{Ad}_{g_\alpha})$ associated to (the irreducible constituent containing $\Ad^+_g$ in) $\Ad_g$ and ``$p$-adic Hasse-Weil-Artin $L$-series" $\mathcal L_{p}(E,\varrho_i)$
attached to the irreducible constituents of $\varrho_{gh}$, and proving that
%\begin{equation}
%\label{str1}
%\int_{\testgamma_\alpha} \testf \cdot \testh \\  \times \  \mbox{(period ratio)} \ \ \  \stackrel{?}{=}  \ \ \ \mathcal L_{p}(E,\varrho_1) \cdot \mathcal L_{p}(E,\varrho_2) \cdot \mathcal L_{p}(E,\varrho_3)
%\end{equation}
\begin{equation}
\label{str1}
\Lp^g(\htg,\testf_2,\testh_1) \\  \cdot \  \mathcal L_p(\mathrm{Ad}_{g_\alpha}) \   \stackrel{?}{=}  \  \mathcal L_{p}(E,\varrho_1) \cdot \mathcal L_{p}(E,\varrho_2) \cdot \mathcal L_{p}(E,\varrho_3)
\end{equation}
and
\begin{eqnarray} 
\label{str2}
\mathcal L_p(\mathrm{Ad}_{g_\alpha})(1)\stackrel{?}{=}   \log_p(u_{g_\alpha}), \quad &&\mathcal L_{p}(E,\varrho_1)(1) \stackrel{?}{=}  \log_{E,p}(P_1), \\ \nonumber 
 \mathcal L_{p}(E,\varrho_3)(1) \in L^\times, \qquad && \mathcal L_{p}(E,\varrho_2)(1)\stackrel{?}{=}  \log_{E,p}(P_2).
 \end{eqnarray} 
 This is the strategy carried out in \S \ref{sec:proof}: see
 Theorem \ref{factor} for the   variant of \eqref{str1} and \eqref{str2} in the special setting
 considered therein.
In general, identities like \eqref{str1}
appear to lie quite deep,  and we are not even able to 
  provide an independent definition of 
  the $p$-adic  $L$-values
  $\mathcal L_{p}(E,\varrho_i)$
   that figure in its putative right-hand side; they
  are perhaps somewhat akin to the ``genuine'' $p$-adic $L$-functions sought for in \cite{hida-genuine}.
  
\section{Heegner points and theta series of imaginary quadratic fields} 
\label{sec:heegner}

Let $K$ be an imaginary quadratic field of discriminant $-D_K$, and let $N_{K/\Q}$ denote the norm map on fractional ideals of $K$. The aim of this chapter is to study
Conjecture $\shi$  in the case where both $g$ and $h$ are theta series 
 associated to $K$. After setting up  the basic notations and assumptions, 
  the cases
  in which the prime $p$ splits or remains inert in $K$
are  treated separately in Sections \ref{sec:proof} and  \ref{p-inert}.

Just as in the introduction, 
let $E$ be an elliptic curve over $\Q$
 and $f\in S_2(N_f)$ denote
 its associated newform. Given a finite order character 
$$
\psi: G_K \lra \C^\times
$$
of conductor $\mathfrak c_\psi \subseteq \cO_K$ and central  
 character $\chi$ (viewed as a Dirichlet
character), let
$$
\theta_\psi  \in M_1(D_K\cdot N_{K/\Q}({\mathfrak c}_\psi),\chi)
$$ 
denote the theta series associated to $\psi$ 
and let $V_\psi:= \Ind_K^\Q \psi$  denote the
 two-dimensional induced representation of
 $\psi$ from $G_K$ to $G_\Q$. It is known that $\theta_\psi$ is
 Eisenstein if and only if $V_\psi$ is reducible, which occurs precisely when
 $\psi = \psi'$, where $\psi'$ denotes the character of $G_K$ defined by the rule $\psi'(\sigma) = \psi(\sigma_0 \sigma \sigma_0^{-1})$. Here $\sigma_0$ is any choice of element in $G_\Q \setminus G_K$.
 


Fix  two  finite  order  characters 
$\psi_g$ and  $\psi_h$ of $K$ of conductors $\mathfrak c_g$ and $\mathfrak c_h$. Assume that the  central character
 $\chi$ of $\psi_h$ is inverse to that of $\psi_g$, and set 
 $$g := \theta_{\psi_g} \in M_1(N_g,\bar\chi), \qquad h:= \theta_{\psi_h}\in M_1(N_h,\chi).$$

Define $\psi_1= \psi_g \psi_h$ and $\psi_2 = \psi_g \psi'_h$. The characters $\psi_1$ and $\psi_2$  are ring class characters of $K$, 
 associated to  orders $\cO_{c_1}$  and $\cO_{c_2}$ in $\cO_K$ 
 of conductors $c_1$ and $c_2$ respectively.
 Such characters are sometimes also called, synonymously, dihedral or anticyclotomic, characters of $K$. Note that $\psi_g$ and $\psi_h$ were not assumed to be necessarily anticyclotomic.

One checks that there is a decomposition of Artin representations and, in parallel to that, a factorisation of $L$-series as follows:
\begin{equation}
\label{eqn:dihedral-decomp}
 V_{gh} = V_{\psi_1} \oplus V_{\psi_2}, \quad  L(E,\varrho_{gh},s) = L(E/K,\psi_1,s)\cdot L(E/K,\psi_2,s).
 \end{equation}
 
Assume also that $\gcd(N_f, \mathfrak c_g \mathfrak c_h) =1$. This implies that for all places of $K$ above any place $v\mid N \infty$ of $\Q$, the local signs of $L(E/K,\psi_1,s)$ and of $L(E/K,\psi_2,s)$ are all equal, and hence $\varepsilon_v(E,\varrho_{gh}) = (\pm 1)^2=1.$ This automatically implies that Hypothesis A holds.
%(when this fails, the computation of the sign of the functional equations is more subtle and  might well lead to  interesting phenomena for further %study).

Let $H$ denote the ring class field of conductor 
$c:=\mathrm{lcm}(c_1,c_2)$ of $K$,  which contains the fields cut out by $\psi_1$ and $\psi_2$. 
The field $L$ of coefficients that is implicit in 
\eqref{eqn:dihedral-decomp} can be taken to be any finite extension of the field generated by the traces of $V_{\psi_1}$ and $V_{\psi_2}$. 




\subsection{Heegner points and elliptic units}\label{sec:Hee}
In  the rank $(1,1)$ setting of Conjecture $\shi$ in which
$ r(E,V_{\psi_1}) = r(E,V_{\psi_2}) = 1$, 
the global points and units that arise on the right hand side are
expected to be expressible in terms of Heegner points and elliptic units.
Not surprisingly, these global constructions are the   key to the theoretical
evidence that has been amassed in support of Conjecture $\shi$ in the rank
$(1,1)$ setting when $p$ is split in $K$.

The Heegner point construction is available under the following

\medskip\noindent
{\bf Heegner hypothesis:} {\em The level of $f$ may be factorised as $N_f = N_f^+ N_f^-$ where 
%$(N_f^+,N_f^-)=1$ and 
\begin{itemize}
\item there exists an integral ideal $\mathfrak N$ of $K$ such that $\cO_K/\mathfrak N \simeq \Z/N_f^+\Z$, 
\item $N_f^-$ is the square-free product of an even number of primes which are
 all inert in $K$.
\end{itemize}}

This hypothesis
implies that the signs which occur in the functional equations for $L(E,V_{\psi_1},s)$ and
$L(E,V_{\psi_2},s)$ are $-1$, and hence that the 
analytic ranks 
 $r_{\an}(E,V_{\psi_1})$ and $r_{\an}(E,V_{\psi_2})$
  are both odd. In particular, Hypothesis $B$ of the introduction is automatically satisfied.
  It is further expected that, in ``generic" instances,
\begin{equation}
\label{r1-hyp}
r_{\an}(E,V_{\psi_1}) = r_{\an}(E, V_{\psi_2}) = 1,
\end{equation}
and that these equalities  certainly hold when the $p$-adic iterated integrals attached to 
the triple $(f,g,h)$ are not identically zero.

 
 The decomposition in \eqref{eqn:dihedral-decomp} can be further refined to 
\begin{equation}
\label{eqn:quad-im-decomp}
\varrho_{gh} = \varrho_1 \oplus \varrho_2 \oplus \varrho_3,
\end{equation}
where
\begin{itemize}
\item for $i=1,2$,  $\varrho_i$ is the unique
  irreducible constituent of $V_{\psi_i}$ 
 for which $r_{\an}(E,\varrho_i)$ is odd;
  \item 
 $\varrho_3$ is the $G_\Q$-stable
  complement of $\varrho_1\oplus \varrho_2$ in $\varrho_{gh}$. 
  \end{itemize}
  This decomposition is uniquely determined by $(f,g,h)$, up to 
   possibly switching $\varrho_1$ and $\varrho_2$.
  When  \eqref{r1-hyp}
is satisfied,   the decomposition \eqref{eqn:quad-im-decomp} 
 satisfies the conditions
 \begin{equation}
 \label{eqn:quad-imag-analytic-ranks}
 r_{\an}(E,\varrho_1)=r_{\an}(E,\varrho_2)=1, \qquad r_{\an}(E,\varrho_3)=0.
 \end{equation}
%% stated in Conjecture $\shi$ of the introduction and required 
%% to ensure   the non-vanishing of the iterated integrals
%% attached to $(f,g,h)$. 
As will be explained shortly, the work of Gross, Zagier, Kolyvagin and Zhang  
     further shows that the same equations as in 
     \eqref{eqn:quad-imag-analytic-ranks} hold  for the 
     corresponding algebraic ranks:
 \begin{equation}
    \label{eqn:quad-imag-algebraic-ranks}
r(E,\varrho_1)=r(E,\varrho_2)=1, \qquad r(E,\varrho_3)=0,
\end{equation}
consistent with the predictions made by the Birch and Swinnerton-Dyer conjecture, and
so we are in the rank $(1,1)$ setting
of Conjecture $\shi$.
It also  leads to the construction
of explicit non-trivial elements in $E(H)_L^{\varrho_1}$ and $E(H)_L^{\varrho_2}$
from {\em Heegner points}, which we now describe. 

Let $X$ be the Shimura curve associated to an Eichler order 
of level $N_f^+$ in the (indefinite, thanks to the Heegner hypothesis)
 quaternion algebra of discriminant $N_f^-$ over $\Q$. Let
\begin{equation} \nonumber
%% \label{eqn:def-pi-E}
\pi_E: \mathrm{Jac}(X)\rightarrow E
\end{equation}
 denote a modular parametrisation of $E$ arising from the fact that $E$
is an abelian variety quotient of  the jacobian of $X$. 



Let $X(H)_{\cO_c} \subset X(H)$ denote the set of
 Heegner points attached to the order of conductor $c$ in $K$.
It is a finite set, and any two points in it are related by the action of an
Atkin-Lehner involution and an element of $G:=\Gal(H/K)$. 
 A {\em Heegner divisor} of conductor $c$ 
 on $X$ is any degree $0$
 divisor supported on $X(H)_{\cO_c}$, and 
a {\em Heegner point} on $E$ of conductor $c$ is the image
of such a divisor. 



 After fixing a choice 
 of $P\in X(H)_{\cO_c}$, any
 ring class character $\psi$ of conductor $c$ gives rise to a degree zero divisor on $X$
 by setting
\begin{equation}
\label{eqn:choice-of-P}
 D_\psi := \left\{ \begin{array}{ll}
 {\color{blue}\sum_{\sigma\in G} \psi(\sigma) P^\sigma,} & \mbox{ if  } \psi \ne 1, \\
  \sum_{\sigma\in G}  P^\sigma -  \#G\cdot \infty & \mbox{ if } \psi=1,
  \end{array} \right.
\end{equation}
  where $\infty$ is the degree one divisor in $\Div(X)\otimes \Q$ defined by
  $\infty = \frac{1}{2g-2} {\mathcal K}_X$, and ${\mathcal K}_X $ denotes the canonical class on $X$.
  The {\em Heegner point attached to $\psi$} is then defined to be
  $$ P_\psi := \pi_E(D_\psi) \in  E(H)_L.$$
  Note that the point $P_\psi$ actually belongs to $E(H)_L^{V_\psi}$, and that the same is true of
  $P_{\bar\psi}$. More precisely, with our definitions the Galois action is given by $P_\psi^\sigma = \bar\psi(\sigma) P_\psi$ and $P_{\bar\psi}^\sigma = \psi(\sigma) P_{\bar\psi}$ for any $\sigma\in G$.
   \begin{lemma} 
 \label{lemma:heegner-point-construction}
 Assume that
   \eqref{eqn:quad-imag-analytic-ranks}
   is satisfied. Then 
\eqref{eqn:quad-imag-algebraic-ranks}
holds as well, and, 
  for $i=1$ and $2$, 
 the Mordell-Weil group
 $E(H)_L^{\varrho_i}$ is generated by the vectors
$P_{\psi_i}$ and 
 $P_{\bar\psi_i}$.
 If $\psi_i \ne \bar\psi_i$, these points are linearly independent and form a basis for
 $E(H)_L^{\varrho_i}$. 
 If $\psi_i = \bar\psi_i$ (i.e., if $\psi_i$ is a genus character) then 
 $P_{\psi_i} = P_{\bar \psi_i}$ and this vector is a basis for
 $E(H)_L^{\varrho_i}$.  
 \end{lemma}
 \begin{proof}
Assumption
   \eqref{eqn:quad-imag-analytic-ranks}
 asserts that  the central 
critical  derivative  $L'(E,\varrho_i,1)$ is non-zero.
 The Gross-Zagier formula and its extension to ring class characters given in 
 \cite{zhang-msri} shows that the vectors $P_{\psi_i}$ and $P_{\bar\psi_i}$ have non-zero canonical
 height and hence are non-zero in $E(H)_L^{V_{\psi_i}}$. The main theorem of 
 \cite{bd-crelle} (and a suitable extension to Heegner points arising from
 Shimura curves as given in \cite{YZZ})
 %%, as described for instance  in \cite{nekovar-euler-shimura}) 
 implies that
 $r(E,V_{\psi_i}) = 1$ and that $E(H)_L^{\varrho_i}$ is generated by the Heegner vectors
 $P_{\psi_i}$ and $P_{\bar\psi_i}$.  The result follows.
  \end{proof}
 
 Note that the points $P_{\psi_i}$ and $P_{\bar\psi_i}$ depend on
  the choice of a point $P \in X(H)_{\cO_c}$ that was made in 
  \eqref{eqn:choice-of-P},
  but only up to multiplication by a root of unity. More precisely, 
  replacing $P$ by $w P^{\sigma_0}$ for $\sigma_0\in G$ and $w$ a
  Fricke involution on $X$  has the 
  effect of
 replacing $P_{\psi_i}$ by $\pm \psi_i(\sigma_0) P_{\psi_i}$ and
 $P_{\bar \psi_i}$ by $\pm \bar\psi_i(\sigma_0) P_{\bar\psi_i}$.
In particular, the  $L$-vector space in $E(H)_L$ spanned 
by  $P_{\psi_i} $ is completely canonical 
 and does not depend on the choice of $P$.

It will also  be convenient to introduce a related basis of $E(H)_L^{\varrho_i}$ attached to a choice of
reflection $\tau$ in the dihedral Galois group $\Gal(H/\Q)$, by setting
\begin{equation}
\label{Heegner-inert}
 P_{\psi_i}^+ = P_{\psi_i} + P_{\psi_i}^\tau, \qquad 
P_{\psi_i}^- = P_{\psi_i} - P_{\psi_i}^\tau.
\end{equation}
Sometimes, the basic Heegner point $P$ used to define
the points $P_{\psi_i}^{\pm}$ can be chosen to be fixed (up to sign) by
the  reflection $\tau$, i.e.,  in such a way that $\tau P = \pm P$. When this is the case, 
$$ P_{\psi_i}^+ = P_{\psi_i} \pm P_{\bar\psi_i}, \qquad 
P_{\psi_i}^- =  P_{\psi_i} \mp P_{\bar\psi_i}.$$
 But this simpler expression for the basis $(P_{\psi_i}^+, P_{\psi_i}^-)$ need
  not be invoked in general.

Let 
$$
%{\color{blue}
\psi_0 := \psi_g/\psi'_g 
%}
$$ 
be the non-trivial ring class character attached to the adjoint representation of $g$.
In  addition to the Heegner points  $P_{\psi_i}$ and 
$P_{\psi_i}^\pm$, Conjecture $\shi$  also involves the 
  elliptic unit $u \in \cO_{H_g}^\times$ and its $\psi_0$-component
   \begin{equation} \nonumber
%% \label{ug-general}
u_{\psi_0} = 
{\color{blue}\sum_{\sigma \in G}  \psi_0(\sigma) u^{\sigma}} \in (\cO_{H_g}^\times)_L^{V_{\psi_0}},
\end{equation}
where the group law in the unit group $\cO_{H_G}^\times$ is written additively, and 
the units $u_{\psi_0}^\pm$ relative to any choice of a reflection $\tau \in \Gal(H/\Q)$
are defined just as in \eqref{Heegner-inert}:
\begin{equation}
\label{ug-pinert}
u_{\psi_0}^+  = u_{\psi_0} +  u_{\psi_0}^\tau, \qquad 
u_{\psi_0}^-  = u_{\psi_0}  -  u_{\psi_0}^\tau.
 \end{equation}
Note that, just as for the points $P_{\psi_i}$ and $P_{\psi_i}^\pm$, the group
$G=\Gal(H/K)$ acts on the element $u_{\psi_0}$ through the character $\bar\psi_0$ 
but does not preserve the line spanned by either $u_{\psi_0}^+$
or $u_{\psi_0}^-$. 




\subsection{Conjecture  $\shi$  when $p$  splits in $K$}
\label{sec:proof}

When $p$ is split in $K$, {\color{blue} the geometric frobenius $\sigma_p$ belongs to $G=\Gal(H/K)$.
The choice of a prime $\wp$ of $K$ above $p$ induces an embedding $K \hookrightarrow \Q_p$ and determines $\sigma_p$ 
as the element  of $G$ satisfying $\psi(\wp)=\bar\psi(\sigma_p)$ for all characters $\psi$ of $G$.}\footnote{Since $\sigma_p$ denotes the geometric (as opposed to the arithmetic) frobenius, we have corrected accordingly some of the statements in this section with respect to the published version.}
The choices of $\alpha_g, \beta_g, \alpha_h$ and $\beta_h$ can then be made so that
$$ \alpha_g = \psi_g(\wp), \qquad \beta_g = \psi_g(\bar\wp), \qquad \alpha_h=\psi_h(\wp), \qquad \beta_h = \psi_h(\bar\wp). $$
With the above choices,
$${\color{blue} \alpha_g/\beta_g = \bar\psi_0(\sigma_p),\qquad \alpha_g \alpha_h = \psi_g\psi_h(\wp) = \bar\psi_1(\sigma_p), \qquad \alpha_g\beta_h = \psi_g\psi_h'(\wp) = \bar\psi_2(\sigma_p).}$$

The following lemma describes the $p$-adic regulator $\det R_{g_\alpha}(E,\varrho_{gh})$
that enters into  Conjecture $\shi$. 
\begin{lemma}
The regulator $\det(R_{g_\alpha}(E,\varrho_{gh}))$ is given by the following expressions:
\begin{enumerate}
\item
If   $r(E,V_{\psi_1})= r(E,V_{\psi_2}) = 1$,
$$ \det R_{g_\alpha}(E,\varrho_{gh}) = \log_{E,p}(P_{\psi_1}) \cdot \log_{E,p}(P_{\psi_2});$$
\item
if   $r(E,V_{\psi_1})\ge 2$ or $r(E,V_{\psi_2})\ge 2$,
$$ \det R_{g_\alpha}(E,\varrho_{gh}) = 0.$$
\end{enumerate}
\end{lemma}
\begin{proof}
The first part follows from the fact that, after choosing a $\sigma_p$-eigenbasis for 
$V_{gh}^{g_\alpha}$, the matrix $R_{g_\alpha}(E,\varrho_{gh})$ takes the form
$$ R_{g_\alpha}(E,\varrho_{gh}) =
\left(\begin{array}{cc} 
 \log_{E,p}(P_{\psi_1}) & 0 \\
 0 &  \log_{E,p}(P_{\psi_2}) \end{array}\right).$$
 In the rank $2$ setting,
assume without loss of generality that $r(E,V_{\psi_1}) = 2$ and that
$r(E,V_{\psi_2}) = 0$. 
The two dimensional  subspace $V_{gh}^{g_\alpha}$ then
intersects 
$V_{\psi_1}$ and $V_{\psi_2}$ in one-dimensional subspaces which determine a decomposition of
$V_{gh}^{g_\alpha}$. Choosing a basis $(v_1,v_2)$ for $V_{gh}^{g_\alpha}$ which is compatible
with this decomposition, and a basis $\Phi_1$, $\Phi_2$ of $\Hom_{G_\Q}(V_{\psi_1},E(H)_L)$,
produces a matrix $R_{g_\alpha}(E,\varrho_{gh})$ whose  second column is zero. 
\end{proof}

The lemma above motivates focussing our attention on the rank $(1,1)$ setting of Conjecture $\shi$ when $p$ is split, which is what shall be done for the remainder of Section 
\ref{sec:proof}. 
Assume as in \S \ref{sec:Hee} that the Heegner hypothesis is satisfied and let 
$$u_{g_\alpha} := u_{\psi_0}, \qquad P_1 := P_{\psi_1}, \qquad P_2 := P_{\psi_2} $$
be the Heegner points constructed in the previous section. 
The main result  of this section is:
\begin{theorem}
\label{main-theta} 
Assume $N=\lcm(N_f,N_g,N_h)$ is square-free and $p\nmid 2 N$. Assume also that Hypotheses C and C' hold. 
If $r_{\an}(E,V_{\psi_1})>1$ or $r_{\an}(E,V_{\psi_2})>1$,  the iterated integrals attached
to the triple $(f,g,h)$ vanish identically. Otherwise,
there exists a finite extension $L$ of $\Q(\psi_g, \psi_h)$, a scalar $\lambda \in L^\times$  and test vectors 
$$(\testf,  \testgamma_{g_\alpha}, \testh)  \in S_2(N)_L[f] \times  M_1(N p,\chi)_L^\vee[g_\alpha]\times M_1(N,\chi)_L[h] $$  
for which
\begin{equation*}
 \int_{\testgamma_{g_\alpha}} \! \testf \cdot \testh  =  \lambda \cdot \frac{\log_{E,p}(P_1) \log_{E,p}(P_2)}{\log_p(u_{g_\alpha})}.
\end{equation*}
\end{theorem}

%% \begin{remark}
%% By Lemma \ref{lemma:heegner-point-construction}, the points 
%% $P_1$ and $P_2$ are non-zero in $E(H)_L$  precisely 
%% when $r_{\an}(E,\varrho_1) = r_{\an}(E,\varrho_2)=1$. 
%% Hence Theorem \ref{main-theta} proves   Conjecture $\shi$
%%  when $p$ is split in $K$.
%% \end{remark}

\begin{remark}\label{rem-formula-precise}
%In the setting in which $h$ is an Eisenstein series an alternative and simpler proof is available
%which leads to a more precise version of the theorem.
%Namely, in the proof one interpolates $h$ through an Eisenstein family (rather than
%a CM cuspidal family) and uses the more explicit Rankin-Selberg formula in place of the Garrett formula.
%The field $L$ can  be shown to be the field $\Q(\psi_g)$ generated by the
%values of $\psi_g$ in that case. 
We believe that in fact $L$ can always be taken to be $\Q(\psi_g,\psi_h)$. It would be interesting to be able to pin down a concrete choice of test vectors $(\testf,  \testgamma_{g_\alpha}, \testh)$ for which the precise value of $\lambda\in L^\times$ could be computed explicitly. For example, let $\chi_K$ denote a quadratic Dirichlet character of prime conductor $D_K$ associated to an imaginary
quadratic field $K$.
Assume $N=D_K$ and that $\psi_g = \psi_h=1$, so that
$$ g = h = \theta_1 \in M_1(D_K,\chi_K)$$ 
is  the Eisenstein theta series associated to the trivial character of $K$. Since $N_f=N_g=N_h=N$, a natural  choice of test vectors, which is unique up to scaling by elements in $L^\times$, is simply $(f,\gamma_{g_\alpha},h)$. The following explicit 
formula has been verified experimentally in a number of examples, and should be amenable to a proof
 by the methods of this section:
\begin{equation}\label{formula-precise}
\int_{\gamma_{g_\alpha}} \! f \cdot h = \frac{|E(\F_p)|^2 \log_{E,p}(P_K)^2}{p(p-1) h_K \log_p(u_\wp)}.
\end{equation}
Here $|E(\F_p)|$ is the number of points on $E$ over the finite field $\F_p$, $h_K$ is the class number of $K$,
$P_K$ is the Heegner point in $E(K)$, and $u_\wp \in K^\times$ is any $p$-unit satisfying $ (u_\wp)=\wp^{h_K}$. 

Formula (\ref{formula-precise}) was first discovered experimentally by the authors (on Christmas day), after
a long theoretical and experimental investigation trying to uncover the arithmetic meaning of these $p$-adic iterated integrals.
It is the prototype for all the formulae in this paper.
\end{remark}




The remainder of this section is devoted to proving Theorem  \ref{main-theta}. 
The proof  rests on a comparison between the Garrett-Hida $p$-adic $L$-function
of the previous section 
and two different types of ``Katz-style" $p$-adic $L$-functions whose
definitions are recalled below.

\medskip
\noindent
{\bf 1.} {\em The Katz two-variable $p$-adic $L$-function of an imaginary quadratic field.} 

Let $\mathfrak c\subseteq \cO_K$ be an integral ideal of the imaginary quadratic field 
$K$ and denote $\Sigma$ the set of Hecke characters of $K$ of conductor dividing $\mathfrak c$. Define $\Sigma_K = \Sigma_K^{(1)}\cup \Sigma_K^{(2)}\subset \Sigma$ to be the disjoint union of the sets
$$
\Sigma_K^{(1)}=\{ \psi \in \Sigma \mbox{ of infinity type } (\kappa_1,\kappa_2),  \mbox{ with } \kappa_1\leq 0, \kappa_2\geq 1\},
$$
$$
\Sigma_K^{(2)}=\{ \psi \in \Sigma \mbox{ of infinity type } (\kappa_1,\kappa_2), \mbox{ with } \kappa_1\geq 1, \kappa_2\leq 0\}.
$$
For all $\psi \in \Sigma_K$, the complex argument
$s=0$ is a critical point for the Hecke $L$-function $L(\psi^{-1},s)$, and Katz's $p$-adic $L$-function is
 constructed by interpolating the (suitably normalized) 
 values $L(\psi^{-1},0)$ as $\psi$ ranges over $\Sigma_K^{(2)}$. 

More precisely, let $\hat{\Sigma}_K$ denote the completion of $\Sigma_K^{(2)}$ with respect to the compact open topology on the space of $\cO_{L_p}$-valued functions on a certain subset of $\mathbf{A}^\times_K$, as described in \cite[\S 5.2]{BDP1}. By the work of Katz \cite{Katz1}, there exists a $p$-adic analytic function
$$
L_p(K): \hat{\Sigma}_K\lra \C_p
$$
which is uniquely characterized by the following interpolation property: for all $\psi \in \Sigma_K^{(2)}$ of infinity
%\footnote{In \cite[Thm.\,4.14]{dS}, he in fact asks the weights to satisfy $k_1>|k_2|$. (In his notation, $k_1=k$ and $k_2=j$.) This restriction is %not imposed in BDP for some reason...} 
type $(\kappa_1,\kappa_2)$,
\begin{equation}\label{int-Katz}
L_p(K)(\psi) = \mathfrak{a}(\psi) \times \mathfrak{e}(\psi) \times \mathfrak{f}(\psi) 
\times \frac{\Omega_p^{\kappa_1-\kappa_2}}{\Omega^{\kappa_1-\kappa_2}} \times L_c(\psi^{-1},0),
\end{equation}
where
\begin{itemize}
\item $\mathfrak{a}(\psi)=(\kappa_1-1)!\pi^{-\kappa_2}$, $\quad \mathfrak{e}(\psi) = (1-\psi(\wp)p^{-1}) (1-\psi^{-1}(\bar\wp))$,  and $\quad \mathfrak{f}(\psi) = D_K^{\kappa_2/2} 2^{-\kappa_2}$; 
\item $\Omega_p\in \C_p^\times$ is a $p$-adic period attached to $K$, as defined in \cite[(2-17)]{BDP2};
\item $\Omega\in \C^\times$ is the complex period associated to $K$ defined in \cite[(2-15)]{BDP2};
\item $L_c(\psi^{-1},s)$ is Hecke's $L$-function associated to $\psi^{-1}$ with the Euler factors at primes dividing $c$ removed.
\end{itemize}

Formula \eqref{int-Katz} is \cite[Proposition 3.1]{BDP2}, which in turn follows from \cite[\S 5.3.0]{Katz2} (cf.\,also
 \cite[Theorem 2.3]{Gr} and \cite[II, Theorem 4.14]{dS};   the reader is cautioned
that the latter reference needs to be slightly modified, for the power of $2\pi $ is not correct there.)

Letting $\mathbf{N}_{K}$ denote the norm of $K/\Q$ regarded as a Hecke character of $K$ of infinity type $(1,1)$, Katz's $p$-adic $L$-function satisfies the functional equation $L_p(K)(\psi) = L_p(K)((\psi')^{-1} \mathbf{N}_K)$ (cf.\,\cite[p.\,90-91]{Gr}). 
%Indeed, recall that Gross' $p$-adic function differs from ours: up to a constant, $L_p^{Gr}(\psi) = %L_p(K)(\psi^{-1})$. This implies that while $L_p^{Gr}$ satisfies a functional equation relating $\psi$ to %$\hat\psi = (\psi')^{-1}\mathbf{N}_K^{-1} = \psi \mathbf{N}_K^{-1}$, our $p$-adic $L$-function %relates then $\psi$ to $\psi \mathbf{N}_K$.

The values of  $p$-adic $L$-functions at classical points lying outside their region of interpolation
are frequently of great arithmetic interest, since  they may encode 
further invariants of the associated motives.
 An instance of this philosophy is   Katz's
  $p$-adic
analogue of the Kronecker limit formula
 (cf.\,\cite[\S 10.4, 10.5]{Katz1}, \cite[p.\,90]{Gr}, \cite[Ch.\,II, \S 5.2]{dS}), 
expressing  the value of $L_p(K)$ at a {\em finite order}
character $\psi$   of $G_K$ in terms of a suitable linear combination of
 $p$-adic logarithms of elliptic units:
\begin{equation}
\label{Katz-GZ}
L_p(K)(\psi)   = \begin{cases}\frac{1}{2} (\frac{1}{p}-1) \cdot \log_p(u_\wp)   & \mbox{ if } \psi=1 \\
 \frac{-1}{24c}  (1-\psi^{-1}(\bar\wp)) (1-\psi(\wp)p^{-1}) \cdot \sum_{\sigma \in G} \psi^{-1}(\sigma) \log_p(\sigma(u)) &  \mbox{ if } \psi\ne 1.
\end{cases}
\end{equation}
%Recall that $\chi_0$ is anticyclotomic and hence $\chi_0^{-1}(\bar \wp) = \chi_0(\wp)$. By \cite{5.2.28}{Katz2}, the local term is $1$ because $\chi_0$ is unramified at $p$.
Here, as above, $u_\wp\in K^\times$ is any element satisfying $ (u_\wp)=\wp^{h_K}$  and $u\in \mathcal U_{\mathrm{CM}}$ is an elliptic unit;
the integer $c>0$ is  the smallest positive integer in the conductor ideal of $\psi$.

\medskip
\noindent
{\bf 2.} {\em The $p$-adic Rankin $L$-function attached to a cusp form and an imaginary quadratic field.}  For any Hecke character $\psi$ of $K$ of infinity 
type $(\kappa_1,\kappa_2)$, let 
$$L(f,\psi,s) := L\left(\pi_f \times \pi_\psi, s-\frac{\kappa_1+\kappa_2+1}{2}\right)$$ denote the $L$-series associated with
 the product of the global automorphic representations attached to the weight two cuspform $f$ and the Hecke character $\psi$, 
 normalised conveniently. As usual, $L(f,\psi,s) = \prod_q L_{(q)}(q^{-s})$ is defined as a product of Euler factors ranging 
 over the set of prime numbers. 
%The Euler factors at primes $q\nmid N$ are exactly the same as that of the
 % Rankin $L$-series $L(\theta_\psi\otimes f,s)$ introduced above, but may differ at primes $q\mid N$: 
 %see \cite[\S 15]{Jac} for more details.

Fix a positive integer $c\geq 1$ which is relatively prime to $pN_f$.
Let $\Sigma_{f,c}$ be the set of Hecke characters $\psi\in \Sigma$ of conductor $c$ and trivial central 
character for which $L(f, \psi^{-1},s)$ is self-dual and has $s=0$ as its central critical point.
 This set can be expressed as the disjoint union of the three subsets 
$$
\Sigma^{(1)}_{f,c} = \{ \psi \in \Sigma_{f,c} \,\mbox{ of infinity type }\,(1,1)\},
$$
$$
\Sigma^{(2)}_{f,c} = \{ \psi \in \Sigma_{f,c} \,\mbox{ of infinity type }\,(\kappa+2,-\kappa), \kappa\geq 0\},
$$
and
$$
\Sigma^{(2')}_{f,c} = \{ \psi \in \Sigma_{f,c} \,\mbox{ of infinity type }\,(-\kappa,\kappa+2), \kappa\geq 0\}.
$$
The three sets $\Sigma^{(1)}_{f,c}$, $\Sigma^{(2)}_{f,c}$ and $\Sigma^{(2')}_{f,c}$ are each dense in the completion $\hat{\Sigma}_{f,c}$ of  $\Sigma_{f,c}$ with respect to the $p$-adic compact open topology as explained in \cite[\S 5.2]{BDP1}. As shown in loc.\,cit., there exists a unique $p$-adic analytic function
\begin{equation*}
L_p(f,K): \hat{\Sigma}_{f,c} \lra \C_p
\end{equation*}
interpolating the critical values $L(f,\psi^{-1},0)$ for $\psi \in \Sigma^{(2)}_{f,c}$, suitably normalized. 
The function  $L_p(f,K)$ is referred to 
as the $p$-adic Rankin $L$-function attached to the pair $(f,K)$. 
By \cite[Proposition 3.11]{BDP2}, the interpolation formula reads 
precisely as follows: for any character $\psi \in 
\Sigma^{(2)}_{f,c}$ of infinity type $(\kappa+2,-\kappa)$,
\begin{equation}\label{int-BDP}
L_p(f,K)(\psi) = \mathfrak{a}(f,\psi)  \times \mathfrak{e}^2(f,\psi) \times \mathfrak{f}(f,\psi)  \times \frac{ \Omega_p^{4\kappa+4}  }{\Omega^{4\kappa+4}} \times L(f,\psi^{-1},0),
\end{equation}
where 
\begin{itemize}
\item $\mathfrak{a}(f,\psi) = \kappa!(\kappa+1)! \, \pi^{2\kappa+1}$, $\quad \mathfrak{e}(f,\psi) = 1-a_p(f) \psi^{-1}(\bar\wp)  + \psi^{-2}(\bar\wp) p\quad $ and

\item $\mathfrak{f}(f,\psi)  = (2/c\sqrt{D_K})^{2\kappa+1}  \cdot \prod_{q\mid c} \frac{q-\chi_K(q)}{q-1}   \cdot \omega(f,\psi)^{-1}$. 
\end{itemize}

Here $\omega(f,\psi)$ denotes the complex number defined in \cite[(5.1.11)]{BDP1}. It is an algebraic scalar of absolute value $1$ relative to any complex embedding of 
$\bar\Q$.
%If $\psi$ is an anticyclotomic character, then $\omega(f,\psi)\in \{\pm 1\}$ because $L(f,\psi^{-1},s)$ is self-dual. 

Consider now a Hecke character $\psi \mathbf{N}_K \in \Sigma^{(1)}_{f,c}$, 
 where $\psi$ has finite order. The character $\psi \mathbf{N}_K$ lies outside 
 the region of interpolation of $L_p(f,K)$ and the main theorem of 
 \cite{BDP1}---cf.\,\cite{Brooks} 
%also \cite[Theorem 3.12]{BDP2} and 
for a proof of it under the more relaxed Heegner hypothesis imposed  here---asserts that
\begin{equation}\label{thm:bdp-gz}
L_p(f,K)(\psi \mathbf{N}_K) =  \left(1-\frac{a_p(f)}{\psi(\bar\wp) p} + \frac{1}{\psi^{2}(\bar\wp) p}\right)^2 \times  \log_{\omega_E}(P_{\bar\psi})^2.
\end{equation}

%% We now turn to the  comparison 
%% of the above two  $p$-adic $L$-functions with 
%% the Garrett-Hida $p$-adic $L$-function of the previous section, which is the key to the proof of  Theorem \ref{main-theta}.
We now turn to the comparison of the Rankin $p$-adic $L$-function with the Garrett-Hida $p$-adic $L$-function of
the previous section: the Katz $p$-adic $L$-function then arises indirectly from the ratio of periods \eqref{eqn-sigmak} 
used in constructing these two $p$-adic $L$-functions.
 
\medskip

Recall the theta series $g$ of weight $1$, level $N_g=D_K N_{K/\Q}(\mathfrak c(\psi_g))$ and character $\chi^{-1}$ appearing in the statement of Theorem \ref{main-theta}. 

Let $\hg$ be the primitive cuspidal Hida family of theta series of tame level $N_g$ and tame character $\chi_g=\chi^{-1}$  constructed in e.g.\,\cite[p.\,235-236]{hida-book} and \cite[\S 5]{Gh}, specialising in weight $1$ to the $p$-stabilisation $g_{\alpha} $ of $g$ satisfying $U_p(g_{\alpha}) = \alpha_g\cdot g_{\alpha}$.  It can be described 
explicitly by fixing a Hecke character $\lambda$ of infinity type {\color{blue} $(0,1)$} and conductor {\color{blue} $\bar\wp$}. 
Let $\Q(\lambda)$ denote the number field generated by the values of $\lambda$ and let $\Q_p(\lambda)$
 be the completion of $\Q(\lambda)$ at the prime determined by an embedding $\Q(\lambda)\hookrightarrow \bar\Q_p$
 which is fixed at the outset. Writing $\cO_{\Q_p(\lambda)}$ for its ring of integers, decompose its group of units
   as $\cO^\times_{\Q_p(\lambda)}=\mu\times W$, where $\mu$ is finite and $W$ is free over $\Z_p$. Let $\langle \cdot\rangle: \cO^\times_{\Q_p(\lambda)} \rightarrow W$ denote the natural projection.

For every integer $k\geq 1$
%, $k\equiv 1$ mod $(p-1)$ in $\mathcal U_g$, 
define
\begin{equation} \nonumber
%% \label{hecke-chars}
\psi^{(p)}_{g,k-1}=\psi_g\langle \lambda \rangle^{k-1}.
\end{equation}

This definition does not depend on the choice of $\lambda$, for any other such choice $\lambda'$ differs from $\lambda$ by a finite character and hence $\langle \lambda' \rangle = \langle \lambda \rangle$. Let now $\psi_{g,k-1}$ be the Hecke character defined by
\begin{equation} \nonumber
\psi_{g,k-1}(\mathfrak q) = \begin{cases} \psi^{(p)}_{g,k-1}(\mathfrak q) & {\color{blue} \mbox{ if } \mathfrak q \ne  \bar\wp} \\ \chi^{-1}(p) p^{k-1}/\psi^{(p)}_{g,k-1}(\wp) & {\color{blue} \mbox{ if } \mathfrak q = \bar\wp}.
\end{cases}
\end{equation}

The Hecke character $\psi_{g,k-1}$ has conductor $\mathfrak c(\psi_g)$ and infinity type $(k-1,0)$ and by construction $\psi_{g,k-1}(\wp)$ is a $p$-adic unit for all $k\geq 1$. For any $k\in \Z_{\geq 1} \cap\mathcal U_g$, the weight $k$ specialisation of 
$\hg$ is $\theta_{\psi^{(p)}_{g,k-1}}$, which is the ordinary $p$-stabilisation of the theta series $g_k=\theta_{\psi_{g,k-1}}$. 
In particular $\psi_{g,0}=\psi_g$ is the character we considered at the outset and $g_1=g$. 

Together with $g$, recall also the eigenforms $f\in S_2(N_f)$ and $h\in M_1(N_h,\chi)$ of the statement. Let $(\testg,\testf,\testh)$ be Watson-Woodbury's test vector of level $N$ associated to these three modular forms as described in Remark \ref{WW-classic}, and let also $\htg$ denote Watson-Woodbury's $\Lambda$-adic test vector discussed in Remark \ref{WW-adic}. Recall that, as discussed in Chapter
 \ref{tri}, $W_v=C_v(\testg_k,\testf,\testh)\in \Q^\times$ are constants independent of $k$ for all $v\mid N$. Moreover, the results of \cite{Wood2} guarantee that in this setting $C_\infty(\testg_k,\testf,\testh)=1$ for all $k$. 

Recall the triple-product $p$-adic $L$-functions introduced in the previous chapter. Denote by  $k\mapsto \Lp^g(\htg,\testf_2,\testh_1)(k)$ the
restriction of $\Lp^g(\htg,\htf,\hth)$ to
$\mathcal U_g\times \{ 2\} \times \{1\}$. Because it does not involve the variation of $f_2$ in a Hida family, this restriction has the virtue of being defined a bit more
generally, without any ordinariness assumptions on $f_2$. 

Proposition \ref{padicL-Garrett} implies, for all $k \in \mathcal U^\circ_g$, that
\begin{equation*}
\begin{split}
 \Lp^g(\htg,\testf,\testh)(k)&  \cdot \Lp^g(\htg^*,\testf,\testh^*)(k)  \\
&   = \mathfrak{e}(k,2,1)^2 \cdot  \mathfrak{f}(k,2,1) \cdot
\mathfrak{a}(k,2,1) 
\cdot  \prod_{v\mid N} W_v   
\cdot  \frac{L(g_k \otimes f \otimes h,\frac{k+1}{2})}{\langle g^*_k,g^*_k\rangle^{2}}.
\end{split}
\end{equation*}
Here the factors on the right are described in 
Proposition \ref{padicL-Garrett}. In particular, $$\mathfrak{f}(k,2,1) =A\cdot 2^B \quad \mbox{ 
and } \quad \mathfrak{a}(k,2,1) = \left(\left(\frac{k-3}{2}\right)!\left(\frac{k-1}{2}\right)! \right)^2 \cdot \pi^{-2k}.$$


For every integer $k\geq 1$ define the Hecke characters
\begin{equation}\nonumber
\Psi_{gh}(k)= {\color{blue} \psi^{-1}_{g,k-1}} \cdot \psi_h^{-1} \cdot \mathbf{N}_K^{\frac{k+1}{2}} \quad \mbox{and} \quad \Psi_{gh'}(k)={\color{blue} \psi^{-1}_{g,k-1}} \cdot  (\psi'_h)^{-1} \cdot \mathbf{N}_K^{\frac{k+1}{2}}.
\end{equation}

For $k=1$ the characters $\Psi_{gh}(1)= \mathbf{N}_K\cdot (\psi_{g} \psi_h)^{-1}$ and $\Psi_{gh'}(1)= \mathbf{N}_K\cdot (\psi_{g}\psi'_h)^{-1}$ lie in $\Sigma^{(1)}_{f,c}$.
%and hence do not belong to region of interpolation $\Sigma^{(2)}_{f,c}$ that we used to define the $p$-adic $L$-function, for
For weights $k\in \mathcal U_g^\circ$, $k>1$, the characters $\Psi_{gh}(k), \Psi_{gh'}(k)$ belong to $\Sigma^{(2)}_{f,c}$ with $c=\mathrm{lcm}(c_1,c_2)$, because their infinity type is $(\kappa+2,-\kappa)$ with $\kappa = \frac{k-3}{2} \geq 0$.

It follows that for weights $k\in \mathcal U_g^\circ$, $k>1$, the values $L_p(f,K)(\Psi_{gh}(k))$, $L_p(f,K)(\Psi_{gh'}(k))$ and $\Lp^g(\htg,\testf,\testh)(k)$ all lie in the region of interpolation. 
An inspection of the Euler factors defining their classical counterparts yields the following relationship between critical values: for every $k\in U_g$ there is a fudge factor $\mathfrak f_0(k)\ne 0$ such that
\begin{equation}\label{eq}
L\left(g_k \otimes f \otimes h,\frac{k+1}{2}\right) =  \mathfrak f_0(k)\cdot L(f, \Psi_{gh}^{-1}(k),0) \cdot L(f, \Psi_{gh'}^{-1}(k),0),
\end{equation}
and we shall prove the theorem by letting $k\in \cU^\circ_g \cap \Z_{>1}$ in \eqref{eq} tend $p$-adically to $1$ within $\mathcal U_g$, so that the triplets $(k,2,1)$ tend to $(1,2,1)$. 

Several of the fudge factors that arise in comparing different $L$-values, 
viewed as functions of weight space, 
satisfy the property described in the following definition:
\begin{definition}
 Let $F$
 be a number field.
A function
$$
\mathfrak f:   \cU^\circ_g \cap \Z_{>1}\, \lra \, \bar\Q
$$
is {\em $F$-admissible} if it extends to an element of the fraction field $\mathcal L_g$ of the algebra $\Lambda_g=\mathcal O(\mathcal U_g)$ of Iwasawa functions  
on $\mathcal U_g$, and if  $\mathfrak f(1)$ is well-defined and belongs to $F^\times$. 
\end{definition}
As in the statement of the definition, an $F$-admissible function  $\mathfrak f$ 
will customarily be identified with the 
 function on $\mathcal U_g \subseteq \Z_p$ which it induces. In order to make the calculations described below easier to follow,
vowels will generally be used to denote non-admissible functions 
(such as rational functions of $p^k$, $\pi^k$ or $k!$), and 
consonants  to denote
 admissible ones (such as  $q^k$ or $a_q(g_k)$ for some prime number $q\ne p$). 


Instances of admissible functions were already encountered in the proof of Proposition \ref{iter=Lp}: using the notation employed there, the
functions $\lambda_d$ were shown in loc.\,cit.\,to be $\Q(g_1,h_1)$-admissible. Recall also that the fudge factor $\mathfrak f_0(k)$ appearing in \eqref{eq} accounts for the discrepancy of the Euler factors at the
 primes $q\mid c D_K N_f$ in the definitions of the above $L$-series. It may be computed explicitly, but we content to note here that it is $\Q(\psi_g,\psi_h)$-admissible because $\mathfrak f_0(k)$ is a rational
   function on $q^k$ for $q\mid c D_K N_f$ with coefficients
    in $\Q(\psi_g,\psi_h)$, with $\mathfrak f_0(k) \ne 0$ for all $k\in \Z$.

For every $k\in \cU^\circ_g$, $k>1$, define
\begin{equation}\label{eqn-sigmak}
\sigma(k):= \langle g^*_k,g^*_k\rangle \times  \left(\frac{\pi \Omega_p}{\Omega}\right)^{2k-2}
\end{equation}
and
$$
\mathfrak{e}_1(k) = \frac{\mathfrak{e}(k,2,1)^2}{ \mathfrak{e}^2(f,\Psi_{gh}(k)) \mathfrak{e}^2(f,\Psi_{gh'}(k))}, \quad \mathfrak f_1(k) =\frac{\prod_{v\mid N} W_v \cdot \mathfrak f_0(k) \cdot  \mathfrak{f}(k,2,1)}{\mathfrak{f}(f,\Psi_{gh}(k)) \mathfrak{f}(f,\Psi_{gh'}(k))}.
$$

\begin{lemma}\label{admis}
The  function $\mathfrak f_1(k)$ is $K(\psi_g,\psi_h)$-admissible and
\begin{equation*}
\begin{split}
\Lp^g(\htg,\testf,\testh)(k) & \cdot \Lp^g(\htg^*,\testf,\testh^*)(k)  \cdot   \sigma(k)^2    \\
  & =  \mathfrak e_1(k)\mathfrak f_1(k)   \cdot  L_p(f,K)(\Psi_{gh}(k))    \cdot  L_p(f,K)(\Psi_{gh'}(k)).
\end{split}
\end{equation*}
\end{lemma}

\begin{proof} We already saw that $\mathfrak f_0(k)$ is  $\Q(\psi_g,\psi_h)$-admissible, and Proposition \ref{propHK} (b) (ii) implies that $ \mathfrak{f}(k,2,1)=A\cdot 2^{B(k)}$ is $\Q$-admissible because $p\ne 2$ and $B(k)$ is a linear function of $k$. It remains to prove that the denominator of $\mathfrak f_1(k)$ is $K(\psi_g,\psi_h)$-admissible. Recall from \eqref{int-BDP} that $\mathfrak{f}(f,\psi)  = (2/c\sqrt{D_K})^{k-2}  \cdot \prod_{q\mid c} \frac{q-\chi_K(q)}{q-1}   \cdot \omega(f,\psi)^{-1}$ for any Hecke character $\psi\in \Sigma^{(2)}_{f,c}$ of infinity type $(\frac{k-3}{2}+2,-\frac{k-3}{2})$. The first two factors are $K$-admissible, because $p$ is assumed not to divide $2 N$. Hence, in order to conclude, we need to show that the functions $\omega(f,\Psi_{gh}(k))$ and $\omega(f,\Psi_{gh'}(k))$, or at least their product, is admissible.
By \cite[(5.1.11)]{BDP1}: 
\begin{equation}\label{last-task1}
\omega(f,\Psi_{gh}(k)) \cdot \omega(f,\Psi_{gh'}(k))  = \frac{\mathbf{N}_K^{2k-2}(\mathfrak b) N^{k-1}}{(\psi_{g,k-1})^2\psi_h \psi'_h(\mathfrak b)\cdot b^{2k-2}},
\end{equation}
where $(\mathfrak b,pNc)=1$ and $\mathfrak b\cdot \mathfrak N =(b)$.
This shows that the function in \eqref{last-task1} is $K(\psi_g,\psi_h)$-admissible, because $p\nmid \mathfrak b \mathfrak N$ and its value at $k=1$ is the non-zero algebraic number \linebreak $((\psi_{g})^2\psi_h \psi'_h(\mathfrak b))^{-1} = (\psi_{g})^2\psi_h \psi'_h(\mathfrak N)$, which does not depend on the choice of $\mathfrak b$.





Finally, the factorisation formula of Lemma \ref{admis} is a direct consequence of the interpolating formulae \eqref{ggg} and \eqref{int-BDP}, combined with \eqref{eq}, once one observes that the factorial terms and powers of $\pi$ on both sides cancel out.
\end{proof}




At this point it is convenient to invoke the following classical formula: 
\begin{lemma}\label{sym-formula} There is a $K$-admissible function $ \mathfrak f_2(k)$ such that
$$\langle g_k^*,g_k^*\rangle = \frac{(k-1)!}{\pi^{k}} \cdot \mathfrak f_2(k) \cdot L(\psi^2_{g,k-1}\cdot  \chi,k).$$
\end{lemma}
\begin{proof}
Since the $L$-series of the adjoint of a CM-form factors as  $$L(\mathrm{Ad}(g^*_k),s) = L(\mathrm{Ad}(g_k),s) = L(\chi_K,s-k+1) L(\psi^2_{g,k-1} \cdot \chi^{-1}_g,s)$$ and $\chi_g=\chi^{-1}$, Lemma \ref{sym-formula} follows directly from  \eqref{Ad-ff} after evaluating at $s=k$ and applying Dirichlet's class number formula $L(\chi_K,1)= 2\pi h_K (|\cO^\times_K|\sqrt{D})^{-1}$.
\end{proof}



It now becomes clear how Katz's $p$-adic $L$-function enters in the calculation: note that 
\begin{equation}\label{aboveformula}
L(\psi^2_{g,k-1} \cdot \chi,k)=\mathfrak f_3(k) \cdot L_c(\psi^2_{g,k-1}\cdot \chi \cdot \mathbf{N}_K^{-k},0),
\end{equation}
where again $\mathfrak f_3(k)$ accounts for the discrepancy between the Euler factors at primes $q\mid c D_K$ defining both $L$-series above, and hence is $\Q(\psi_g,\psi_h)$-admissible. 

Define $$\Psi_g(k) := {\color{blue} \psi^{-2}_{g,k-1}} \,\chi^{-1} \,\mathbf{N}_K^k.$$ 

\begin{lemma}
\label{sym-formula-Lp} 
The following equality holds for every $k\in \cU^\circ_g$, $k>1$:
\begin{equation*}
\begin{split}
L_p(K)( \Psi_{g}(k)) & =   \frac{\mathfrak e(\Psi_{g}(k)) \cdot  \mathfrak f(\Psi_{g}(k))}{\mathfrak f_2(k) \cdot \mathfrak f_3(k)} \times \langle g^*_k,g^*_k\rangle \times  \left(\frac{\pi \Omega_p}{\Omega}\right)^{2k-2}.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
In light of Katz's interpolation formula \eqref{int-Katz}, since the infinity type of $\Psi_{g}(k)$ is $(\kappa_1,\kappa_2)=(k,-k+2)$, 
it follows that
\begin{equation*}
 L(\psi^2_{g,k-1}\cdot \chi,k) \, = \,  \frac{\pi^{-k+2}}{(k-1)!} \cdot  \frac{1}{\mathfrak e(\Psi_{g}(k))} \cdot   \frac{\mathfrak f_3(k) }{\mathfrak f(\Psi_{g}(k)) }  \cdot \frac{\Omega^{2k-2}}{\Omega_p^{2k-2}} \cdot L_p(K)( \Psi_{g}(k)).
\end{equation*}
Invoking Lemma \ref{sym-formula} and \eqref{aboveformula}, 
 one finds that the powers of $\pi$ and the factorials (which, as functions of $k$, are not admissible) cancel out in the left and right hand sides of
 the formula in Lemma \ref{sym-formula-Lp},
which then follows.
\end{proof}


For all integers $k\in \cU^\circ_g$, $k>1$, define $$\omega(k) =  I(\testg^*_k,\testf,\testh^*)\,/\,I(\testg_k,\testf,\testh)
%= J(\testg^*_k,\testf,\testh^*)/J(\testg_k,\testf,\testh)
$$ as in \eqref{J-omega} and
\begin{equation} \nonumber
%% \label{foo}
\mathfrak{f}(k) = \frac{\mathfrak f_1(k)}{\omega(k)} \left( \frac{\mathfrak{f}(\Psi_{g}(k))}{\mathfrak f_2(k) \mathfrak f_3(k)} \right)^2.
\end{equation}

Let $\mathfrak g(\chi)\in \bar\Q \subset \C_p $ denote the Gauss sum associated to the character $\chi$ and define the number field $L_0=K(\psi_g,\psi_h,\lambda_g,\lambda_h)$, where $\lambda_g$ and $\lambda_h$ are the pseudo-eigenvalues of the eigenforms $g$ and $h$ in the sense of \cite{atkin-li}, characterized by the equations $W_{N_g}(g) = \lambda_g g^*$ and $W_{N_h}(h) = \lambda_h h^*$. By \cite[Theorem 1.1]{atkin-li}, $L_0$ is a finite extension of $\Q$. Theorem \ref{main-theta} will follow from the next result:

\begin{theorem}
\label{factor} 
The function 
$\mathfrak{f}(k)$ is $L_0$-admissible and the following factorisation of $p$-adic $L$-series is valid
\begin{equation}\label{fact0}
\begin{split}
\Lp^g(\htg,\testf,\testh)(k) ^2\,\cdot &  \, L_p(K)(\Psi_{g}(k))^2  = \\
&  \mathfrak{f}(k) \,\cdot \, L_p(f,K)(\Psi_{gh}(k)) \, \cdot \,L_p(f,K)(\Psi_{gh'}(k)).
\end{split}
\end{equation}
Moreover, there exist explicit elements $\lambda_1$, $\lambda_2$, $\lambda_3\in \Q(\psi_g,\psi_h)^{\times}$ such that
\begin{equation}
\label{out}
\begin{split}
 L_p(f,K)(\Psi_{gh'}(1)) & = \lambda^2_1\cdot \log_{E,p}(P_1)^2,  \\
   L_p(f,K)(\Psi_{gh}(1)) &= \lambda_2^2\cdot \log_{E,p}(P_2)^2, \\ 
  L_p(K)(\Psi_{g}(1)) & = \lambda_3\cdot \log_p(u_{g_\alpha}). 
\end{split}
\end{equation} 
\end{theorem}

\begin{proof} We first prove \eqref{out}: note that $\Psi_{g}(1) = (\psi_g)^{-2}\cdot \chi^{-1} \cdot \mathbf{N}_K = \frac{\psi'_g}{\psi_g} \mathbf{N}_K$.
 By the functional equation of Katz's $p$-adic $L$-function quoted above, since ${\color{blue}\bar\psi_0 = \frac{\psi'_g}{\psi_g}}$ is anticyclotomic we have
  $L_p(K)(\bar\psi_0 \mathbf{N}_K) = L_p(K)(\bar\psi_0)$
   and hence \eqref{Katz-GZ} applies, {\color{blue} yielding $L_p(K)(\Psi_{g}(1)) = \lambda_3\log_p(u_{g_\alpha})$} for some explicit value of $\lambda_3\in \Q(\psi_g,\psi_h)^{\times}$. 
Moreover, 
$$
{\color{blue}
\Psi_{gh}(1) = (\psi_g)^{-1} \cdot \psi_h^{-1} \cdot \mathbf{N}_K = \bar\psi_1\cdot \mathbf{N}_K, \qquad 
   \Psi_{gh'}(1) = (\psi_g)^{-1} \cdot (\psi_h')^{-1} \cdot \mathbf{N}_K = \bar\psi_2\cdot \mathbf{N}_K,
}
$$
    and hence
     \eqref{thm:bdp-gz} can be applied, thereby proving \eqref{out}. 

In order to prove the admissibility of $\mathfrak f$ and the factorisation formula, recall we already saw in Lemma \ref{admis} that $\mathfrak f_1$ is $K(\psi_g,\psi_h)$-admissible, and likewise for $\mathfrak f_2$ and $\mathfrak f_3$. An inspection of the fudge factor $\mathfrak{f}(\Psi_{g}(k))$ shows that it also is. Hence it remains to prove that $\omega(k)$ is $L_0$-admissible. Let us first show that $\omega$ interpolates to a meromorphic Iwasawa function on $\mathcal U_g$ such that \eqref{fact0} holds and $\omega(1)\in \C_p^\times$ is a non-zero $p$-adic number, leaving for later that $\omega(1)$ actually lies in $L_0^\times$. 

Note that Lemmas \ref{admis} and \ref{sym-formula-Lp} give rise to the following identity for all integers $k\geq 0$:
\begin{equation} 
\label{fact1}
\begin{split}
\Lp^g(\htg, &\testf,\testh)(k)   \cdot \Lp^g(\htg^*,\testf,\testh^*)(k)\,\cdot  
 L_p(K)(\Psi_{g}(k))^2 
 \\ 
 & = \mathfrak e(k) \cdot \mathfrak f_1(k) \cdot \left( \frac{\mathfrak{f}(\Psi_{g}(k))}{\mathfrak f_2(k) \mathfrak f_3(k)} \right)^2 \, \cdot \, L_p(f,K)(\Psi_{gh}(k))\,  \cdot \,L_p(f,K)(\Psi_{gh'}(k)),
\end{split}
\end{equation}
where 
$$
\mathfrak e(k) := \mathfrak e(\Psi_{g}(k))^2 \mathfrak e_1(k)  
= \left(\,\frac{\mathfrak e(\Psi_{g}(k)) \cdot \mathfrak{e}(k,2,1)}{ \mathfrak{e}(f,\Psi_{gh}(k)) \mathfrak{e}(f,\Psi_{gh'}(k))}\,\right)^2.
$$

An inspection  of  the factors appearing above shows that
\begin{equation}\label{e=1}
\mathfrak e(k)=1 \quad \mbox{ for all } k\geq 0.
\end{equation}
More precisely, one checks that the four factors in the numerator $\cE(g_k,f,h)$ of $\mathfrak{e}(k,2,1)$ are equal to the four factors appearing in $\mathfrak{e}(f,\Psi_{gh}(k)) \mathfrak{e}(f,\Psi_{gh'}(k))$, and hence cancel out. In addition, the two factors showing up in $\mathfrak e(\Psi_{g}(k))$ are identical to those in the denominator  of $\mathfrak{e}(k,2,1)$, and thus also cancel out. 

Since the unit $u_{g_\alpha}$ and the points $P_1$ and $P_2$ are non-torsion, it also follows from \eqref{out} and \eqref{fact1} 
that neither $\Lp^g(\htg,\testf,\testh)(k)$ nor $\Lp^g(\htg^*,\testf,\testh^*)(k)$ vanish at $k=1$. Note that \eqref{int-Garrett} implies that 
\begin{equation}\label{omm}
\omega(k) = \Lp^g(\htg^*,\testf,\testh^*)(k)\,/\,\Lp^g(\htg,\testf,\testh)(k),
\end{equation}
from which it follows that $\omega \in \mathcal L_g$, with no poles at $k=1$ and satisfying $\omega(1)\ne 0$. The factorisation formula claimed in the theorem is also a consequence of \eqref{fact1}, \eqref{e=1} and \eqref{omm}.

To finish the proof, we need to show that $\omega(1)\in L_0$. To do that,
%note first that $\omega(k,\ell,m) =  J(\testg^*_k,\testf,\testh^*)/J(\testg_k,\testf,\testh)$ because $\langle \testg^*_k,\testg^*_k\rangle=\langle \testg_k,\testg_k\rangle$. By %\cite[Lemma 2.12]{DR1}, both the numerator and denominator in this expression belong to $\Q(g_k,f_\ell,h_m)$, and hence so does $\omega(k)$. 
let $W_N$ denote the Atkin-Lehner operator induced by the matrix $\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}$. As shown in \cite{atkin-li} and \cite[Ch.\,X]{hida-book}, it commutes with the Shimura-Maass operator, and its adjoint with respect to the Petersson scalar product is $-W_N$. 
This implies that 
$$I(\testg_k,\testf,\testh) = I(W_N(\testg_k),W_N(\testf),W_N(\testh)).$$

By the formulae in \cite{atkin-li}, $W_N(\testh(q))$ is a linear combination with coefficients in $\Q(h)$ of the modular forms $\lambda_h\cdot h^*(q^d)$ where $d$ ranges over the positive divisors of $N/N_h$. The analogous claim, with similar notations, holds for $g_k$ and $f$. Since $\lambda_f = \pm1$, it thus follows from e.g.\,\cite[Lemma 2.12]{DR1} that the ratio 
$$
\omega(k) = I(\testg^*_k,\testf,\testh^*)\,/ \,I(W_N(\testg_k),W_N(\testf),W_N(\testh))
$$
lies in $\Q(g_k,f,h,\lambda_f,\lambda_g,\lambda_h) = \Q(\psi_g,\psi_h,\lambda_g,\lambda_h)=L_0$.  Theorem \ref{factor} follows.
Let us note in passing that in those cases where $a_{N_g}(g_k)$ and $a_{N_h}(h)$ are both non-zero, the pseudo-eigenvalues of $f$, $g_k$ and $h$ are
$$
\lambda_f= -a_{N_f}(f), \quad \lambda_g=N_g^{\frac{k-2}{2}}\mathfrak g(\chi^{-1}) / a_{N_g}(g_k), \quad \lambda_h= \mathfrak g(\chi) / (\sqrt{N_h}\cdot a_{N_h}(h))
$$ 
by \cite[Theorem 2.1]{atkin-li}.
Note that, since $p\nmid N$ and the coefficients $a_{N_g}(g_k)$ are Iwasawa functions on $\mathcal U_g$, the above formulae give rise to a rather explicit expression of $\omega(k)$ as  $p$-adic continuous function.
\end{proof}

To conclude the proof of Theorem \ref{main-theta}, recall that Proposition \ref{iter=Lp} recasts $\Lp^g(\htg,\testf,\testh)(1)$ as the $p$-adic iterated integral.
 Combined with Theorem  \ref{factor} 
we deduce that
\begin{equation}
\label{step2}
\left( \int_{\testgamma_{\alpha}} \! \testf \cdot \testh\right)^2 \cdot \log_p(u_{g_\alpha})^2 \,=\, \mathfrak{f}(1) \log_{E,p}(P_1)^2 \log_{E,p}(P_2)^2
\end{equation}
up to {\em squares} in $\Q(\psi_g,\psi_h)^\times$. Set $L=L_0(\sqrt{\mathfrak f(1)})$. Taking square-roots in \eqref{step2}
shows that  
$$
\int_{\testgamma_{\alpha}} \! \testf \cdot \testh = \lambda\cdot  \frac{  \log_{E,p}(P_1) \log_{E,p}(P_2)}{\log_p(u_{g_\alpha})}
$$
for some $\lambda \in L^\times$,
thereby proving Theorem \ref{main-theta}.






\subsection{Numerical evidence for inert primes}
\label{p-inert}
The proof of Conjecture $\shi$
 based on   Katz-style 
 $p$-adic $L$-series and their factorisations
 breaks down when  
the  prime $p$ is {\em inert} in $K$. 
Yet the   units and points that arise   can still be expressed in terms
of elliptic units and Heegner points, at least in the rank
$(1,1)$ setting, making
 this   a  natural first testing ground for Conjecture $\shi$. 

\subsubsection{General remarks}

Retain all the notations and assumptions of
Section \ref{sec:proof}, but assume 
now that the prime $p$ is {\em inert} in $K$. 

Let $K_p$ denote the completion of $K$ at $p$, a quadratic 
unramified extension of $\Q_p$. Since the frobenius element $\sigma_p$ is a reflection in the dihedral group $\Gal(H/\Q)$, the prime $p$ splits completely in $H/K$ and the
 choice of $\sigma_p$ amounts to choosing 
an embedding $H\hookrightarrow K_p$, which shall be fixed from now on. 

Because  $\varrho_g$ and $\varrho_h$ have dihedral projective image,
it follows that 
%$\sigma_p$ acts on  $V_g$ (resp.~on $V_h$)  with eigenvalues $\lambda$ and $-\lambda$ (resp.~$\lambda^{-1}$ and $-\lambda^{-1}$), 
%Label the four values 
$$\alpha_g=\lambda,   \quad \beta_g=-\lambda, \qquad \alpha_h=\lambda^{-1}, \quad \beta_h= -\lambda^{-1}$$
for a suitable root of unity $\lambda$, so that   
$$\alpha_g\alpha_h  = 1, \qquad \alpha_g \beta_h  = -1,$$
as shall be assumed from now on.
The
 decomposition 
 $$V_{gh}\simeq V_{\psi_1} \oplus V_{\psi_2}$$ 
 of $\Gal(H/\Q)$-modules continues to hold, but now 
 $\sigma_p$ acts on both $V_{\psi_1}$ and $V_{\psi_2}$ as reflections,
  with eigenvalues $ 1$ and $-1$. 
   
The asymmetry in the roles of $g$ and $h$ in the definition of the $p$-adic iterated integral  continues to determine a pair $(\psi_1,\psi_2)$ of global characters of
$G_K$ by requiring that $\psi_1$ and $\psi_2$ ``each be divisible by $\psi_g$". 
This pair is uniquely defined  up to the action of $\Gal(K/\Q)$, since replacing $\psi_g$ by $\psi_g'$   has the effect of
replacing $(\psi_1,\psi_2)$ by $(\psi_2',\psi_1')$. Fix  $v_{\psi_j}\in V_{\psi_j}$
to be a  non-zero
vector  in the one-dimensional $\psi_j$-eigenspace
 for the action of $G_K$ on $V_{\psi_j}$. With this choice in hand, set
 $v_{\psi_j'} := \sigma_p v_{\psi_j}$, as in the previous section. 
Given any $P\in \Hom(V_{\psi_j},E(H)_L)$, we shall  write
$$P_{\psi_j}:= P(v_{\psi_j}), \qquad P_{\psi_j'} := P(v_{\psi_j'}).$$
\begin{lemma}
The regulator $\det(R_{g_\alpha}(E,\varrho_{gh}))$ is given by
\begin{enumerate}
\item 
If    $r(E,V_{\psi_1})= r(E,V_{\psi_2}) = 1$,
$$ \det R_{g_\alpha}(E,\varrho_{gh}) = \log_{E,p}(P_{\psi_1}) \cdot \log_{E,p}(Q_{\psi_2})
- \log_{E,p}(P_{\psi_1'}) \cdot \log_{E,p}(Q_{\psi_2'}).$$
In particular, if  $\psi_1$  (say) 
is a quadratic (genus) character, 
so that the point $P_{\psi_1}$ is a $\sigma_p$-eigenvector with eigenvalue $\iota=\pm 1$, 
then
$$ \det R_{g_\alpha}(E,\varrho_{gh}) = \log_{E,p}(P_{\psi_1}) \cdot \log_{E,p}(Q_{\psi_2} - \iota  Q_{\psi_2'}).$$
\item 
If   $r(E,V_{\psi_1})= 2$ and  $r(E,V_{\psi_2})=0$,
 $$ \det R_{g_\alpha}(E,\varrho_{gh}) = \log_{E,p}(P_{\psi_1}) \cdot \log_{E,p}(Q_{\psi_1'})
- \log_{E,p}(P_{\psi_1'}) \cdot \log_{E,p}(Q_{\psi_1}),$$
where $(P,Q)$ is a basis for $\Hom_{G_\Q}(V_{\psi_1},E(H)_L)$.
\end{enumerate}
\end{lemma}

Observe in particular that the regulator vanishes in the special case where
there are no global points  $(P_1,P_2)\in E(H)_L^{\varrho_1} \times E(H)_L^{\varrho_2}$ 
belonging to distinct eigenspaces for the action of $\sigma_p$; this exceptional scenario in which the regulator is degenerate occurs precisely when  
\begin{equation}
\label{non-adm}
\dim(\varrho_1) = \dim(\varrho_2)=1 \mbox{ (so that } \psi_1^2 = \psi_2^2=1) \mbox{ and } \varrho_1(p) = \varrho_2(p).
\end{equation}
If \eqref{non-adm} holds, Conjecture $\shi$ predicts that the  iterated integrals
attached to $(f,g,h)$ vanish, in spite of the running hypothesis
 $r_{\an}(E,\varrho_1) = r_{\an}(E,\varrho_2)=1$. 
See Example \ref{ex:155} below 
%associated to the curve $E_a$ and prime $p = 7$
for a numerical verification of this interesting phenomenon.

Suppose for the rest of this section that \eqref{non-adm} is not satisfied, 
so that, switching $\varrho_1$ and $\varrho_2$ if necessary,
 it may be  assumed that $1$ and $-1$
are eigenvalues of $\varrho_1(\sigma_p)$   and $\varrho_2(\sigma_p)$ respectively. 
 Recall furthermore the  points $P_{\psi_i}^\pm$ defined from   Heegner points
  in Lemma \ref{lemma:heegner-point-construction}  
and \eqref{Heegner-inert}, and the unit $u_{\psi_0}^-$ defined in 
 \eqref{ug-pinert} from elliptic units.
 Assuming \eqref{r1-hyp} or 
 \eqref{eqn:quad-imag-analytic-ranks},
%% Conjecture $\shi$ predicts that some iterated integrals associated to 
%%  the triplet $(f,g_\alpha,h)$ are non-zero, and 
%% Conjecture $\shi$ proposes a formula for their values up to $L^\times$;  namely,
 Conjecture $\shi$
 predicts
 that there is some choice of test vectors $\testf$, $\testgamma$ and $\testh$ for which
\begin{equation}
\label{conj:pinert}
\int_{\testgamma} \! \testf \cdot \testh = 
\begin{cases}
 \dfrac{\log_{E,p}(P_{\psi_1}^+) \log_{E,p}(P_{\psi_2}^-) - \log_{E,p}(P_{\psi_1}^-) \log_{E,p}(P_{\psi_2}^+)}{\log_p(u_{\psi_0}^-)}
 & \mbox{ if } \psi_1^2, 
 \   \psi_2^2\ne 1, \\
& \mbox{ and } V_{\psi_1} \ne V_{\psi_2},  \\
& \\ 
\dfrac{\log_{E,p}(P_{\psi_1}^+) \log_{E,p}(P_{\psi_2}^-)}{\log_p(u_{\psi_0}^-)} & \mbox { otherwise. }
\end{cases}
\end{equation}




\subsubsection{Numerical evidence}

The following experiment in support of 
\eqref{conj:pinert} 
gave  the authors their 
first evidence that 
the   $p$-adic iterated integrals continue
to have geometric meaning beyond the setting 
treated in Section \ref{sec:proof}.
\begin{example}
\label{ex:83}
Consider the elliptic curve
$$ E: y^2 + xy + y = x^3 + x^2 + x $$
of conductor $83$ labelled $83a$ in Cremona's tables and write $f$ for the associated weight $2$ newform.
Let $\chi$ be the odd quadratic character
of conductor $83$. The space $S_1(83,\chi)$ has dimension $1$ and is spanned by the theta
series $g:= \theta_{\psi_g}$ attached to  a cubic character $\psi_g$ of $G = \Gal(H/K)$,
where  $H$ is the Hilbert class field of  $K = \Q(\sqrt{-83})$, which has
class number $3$.
The representation $\rho_g$ is equal to the induced representation $V_{\psi_g}$.  Let $h = E(1,\chi) \in M_1(83,\chi)$ be the Eisenstein series attached to the pair $(1,\chi)$,
whose Galois representation $\varrho_h$ is induced from the trivial character
$\psi_h = 1$ of $G$. 
Since $g$ and $h$ have rational Fourier coefficients it will be enough to take
$L=\Q$ (although the representations are in fact defined over $\Q(\zeta_3)$). The
  representations $V_{gg}$ and $V_{gh}$ decompose into irreducible constituents as follows:
$$
\label{eqn:decomp-Vgg-exotic}
\begin{array}{lllllcl}
V_{gg} & = & V_{\psi_g^2} \oplus V_{\psi_g \cdot \psi_g^{-1}} & = & V_{\psi_g} \oplus \Q \oplus \Q(\chi)& =:&\varrho_1 \oplus \varrho_2 \oplus \varrho_3, \\
V_{gh} & = & V_{\psi_g \cdot \psi_h} \oplus V_{\psi_g \cdot \psi_h^{-1}}  &  = & V_{\psi_g} \oplus V_{\psi_g} &=& \varrho_1 \oplus \varrho_1.
\end{array}
$$
A direct calculation shows that
$$  r(E,\varrho_1) = r(E,\varrho_2) =  1, \qquad r(E,\varrho_3) = 0,$$
and similarly for the analytic ranks.
More precisely, the Mordell-Weil group
$$E(H)_\Q = E(H)_{\Q}^{V_{\psi_g}} \oplus E(\Q)_\Q$$
is of rank $3$, and 
is generated by 
the Heegner point
\begin{equation}
\label{eqn:ex1-heegner}
 P = (t, -t^2-2)
 \in E(H), \qquad \mbox{ where } t^3 - t^2 + t - 2=0, 
 \end{equation}
along with its Galois translates $P^\sigma$ and $P^{\sigma^2}$, where
$\sigma $ is a generator of $\Gal(H/K)$. 
The vector spaces  $E(\Q)_\Q$ and $E(H)_\Q^{V_{\psi_g}}$ are generated by
$$
P_\Q := P + P^\sigma + P^{\sigma^2} =  (1,-3),  
\qquad \mbox{ and } \qquad \left\{ \begin{array}{l}
P_{\psi_g}^+ :=  
2P - P^\sigma - P^{\sigma^2}, \\
P_{\psi_g}^- := P^\sigma - P^{\sigma^2}.
\end{array}\right.
$$
Let $p = 5$, which is in inert in $K$. 
The cubic polynomial in \eqref{eqn:ex1-heegner}
has exactly one root over $\Q_5$.
There is therefore   a unique  embedding of $H$ into $\C_5$ for which
the Heegner point $P$ maps to $E(\Q_5)$;   the images of the points $P^\sigma$ and 
$P^{\sigma^2}$ are then  interchanged by the frobenius element $\sigma_5$.

The Stark unit that enters into the denominator of the conjectural expression for the
$p$-adic iterated integral is the   elliptic unit
defined by
$u_{\psi_0}^-  = u^{\sigma}/u^{\sigma^2},$ so that
$$\log_p(u_{\psi_0}^-) = 
 \log_p(u^\sigma) - \log_p(u^{\sigma^2}),$$
where $u \in H$ is the unique root of the polynomial 
$$ x^3 - 2 x^2 -  2 x - 1$$ 
whose image under the previously chosen embedding of $H$ belongs to $\Q_5$.
  
Let $\gamma_{\pm 1} \in S_1^{\oc,\ord}(83,\chi)^\vee$ be the dual
basis  to $g_{\pm 1} = g(q) \mp g(q^5)$ in the space of $5$-adic overconvergent modular
forms, which   maps $g_{\pm 1}$ to $1$ and all other generalised eigenspaces
in $S_1^{\oc,\ord}(83,\chi)$ to zero. 
A numerical calculation of $5$-adic iterated integrals shows to $70$ digits of $5$-adic precision 
\[
\begin{array}{l}
\int_{\gamma_{\pm 1}} f \cdot g  =   -1688341751390720654446425615337716320464561776401, \\
\int_{\gamma_{\pm 1}} f \cdot h  =   -2981481189608571355501040629889815717824676581453.
\end{array}
\]
Let
${\mathcal E}(g_{\pm 1},f,g)$ denote
 the algebraic factor defined at the start of Section \ref{sec:exotic-points-evidence}.
 %% \cite[Theorem 3.1, Eqn (6)]{DR1}. 
 In all the  examples where $p$ is inert, 
  this  factor is always equal to 
\[ 
{\mathcal E}(g_{\pm 1},f,g):=
\left(1 - \frac{a_p(f)}{p} + \frac{1}{p}\right) \left(1 + \frac{a_p(f)}{p} + \frac{1}{p}\right) \in \Q.\]
The following identities were checked to $70$ digits of $5$-adic precision:
\begin{align}
\label{eqn:ex1-check1}
\int_{\gamma_{\pm 1}} f \cdot g & = \frac{5 \cdot {\mathcal E}(g_{\pm 1},f,g) \cdot \log_{E,5}(P_\Q) \log_{E,5}(P_{\psi_g}^-)}{24 \cdot \log_5(u_{\psi_0}^-)}, \\
\label{eqn:ex1-check2}
\int_{\gamma_{\pm 1}} f \cdot h & = -\frac{5 \cdot {\mathcal E}(g_{\pm 1},f,h) \cdot \log_{E,5}(P_{\psi_g}^+) \log_{E,5}(P_{\psi_g}^-)}{24 \cdot \log_5(u_{\psi_0}^-)},
\end{align}
in perfect agreement with the prediction (\ref{conj:pinert}) for both $\gamma_{+1}$ and   $\gamma_{-1}$. Note that $u_{\psi_0}^-$ is a unit in the $(-1)$-eigenspace for
$\sigma_5$ while the numerators involve points in both   eigenspaces, so that the resulting ratio belongs
to $\Q_5$.
%  Thus the righthand side of the formula is independent of the choice of embedding of $H$ into $K_p$, as is evidently the lefthand side since $f$,\,$g_{\pm 1}$ and $h$ are defined over $\Q$,cf. Remark 4 in the Introduction.

Although the Heegner point $P$ is only defined up to
sign, it is worth noting
that  the right-hand side of  \eqref{eqn:ex1-check1} and 
\eqref{eqn:ex1-check2} are unchanged if $P$ is replaced by $-P$. 


Formulae similar to 
\eqref{eqn:ex1-check1} and 
\eqref{eqn:ex1-check2} (with the terms $5$ and $24$ in the numerator and denominator
replaced by $p$ and $4(p+1)$ respectively) 
were verified  to $15$   digits of $13$-adic precision (respectively, $10$ digits of $19$-adic precision) replacing 
$p = 5$ by the inert primes $p = 13$ and  $p = 19$.
Similar experiments were also performed with  the class number
$3$ field $K = \Q(\sqrt{-331})$ and the inert prime $p = 7$, where
 the same formulae (mutatis mutandis)
were verified to $10$ significant $7$-adic digits.
\end{example}
\medskip \medskip

The next example  also involves Heegner points but  is one of the most illuminating in this paper, insofar as it concerns a rank $(1,1)$ setting of Conjecture $\shi$ in which nonetheless
the regulator $R_{g_\alpha}(E,\varrho_{gh})$ is not factorisable. 
It 
illustrates a feature of Conjecture $\shi$
which initially surprised the authors, namely, that the numerator on the right hand side of Conjecture
$\shi$
 needs not, in contrast with what one might expect from  Theorem  \ref{main-theta}, 
be the product of logarithms of two points but may involve a linear combination 
of such products. 
% This ran counter to the intuition the authors had first gainedin their proof of Conjecture $\shi$  in the setting of Section \ref{sec:proof}. 

\begin{example}
\label{ex:79}
Consider the curve
\[ E : y^2 + x y + y = x^3 + x^2 - 2 x \]
of conductor $79$ labelled $79a$ in Cremona's tables and
write $f$ for the associated weight $2$ newform. 

Let $\chi$ be the odd Dirichlet character of order $2$ and
conductor $79$ associated to the imaginary quadratic field
$K = \Q(\sqrt{-79})$, of class number $5$.  As in the previous example,
denote by $H$ the Hilbert class field of $K$ and set  $G = \Gal(H/K)$.
The space
$S_1(79,\chi)$ has dimension $2$ and is spanned by a pair of dihedral eigenforms
with coefficients in the field $L=\Q(\sqrt{5})$ generated by the golden ratio
$\omega = (1+\sqrt{5})/2$, namely
\[ g \ \ =\ \  q + \left(-1+\omega\right) q^2 + \left(1-\omega\right) q^4 -\omega  q^5
 - q^8 + q^9 - q^{10} + \cdots \]
and its Galois conjugate $g^\dagger$. 
Let $h = E(1,\chi) \in M_1(79,\chi)$ be the Eisenstein series attached to the pair 
$(1,\chi)$.
After fixing a quintic character $\psi$ of $G$, it can be assumed that
$$ V_g = V_{\psi^2} = V_{\psi^3}, \qquad V_{g^\dagger} = V_{\psi} = V_{\psi^4}, \qquad V_h = V_{\psi_h} = L \oplus L(\chi_K), $$
where $\psi_h$ is the trivial character of $G_K$.
The representations
 $V_{g^\dagger g}$ and $V_{g^\dagger h}$ decompose   into irreducible summands as follows:
\[ 
\begin{array}{lllllll}
V_{g^\dagger g} & = & V_{\psi \cdot \psi^{-2}}  \oplus V_{\psi \cdot \psi^{2}}  & = & V_{\psi} \oplus V_{\psi^2}
& =: & \varrho_1\oplus \varrho_2, \\
V_{g^\dagger h} & = & V_{\psi \cdot \psi_h} \oplus V_{\psi \cdot \psi_h^{-1}} &  = & V_{\psi}  \oplus V_{\psi}
& = & \varrho_1 \oplus \varrho_1.
\end{array}
\]
A Heegner point calculation reveals that the Mordell-Weil group $E(H)$ has rank $5$, and is generated by the Heegner point
$$ P = \left(t, \frac{-t^4 - 3 t^3 + 2t^2 + 7t - 6}{3} \right), \quad  \mbox{ where } \quad  t^5 + t^4 - 5t^3 - 3t^2 + 8t - 3 = 0, $$
and its Galois translates $P^\sigma$, $P^{\sigma^2}$, $P^{\sigma^3}$ and $P^{\sigma^4}$, 
where 
$\sigma$ is a generator of $\Gal(H/K)$, chosen in such a way that $\psi(\sigma) = \zeta_5$ 
with $\zeta_5$ a primitive fifth root of unity satisfying
$\zeta_5+ \zeta_5^4 = -\omega$. Let $\omega' = (1-\sqrt{5})/2$ denote
 the conjugate of the golden ratio. 
The $L$-vector space $E(H)_L^{\varrho_1}$ is then spanned by the points
\begin{eqnarray*}
 P_1^+ &=&   2 P - \omega P^\sigma - \omega' P^{\sigma^2} - \omega' P^{\sigma^3} - \omega P^{\sigma^4}, \\
 P_1^-  &=& (\zeta_5 - \zeta_5^4) \left( P^\sigma - \omega P^{\sigma^2} + \omega P^{\sigma^3} - P^{\sigma^4} \right), 
 \end{eqnarray*}
 and $E(H)_L^{\varrho_2}$ has a similar basis $(P_2^+, P_2^-)$ obtained by replacing 
 $\zeta_5$ by $\zeta_5^3$, and thus
 $\omega$ by its conjugate $\omega'$, in the above expressions
  for $P_1^+$ and $P_1^-$. 
 
 Let $p = 29$ be the smallest prime which is  inert in $K$ but split in $L/\Q$.
(The latter property is not essential to the conjectures, but makes it easier to  calculate the
$p$-adic iterated integrals.) 
Fix the unique embeding of $H$ into $\C_p$ for which 
the image of $P$ belongs to $E(\Q_p)$,
and for which $\sigma_p$ interchanges $P^\sigma$ with $P^{\sigma^4}$ 
and $P^{\sigma^2}$ with $P^{\sigma^3}$. 


For the elliptic unit, one may start with $u$ as the unique root of $x^5 - 1014 x^4 + 299243 x^3 - 21035 x^2 + 195 x - 1$ in $H \cap \Q_{29}$ and
define $u_{g^\dagger_{\pm 1}}$ in exactly the manner as 
$P_2^-$ was defined above (only replacing $P$ by $u$ and using additive  notation for $(\cO_H^\times)_L$), i.e., 
$$ u_{g^\dagger_{\pm 1}} : =   (\zeta_5^3 - \zeta_5^2) \left(u^\sigma - \omega u^{\sigma^2} + \omega u^{\sigma^3} - u^{\sigma^4}\right).$$
Note that $u_{g^\dagger_{\pm 1}}$ may be alternatively denoted $u_{\psi_0}^-$.

Let $\gamma_{\pm 1}^\dagger \in S_1^{\oc,\ord}(79 \cdot 23,\chi)^\vee$ be the dual
element to $g^\dagger_{\pm 1}$, defined as before. A numerical calculation of
$p$-adic iterated integrals reveals that
\begin{eqnarray*}
\int_{\gamma^\dagger_{\pm 1}} f \cdot h & = & 371806430046362 \bmod{29^{10}}\\
\int_{\gamma^\dagger_{\pm 1}} f \cdot g & = &  259114157339642 \bmod{29^{10}}.
\end{eqnarray*}
The following identity was then verified to $10$ digits of $29$-adic precision:
\begin{equation}
\label{eqn:ex2-check1}
\int_{\gamma^\dagger_{\pm 1}} f \cdot g = \frac{p \cdot {\mathcal E}(g^\dagger_{\pm 1},f,g) \cdot (\log_{E,29}(P_1^+) \log_{E,29} (P_2^-) - \log_{E,29}(P_1^-) \log_{E,29} (P_2^+)) }{4 \cdot 5 \cdot \sqrt{5} \log_{29}(u_{g^\dagger_{\pm 1}})}
\end{equation}
where here $\sqrt{5}$ denotes the unique $29$-adic square root of $5$
 which is congruent to 
$ 18$ modulo $29$.
%% April 14, 2014: choice zeta_5 -> zeta_5^3 rather than zeta_5^2 switched sign on both P_2^- and unit on denominator.


Turning to $\int_{\gamma^\dagger_{\pm 1}} f \cdot h$,
 notice that ${\mathcal E}(g^\dagger_{\pm 1},f,h) = {\mathcal E}(g^\dagger_{\pm 1},f,g)\,(=  864/841)$
and the transcendental factor $\log_{29}(u_{g^\dagger})$ expected on the righthand side of the formula is the same as for 
 $\int_{\gamma^\dagger_{\pm 1}} f \cdot g$. Thus these factors should cancel upon taking a ratio and indeed 
 one observes (modulo $29^{10}$) 
  the   identity 
\begin{eqnarray}
\label{eqn:gangnam-style}
\frac{\int_{\gamma^\dagger_{\pm 1}} f \cdot g}{\int_{\gamma^\dagger_{\pm 1}} f \cdot h} &=&  \frac{1}{2} \cdot
\frac{\log_{E,29}(P_1^+) \log_{E,29} (P_2^-) - \log_{E,29}(P_1^-) \log_{E,29} (P_2^+)}
{\log_{E,29}(P_1^+) \log_{E,29} (P_1^-)} \\
&=& \frac{1}{2}\left( \frac{\log_{E,29}(P_2^-)}{\log_{E,29}(P_1^-)} - 
\frac{\log_{E,29}(P_2^+)}{\log_{E,29} (P_1^+)} \right), \nonumber
\end{eqnarray}
%% April 14, 2014: here the sign on P_2^- has changed, but then need to multi through numerator by -1 to get
%% it unchanged and sign on P_1^- changed. So overall -1/2 -> 1/2.
which suggests that
\begin{equation}
\label{eqn:ex2-check2}
\int_{\gamma^\dagger_{\pm 1}} f \cdot h = 
\frac{p \cdot {\mathcal E}(g^\dagger_{\pm 1},f,h) \cdot \log_{E,29}(P_1^+) \log_{E,29} (P_1^-)}{2 \cdot 5 \cdot \sqrt{5} \log_{29}(u_{g^\dagger_{\pm 1}})}.
\end{equation}
%% April 14, 2014: submitted version seemed to have a typo P_2 rather than P_1 here.
%% Also I believe submitted version may have had sign error due to confusion over which root of 5 taken.
Both equations \eqref{eqn:ex2-check1} and 
\eqref{eqn:ex2-check2} are in perfect agreement with Conjecture $\shi$. 



\begin{remark}
It was a ratio identity like the one in \eqref{eqn:gangnam-style} 
 (for a simpler ratio in Example \ref{ex:83}) which first convinced the
authors that the $p$-adic iterated integral had geometric meaning in the setting where  $p$
is inert, even
prior to their full understanding of the logarithm of the Stark unit
 occurring in the denominator
of the   identity of Conjecture $\shi$.
\end{remark}
\end{example}
\medskip \medskip

We now turn to our first numerical example in the rank $2$ setting,
illustrating a notable feature of Conjecture $\shi$, namely that its right hand side
can be a non-trivial $p$-adic combination of logarithms of global points in a ``genuine rank two" setting
involving a primitive $L$-function with a double zero at the center.



\begin{example}\label{ex:57}
We take the curve
\[ E: y^2 + xy + y = x^3 - 7x + 5 \]
of conductor $57$ labelled $57b$ in Cremona's tables and write $f$ for the attached
newform of weight $2$.

Let $\chi$ be the odd Dirichlet character of order $2$ and conductor $23$ associated to the
imaginary quadratic field $K = \Q(\sqrt{-23})$, of class number $3$. The space $S_1(23,\chi)$
has dimension $1$ and is spanned by the theta series $g := \theta_{\psi_g}$ attached to 
a cubic character $\psi_g$ of $G = \Gal(H/K)$, where $H$ is the class field of $K$. 
%Let $h = E(1,\chi)\in M_1(23,\chi)$ be the Eisenstein series.  
As in Example \ref{ex:83} we find
$$ V_{gg} = V_{\psi_g} \oplus L \oplus L(\chi), \qquad  \mbox{ with } L = \Q(\zeta_3), $$
but a calculation reveals that here
$$ r(E,V_{\psi_g}) = 2, \qquad r(E,L) = r(E,L(\chi)) = 0, $$
and likewise for the analytic ranks.
Here the Heegner hypothesis is not satisfied, but   the descent code in {\sc Magma} 
can be used to
construct the required global points in $E(H)_L^{V_{\psi_g}}$. Over the cubic subfield
\[ \Q(a), \qquad  a^3 - a^2 + 2 a - 1 = 0 \]
of $H$, 
the curve $E$ has rank $2$ with generators for the Mordell-Weil lattice
$$P := (9 a^2 - 4 a + 17 , -45 a^2 + 20 a - 80), \quad Q := (a^2 + 3 , 2 a^2 + 2). $$
We embed $\Q(a)$ into $H = \Q(b)$, where $b^6 - 6 b^4 + 9 b^2 + 23 = 0$, by taking
$a = (b^4 - 2 b^2 + 1)/9$. Taking $\sigma$ as a generator for $G$ we compute four points
\[
\begin{array}{ll}
P_{\psi_g}^+ := 2 P - P^\sigma - P^{\sigma^2},  & \quad P_{\psi_g}^- := P^\sigma - P^{\sigma^2} ,\\
Q_{\psi_g}^+ := 2 Q - Q^\sigma - Q^{\sigma^2},   &\quad  Q_{\psi_g}^- := Q^\sigma - Q^{\sigma^2},
\end{array}
\]
and similarly a unit $u_{\psi_0}^-$ starting from the root
$(b^4 - 5 b^2 + 4)/9$ of $x^3 - x - 1$ in $H$. We take $p = 19$ and find we may embed $H$ into $\C_{19}$ so
that $P$, $Q$ and $u$ are defined over $\Q_{19}$.

To compute the relevant $19$-adic iterated integral we must work in $19$-adic level $3 \cdot 23$ and so
choose a $3$-stabilisation $\breve{g}$ of $g$. We find $x^2 - a_3(g) x + \chi(3) = (x - \zeta_3)(x - \zeta_3^2)$
and take $\breve{g}$ to be the $3$-stabilisation with $U_3$-eigenvalue $\zeta_3$.  (With the other choice
the formula is exactly the same, except the algebraic factor is replaced by its complex conjugate.) We embed
$\Q(\zeta_3)$ into $\Q_{19}$ so that $\zeta_3 \equiv 7 \bmod{19}$, and find
\[ \int_{\breve{\gamma}_{\pm 1}} f \cdot g = -25103076413984358720047537708218 \bmod{19^{25}}. \]
%% -3469380219741068153737605 \bmod{19^{20}}.\]
One checks to $25$ digits of $19$-adic precision that
$$ \int_{\breve{\gamma}_{\pm 1}} f \cdot g  = \frac{63 \zeta_3 + 18}{26 \cdot 19} \cdot 
\frac{ \log_{E,19}(P_{\psi_g}^+) \log_{E,19}(Q_{\psi_g}^-) - \log_{E,19} (P_{\psi_g}^-) \log_{E,19} (Q_{\psi_g}^+)}
{\log_p u_{\psi_0}^-}, $$
as predicted by Conjecture $\shi$.
\end{example}



In the next example,  the relevant 
 Heegner point arises from a  Shimura curve rather than a classical modular curve
 parametrisation. Note that Conjecture $\shi$ in principle
 allows the efficient computation of these  Heegner
points in terms of the  Fourier expansions of classical 
modular forms rather than
working with their Jacquet-Langlands lifts. For the purpose of verifying Conjecture $\shi$,
the authors did not
actually calculate the Heegner points arising from
 the  Shimura curve parametrisation, which raises significant computational challenges, 
but rather found appropriate global points by general descent methods.

\begin{example}\label{ex:35}
Consider the curve
\[ E: y^2 + y = x^3 + x^2 + 9 x + 1 \]
of conductor $35$ labelled $35a$ in Cremona's tables and write $f$ for the associated weight $2$ newform.
Let $K = \Q(\sqrt{-23})$  be the smallest imaginary quadratic field
of class number $3$, and note that both primes dividing $35$ are inert in $K$. This is the simplest setting where there is no 
Heegner point on $E$ arising from  a modular curve, and in which one must rather work
with the Shimura curve attached to the indefinite quaternion algebra ramified at $5$ and $7$.

Let $\chi$ be the quadratic Dirichlet character of conductor $23$. 
Then $S_1(23,\chi)$ is one-di\-men\-sion\-al and
is spanned by the single dihedral form
\[ g = q - q^2 - q^3 + q^6 + q^8 - q^{13} - q^{16}  + \cdots.\]
Let   $H$ be the Hilbert class field of $K$,    set
$G = \Gal(H/K)$, and let  $\varrho_g$ be the representation
 induced from a cubic character
$\psi_g$ of $G$ with image contained in $L = \Q(\zeta_3)$.

The representation $V_{gg}$ decomposes as
\[ 
\begin{array}{lllll}
V_{gg} & = & V_{\psi_g \cdot \psi_g} \oplus V_{\psi_g \cdot \psi_g^{-1}} & = & V_{\psi_g} \oplus L(\chi)  \oplus L.\\
\end{array}
\]
A direct calculation reveals that 
$$  r(E,V_{\psi_g}) = r(E,L(\chi)) = 1, \qquad r(E,L)=\mbox{rank}(E(\Q))=0,$$
and the same is true of the analytic ranks.
Let  $P_K$ be  a generator of 
 $E(K)$  up to torsion, which lies
  in the $-1$ eigenspace for the action of the involution in
$\Gal(K/\Q)$, and let $P$ be a $\Q[\Gal(H/K)]$-module generator of
$E(H)\otimes\Q$. 
The points   $P_K$ and $P$
were not constructed    using Lemma \ref{lemma:heegner-point-construction},
since   the Heegner point  evoked in this lemma is difficult to compute
directly in practice. 
Instead,  one just
takes 
$$P_K := \left(-5 , \frac{-5 \sqrt{-23} - 1}{2}\right), \quad
P := \left(\frac{-7 t^4 + 14 t^2 + 29}{36} , \frac{7 t^4 - 14 t^2 - 68}{24}\right) \in E(H)$$
where $t$ is  a normal basis generator   for 
$H$ over $\Q$  satisfying the equation
$$t^6 - 6 t^4 + 9 t^2 + 23 = 0.$$ 
The point  $P$  just turned out to be 
the first point in a basis for the rank $3$ Mordell-Weil group $E(H)$ computed by {\sc Magma}.
 
Let $p=5$ be the smallest prime which is inert in $K$.
After a suitable choice of embedding of $H$ into $K_5$ for which $P$
maps to $E(\Q_5)^-$, the frobenius element $\sigma_5$ interchanges 
$P^{\sigma}$ and $P^{\sigma^2}$.  With these conventions, 
we may set
 $$P_{\psi_g}^+ := 2P - (P^\sigma + P^{\sigma^2}).$$

Both $f$ and $g$ are $5$-adic  overconvergent modular forms
of level $7$ and $23$ respectively.
Stabilise $g$ to an eigenform $\breve{g} = g(q) - g(q^7)$ of level $161 = 7 \cdot 23$ 
with $U_7$ eigenvalue $1$.
Then $\breve{g}$ has two  stabilisations 
$$\breve{g}_{\pm 1} = \breve{g}(q) \mp \breve{g}(q^5) $$ 
at $5$, which  can be viewed either as classical 
modular forms of level $161 \cdot 5$ or as $5$-adic overconvergent forms of
level $161$. 
Write $\breve{\gamma}_{\pm 1} \in S_1^{\oc,\ord}(7 \cdot 23 ,\chi)^\vee$ for the usual dual element to $\breve{g}_{ \pm 1}$. A direct evaluation of the $5$-adic iterated integral
reveals that
\[
\int_{\breve{\gamma}_{+1}} f \cdot g = - \int_{\breve{\gamma}_{-1}} f \cdot g = 376357248757242028241 \bmod{5^{30}}.
\]
%% (Note that for the purposes of numerical computation one only needs to stabilise the form $g_\alpha$ in a triple $(f,g_\alpha,h)$ to a true
%% eigenform in level $Np$, where $N$ is the common overconvergent level of $f,g_\alpha,h$)

Starting with the elliptic unit $u$ which is the unique root of  $x^3 + 6 x^2 + 29 x - 1$ in $H \cap \Q_5$,
the   unit $u^-$ is defined by $u^- = (u^\tau / u^{\tau^{2}})$.
%  (and so $u_g = (u^-)^{\sqrt{-3}}$).
The following identity was then  checked to  $30$ digits of $5$-adic precision:
\[ \int_{\breve{\gamma}_{+1}} f \cdot g = \frac{3 \cdot 7 \cdot \log_{E,5}(P_K) \cdot \log_{E,5}(P_{\psi_g}^+) }{2 \cdot 5 \cdot \log_5(u^-)}\],
as predicted by (\ref{conj:pinert}). 
Note that  once the point $P_K$ is known,
this identity leads to an explicit analytic formula for  a global
point
  on $E$ over the Hilbert class field $H$, namely $P_{\psi_g}^+$,
  without passing through the calculation of the Shimura curve parametrisation of $E$, as would be necessary if one were to resort to  Heegner points instead of $p$-adic iterated integrals.
\end{example}
\medskip \medskip

 \section{Stark-Heegner points and theta series of real quadratic fields} 
\label{sec:stark-heegner}

\hiddensubsection{Theoretical preliminaries}
\label{sec:stark-heegner-statement}
This chapter focusses on the case where $g$ and $h$ are theta series attached to characters---denoted $\psi_g$ and $\psi_h$ respectively---of mixed signature $(+,-)$ of the  same 
{\em real quadratic} 
 field $K$. The self-duality assumption on $V_{gh}$ implies that 
$$ \psi_1 := \psi_g \psi_h, \qquad \psi_2 := \psi_g \psi_h'$$
are dihedral characters of $K$, which are totally even and totally odd respectively 
and that
$$ V_g = V_{\psi_g}, \qquad V_h = V_{\psi_h}, \qquad
 V_{gh} =  V_{\psi_1} \oplus V_{\psi_2}.$$

As in the previous chapter, the Hasse-Weil-Artin $L$-function factors as 
$$L(E,\varrho_{gh},s)=L(E/K,\psi_1,s) \cdot L(E/K,\psi_2,s),$$ and Hypothesis A is fulfilled as soon as the conductors of $\psi_g$ and $\psi_h$ are coprime to $N_f$. In all examples where Conjecture $\shi$ is tested below, Hypothesis B holds because   $L(E/K,\psi_1,1)=L(E/K,\psi_2,1)=0$.

 Let $p$ be a rational prime, at which the representations $V_g$ and $V_h$
 are unramified, and let $\alpha_g$ and  $\beta_g$ (resp. $\alpha_h$ and $\beta_h$) denote as before
    the frobenius eigenvalues
     for $V_g$  (resp.~$V_h$) attached to this prime. In order for $g$ to satisfy Hypotheses C
     and C', it is necessary and sufficient to assume that 
  \begin{equation}
\label{eqn:p-is-inert}
 \mbox{ $p$ is inert in $K$.}
 \end{equation}
  In that case, by the remarks surrounding 
  \eqref{eqn:gs-unit-real-K},
   the Stark unit $u_{g_\alpha} =: u_K$  of Conjecture $\shi$ 
   can be taken to be equal to the fundamental unit of $K$, and in particular it depends only on $K$ and not on $\psi_g$.
 
Assumption \eqref{eqn:p-is-inert}
 also implies that the frobenius element $\sigma_p$ corresponds to  reflections in the dihedral 
 groups cut out by the representations $V_{\psi_1}$ and $V_{\psi_2}$. 
As in Section 
\ref{p-inert},
 it is  assumed that
 $$ \alpha_g \alpha_h =1, \qquad \alpha_g\beta_h = -1, \qquad
 \beta_g \alpha_h =  -1, \qquad \beta_g \beta_h = 1.
$$
The  eigenvalues of $\sigma_p$ on  $V_{\psi_1}$ are $(1,-1)$, and likewise for
$V_{\psi_2}$. 
 Note in particular that in this setting, the representation $V_{gh}$ is never 
 regular at $p$, even though the representations $V_g$, $V_h$, $V_{\psi_1}$ and $V_{\psi_2}$ are 
 all regular.   
 
  
Consider first  the ``generic" setting where neither
  of the characters $\psi_1$ or
 $\psi_2$ is a quadratic (genus) character, so that the representations $V_{\psi_1}$ and
 $V_{\psi_2}$ are both {\em irreducible}.  
 %% Conjecture  $\shi$ of the introduction then predicts  that,
 %% for all $\gamma_\alpha \in M_1^{\ord}(N, \chi)_L^\vee [g_\alpha]$, and all test vectors 
 %% $\testf$ and $\testh$ in the relevant Hecke eigenspaces, the non-vanishing of
 %% the associated $p$-adic iterated integral 
%% implies that $V_{\psi_1}$ and $V_{\psi_2}$ each occur with multiplicity one in the Mordell-Weil group of $E$. 
Conjecture $\shi$ then predicts the
 existence of     points  $P_{1} \in E(H)_L^{V_{\psi_1}}$ and
  $P_{2} \in E(H)_L^{V_{\psi_2}}$ whose projections onto the different $\sigma_p$-eigenspaces
\begin{eqnarray*}
P_{1}^+ \in (E(H)_L^{V_{\psi_1}})^{\sigma_p=+1}, & &  \qquad P_{2}^- \in (E(H)_L^{V_{\psi_2}})^{\sigma_p=-1}, \\
 P_{1}^- \in (E(H)_L^{V_{\psi_1}})^{\sigma_p=-1},  & & \qquad P_{2}^{+} \in (E(H)_L^{V_{\psi_2}})^{\sigma_p=+1},
 \end{eqnarray*}
 are related to a suitable  $p$-adic iterated integral by
 \begin{equation}
 \label{eqn:real-inert}
  \int_{\testgamma_\alpha}  \testf\cdot \testh  \stackrel{?}{=} \frac{ \log_{E,p}(P_1^+) \log_{E,p}(P_2^-) -
 \log_{E,p}(P_1^-) \log_{E,p}(P_2^+) }{ \log_p(u_{K})}.
 \end{equation}
This conjecture becomes slightly easier to verify 
when one of the characters $\psi_1$ or
$\psi_2$ is a quadratic (genus) character. Assume without loss of generality that
this is the case for $\psi_1$, so that 
$$V_{\psi_{1}} = L(\chi_1) \oplus L(\chi_2)$$ 
becomes  reducible as a $G_\Q$-module, breaking up into two one-dimensional representations attached to quadratic Dirichlet characters
$\chi_1$ and $\chi_2$ satisfying $\chi_1\chi_2 = \epsilon_K$, so that in particular
$\chi_1(p) = -\chi_2(p)$. 
When the iterated integral of the left-hand side 
of \eqref{eqn:real-inert}
is non-zero, Conjecture $\shi$ predicts that 
precisely one of the characters $\chi_1$ or $\chi_2$ occurs in 
the Mordell-Weil group of $E$.  Assume that it is $\chi_1$ 
which arises, and set $\iota := \chi_1(p) = \pm 1$. 

One then has
 $$ P_1  \in E(H)_L^{\chi_1}.$$
Conjecture $\shi$ predicts that, for a suitable choice of test vectors,
\begin{equation}
\label{eqn:real-inert-genus}
\int_{\testgamma_\alpha}  \testf\cdot \testh  \stackrel{?}{=} \frac{ \log_{E,p}(P_1) \log_{E,p}(P_2^{-\iota})}{ \log_p(u_{K})}.
\end{equation}

Finally consider the case in which $\psi_1^2 =\psi_2^2 = 1$. The representation $V_{gh}$  then decomposes
as a sum of one dimensional summands, and Conjecture $\shi$  takes the  form 
\begin{equation}
\label{eqn:real-inert-genus-bis}
\int_{\testgamma_\alpha}  \testf\cdot \testh  \stackrel{?}{=} \frac{ \log_{E,p}(P_1) \log_{E,p}(P_2)}{ \log_p(u_{K})},
\end{equation}
{\em provided } the points $P_1$ and $P_2$, which both belong to quadratic extensions of
$\Q$, belong to {\em distinct} eigenspaces for $\sigma_p$; in the case where
they are in the same eigenspace,
Conjecture $\shi$    predicts a
subtle vanishing of the $p$-adic iterated  integral arising from a degeneracy in the regulator term.
This case is illustrated in Example \ref{ex:155} below.
 

\subsection{Numerical evidence}
\label{sec:stark-heegner-evidence}

Weight one modular forms whose Artin representations can be induced {\it only} from characters of
class groups of real quadratic fields are rather rare in small level. 
The first such occurrence is in level
$145$, where one finds such a form coming from a  quartic
 character of $\Q(\sqrt{5})$ with conductor a prime of norm $29$.
The next example is concerned with this modular form.

 \medskip\medskip 
 \begin{example}\label{ex:145}
 Let $K=\Q(\sqrt{5})$ and let 
 $$\psi_g = \psi_h^{-1}$$
 be the   quartic   character of $K$ which is ramified at the prime
 $ \lambda = \left(29, \frac{11+\sqrt{5}}{2}\right)$ of norm $29$.
 Let $\chi_5$  and $\chi_{29}$ be the quadratic 
 and quartic Dirichlet characters of conductors 
 $5$ and $29$ respectively, and let $\chi:= \chi_5\chi_{29}$ be the odd quartic character of conductor   $5\cdot 29$.
 Both $\psi_g $ and $\chi$ take values in the field $ L:= \Q(i)$ and
 are unique up to  conjugation by $\Gal(L/\Q)$.  Fix a choice of these characters in such a
 way that the
   theta series
 $$ g \in S_1(5\cdot 29, \chi^{-1}), \qquad h \in S_1(5\cdot 29,\chi)$$
 associated to $\psi_g$ and $\psi_h$
 belong to   spaces of weight $1$ 
 modular forms  with those characters (and Fourier coefficients in $L$). 
The Artin representation $V_{gh}$ is given by
 $$ V_{gh} =  L \oplus L(\chi_5) \oplus V_{\psi},$$
 where  
 $\psi$ is the unique  (up to conjugation)
 quartic ring class character of $K$ of conductor $29$.
 Let $H$ be the abelian quartic extension of $K$ cut out by 
 $\psi$, and write
  $\Gal(H/K)=:\langle \sigma\rangle$.  
The field $H$ is Galois over $\Q$, with Galois group $G=D_8$, the dihedral group of order 
$8$. The subfield of $H$ fixed by the center $\langle \sigma^2\rangle$  of $G$
is the totally real
biquadratic field $M:= \Q(\sqrt{5},\sqrt{29})$, and $H$ is the quadratic extension of $M$ 
given by
$$ H = M(\sqrt{\delta}) = M(\sqrt{\delta'}), \qquad \delta = \left(\frac{-29+3\sqrt{29}}{2}\right),
 \quad \delta'  = \left(\frac{-29-3\sqrt{29}}{2}\right).$$
Note that $\delta$ and $\delta'$ are  
  totally negative elements  of $M$ and that
$  \delta \delta' =5\cdot 29.$


Let   $f$ be the weight two cusp form of level $17$ attached to the unique 
  elliptic curve 
  $$E: \ \  y^2 +xy + y = x^3 -x^2 -x-14 $$ 
  of conductor $17$ denoted $17a$ in the tables of Cremona.
  The conductor of $E$  is inert in the quadratic field $K$. An explicit calculation 
  shows that 
 $$ E(\Q)_L = 0, \quad\qquad E(K)_L^{\chi_5} = L \cdot P_K, \quad\mbox{where}\quad P_K = \left(\frac{379}{20} , \frac{-1995 + 7218\sqrt{5}}{200}\right).$$
  The group $G$ admits four distinct one-dimensional representations, on which 
$\sigma^2$ acts trivially, and 
a unique irreducible representation of dimension two with rational
traces,
which is isomorphic
 to $V_{\psi}$ and  on which $\sigma^2$ acts as multiplication by $-1$.
Hence
$$ E(H)_\Q^{V_{\psi}} = E(H)_\Q^{\sigma^2=-1}.$$
Considerations involving signs in the functional equations of $L$-series suggest that the
representation $V_{\psi}$ occurs with {\em odd multiplicity} in  
$E(H)$ (cf. for example the introduction of \cite{darmon-hpxh}).
Note that the field $H$,  a totally odd cyclic quartic 
 extension of $K$,   contains no imaginary quadratic subfields.
The classical Heegner point construction is therefore unavailable to produce the global points in 
$E(H)_\Q^{V_{\psi}}$ whose existence is predicted by the Birch and Swinnerton-Dyer conjecture. 
The Stark-Heegner point  algorithms described in \cite{DP}
can be used as a substitute for the method based on Heegner points, and
yield four non-trivial points in $E(H)$ given by 
\begin{center}
\resizebox{14cm}{!}
{
\begin{tabular}{ccc}
$P$  &=&  $\left(\frac{-220777-17703\sqrt{145}}{5800}, \frac{214977+17703\sqrt{145}}{ 11600} + \frac{28584525+3803103\sqrt{5}+1645605\sqrt{29}+2364771\sqrt{145}}{290000}\sqrt{\delta} \right)$ \\
$P^\sigma$    &=&  $\left(\frac{-220777+17703\sqrt{145}}{5800}, \frac{214977-17703\sqrt{145}}{ 11600} + \frac{28584525+3803103\sqrt{5}-1645605\sqrt{29}-2364771\sqrt{145}}{290000}\sqrt{\delta'} \right)$ \\
 $P^{\sigma^2}$   &=&  $\left(\frac{-220777-17703\sqrt{145}}{5800}, \frac{214977+17703\sqrt{145}}{ 11600} - \frac{28584525+3803103\sqrt{5}+1645605\sqrt{29}+2364771\sqrt{145}}{290000}\sqrt{\delta} \right)$ \\
 $P^{\sigma^3}$     &=&  $\left(\frac{-220777+17703\sqrt{145}}{5800}, \frac{214977-17703\sqrt{145}}{ 11600} - \frac{28584525+3803103\sqrt{5}-1645605\sqrt{29}-2364771\sqrt{145}}{290000}\sqrt{\delta'} \right).$
\end{tabular}
}
\end{center}
These four Galois conjugate points are constructed from  periods 
 arising from a $p$-adic integration on $\cH_{17}\times \cH$
     based 
 on the {\em minus} modular symbol of $f$, as described in 
 \cite{darmon-hpxh}.
 The conjectures of loc.cit.~predict that these points should lie
 in   the minus eigenspace for  complex conjugation, which coincides
 in this case with the central
 involution $\sigma^2$ of $\Gal(H/\Q)$. 
 It can indeed be observed that   $\sigma^2 P = -P$ and that $\sigma^3 P = -\sigma  P$. In particular, the Stark-Heegner point $P$ and its conjugates
generate a rank two subgroup of $E(H)$ which is isomorphic to   $V_{\psi}$.
This shows that  $r(E,V_{\psi})\ge 1$, and  the belief
that Stark-Heegner points 
ought to behave like their quadratic imaginary counterparts strongly
suggests 
 that in fact $r_{\an}(E,V_{\psi}) = 1$.
If such is the case, Conjecture $\shi$ 
predicts the non-vanishing of some $p$-adic iterated   integral attached 
to $(f,g,h)$.
   
      The global point  $P$ and its conjugates
  were obtained from a $17$-adic  calculation.
This is an essential limitation of \cite{darmon-hpxh},  where one is
forced   to work $p$-adically at a prime $p$
 dividing the conductor of $E$.
   On the other hand, the $p$-adic iterated 
   integral appearing on the left of \eqref{eqn:real-inert-genus}
  can be calculated using any prime $p$ which is inert in $K$. It turns out to be expedient (albeit, not essential, as in the calculations relying on the algorithms of \cite{DP})
  to work at $p=17$. This is because, with this choice of $p$, the classical modular form 
  $f$ of level $17$ is $p$-adically of level $1$, and the $p$-adic iterated integral can therefore be calculated by applying the ordinary projection algorithms of 
  \cite{lauder} to a space of ordinary overconvergent $17$-adic modular forms of weight
  one and relatively modest level $5\cdot 29$. 
  
It can be checked that, for a suitable embedding of $H$ into the quadratic extension $\Q_{17^2}$ of $\Q_{17}$, the points $P$ and $\sigma^2 P$ are defined over $\Q_{17}$ while the points $\sigma P$ and $\sigma^3 P$ are defined over $\Q_{17^2}$ 
 and are interchanged by the frobenius element $\sigma_{17}$. 
Fix such an embedding from now on, and 
 let $\gamma_{\pm}\in S_1(\Gamma_1(5\cdot 29), \chi)_L^\vee[g_\pm]$ be the unique element
  satisfying $$ \gamma_\pm(g^*_\pm) = 1.$$
Here  $g_\pm = g_{\pm \zeta_8^3}$  denotes the $17$-stabilisations of $g$.
 A numerical evaluation of the $17$-adic iterated integrals, to $16$ significant digits shows that
$$
\int_{\gamma_\pm} f \cdot h = 1259389260500681328 \times 17 \bmod{17^{16}},
$$
 and that
\begin{equation}
\label{eqn:exciting-formula}
 \int_{\gamma_\pm} f\cdot h  = \frac{\log_{E,17}(P_K) \log_{E,17}(P)}{3 \cdot 17\cdot\log_{17}(u_K)}
 \end{equation}
    to an accuracy of $16$ significant $17$-adic digits. Since  $\sigma_{17} P_K = -P_K$,
    while $P$ generates
    $E(H)_L^{V_\psi}\cap E(\Q_{17})$,
      this is
   in perfect agreement with the prediction  of \eqref{eqn:real-inert-genus}.
    \end{example}

   \begin{remark}
   It is worth emphasising that the Stark-Heegner point $P$ is {\em a priori } quite
    mysterious and 
  not known to 
   correspond to a global point,  
   while  the $17$-adic iterated integral appearing on the 
   left-hand side of 
   \eqref{eqn:exciting-formula}  
    can be related to the $17$-adic logarithm  of the 
   {\em global} cohomology class $\kappa(\testf,\testg,\testh)\in H^1(H,V_{17}(E))$   
   constructed in \cite{DR2} as 
a 
    limit  of $17$-adic  \'etale   Abel-Jacobi  
  images of suitable ``generalised Gross-Schoen diagonal cycles".
  Since both the $17$-adic iterated integral appearing on the left hand side of
  \eqref{eqn:exciting-formula} and the ``integrals on $\cH_{17}\times \cH$" used to compute the Stark-Heegner point $P$ are given by explicit $17$-adic analytic expressions,
  a proof of formula  \eqref{eqn:exciting-formula} does not seem entirely out of reach.
  Such a proof would represent substantial progress in understanding the global
  properties of Stark-Heegner points, perhaps making it possible 
   to use them in   Euler system 
  arguments of the kind  exploited 
  originally by Kolyvagin. This would extend the approach described in Theorems A and C
  of   \cite{DR2} from  ``rank zero"
  to certain ``rank one" settings.
  \end{remark}
 \medskip 
 
The next example involves   forms $g$ and $h$
 with  representation having projective image $D_4$. Such odd representations are induced from class
characters of {\it both} real and imaginary quadratic fields, so this example would be equally at home in Chapter \ref{sec:heegner}.
Its principal novelty   is that it illustrates   situations where the $p$-adic
iterated integrals vanish
in rank $(1,1)$ settings because  the regulator term of Conjecture $\shi$ 
is $0$.
 
\begin{example}\label{ex:155}
Consider the curves
\[
\begin{array}{l}
E_a : y^2 + y = x^3 - x^2 + 10 x + 6\\
 E_b :  y^2 + x y + y = x^3 + x^2 - x - 2\\
E_c  : y^2 + y = x^3 - x^2 - x + 1
\end{array}
\]
of conductor $155$ labelled $155a$, $155b$,  and $155c$ in Cremona's tables and write $f_a$, $f_b$  and $f_c$ for 
the associated weight two newforms. 
%(For the curve $155b$ we have $L(E,V_{gg},1) \ne 0$ where $V_{gg}$ is the representation below, and so the hypothesis of our conjecture is not satisfied.)

Let $\chi_1$ be the trivial character, let  $\chi_5$ and $\chi_{31}$ denote
the (even and odd, respectively)
characters of conductor $5$ and $31$, and set  $\chi:= \chi_{155} := \chi_5 \cdot \chi_{31}$, which takes values in $L = \Q$.
The space $S_1(155,\chi)$ has dimension $3$ and contains three dihedral forms:
 one form $g$ with integer Fourier coefficients
\[ g =  q + q^4 - q^5 - q^9 + q^{16} - 2 q^{19} - \cdots  \]
and a second pair of complex conjugate forms each defined over the cyclotomic field $\Q(\zeta_3)$. 
The form $g$ has associated linear representation $\rho_g$ with projective image $D_4$, and $\rho_g$ can be described in the following equivalent ways:
\begin{enumerate}
\item It is induced
from a quartic character of the Hilbert class field of the imaginary quadratic field $\Q(\sqrt{-155})$;
\item it is induced from a  quadratic  character of the imaginary quadratic field
 $\Q(\sqrt{-31})$, whose   conductor is a prime ideal of norm $5$;
 \item it is induced from a quadratic character of the real quadratic field 
 $\Q(\sqrt{5})$, whose   conductor is a prime ideal of norm $31$.
 \end{enumerate}
  (The remaining pair of complex
conjugate forms are associated to representations which arise  from
quartic  characters  of $\Q(\sqrt{-31})$ with conductor a prime ideal of norm
$5$---they are not  considered   further in this example.) 

The representation 
$V_{gg}$ decomposes as
 $$ V_{gg} =  L \oplus L(\chi_5) \oplus L(\chi_{31}) \oplus L(\chi_{155}),$$
%% We have the decomposition of representation
%% \[ \rho_{\bar{g}} \otimes \rho_g  =  {\bf 1} \oplus \varepsilon_{5} \oplus \varepsilon_{-31} \oplus \varepsilon_{-155} \]
%% {\bf To DR: improve notation here and explain what these are.}
and the corresponding ranks are given by
\[
\begin{array}{llll}
r(E_a/\Q) = 1, & r(E_a,\chi_5) = 0, & r(E_a,\chi_{31}) = 1, & r(E_a,\chi_{155}) = 0\\
r(E_b/\Q) = 0, & r(E_b,\chi_5) = 0, & r(E_b,\chi_{31}) = 0, & r(E_b,\chi_{155}) = 0\\
r(E_c/\Q) = 1, & r(E_c,\chi_5) = 1, & r(E_c,\chi_{31}) = 0, & r(E_c,\chi_{155}) = 0,\\
\end{array}
\]
and likewise for the analytic ranks.
In particular,
the central critical value $L(E_b,\varrho_{gg},1)$ is non-zero, and hence the curve $E_b$ fails to  satisfy Hypothesis
B of the introduction. It will therefore be ignored from now on. 

As predicted by the BSD conjecture,
the points on $E_a$ and $E_c$ over $\Q$ and the   fields 
$\Q(\sqrt{-31})$ and $\Q(\sqrt{5})$ are given by
\begin{equation*}
\begin{split}
P_a   = (2,5),  \qquad &
P_{a,31} =  \left(\frac{-1630}{729}, \frac{40175 \sqrt{-31} - 19683}{39366} \right),  \\
  P_c  =  (1,0), \qquad &
P_{c,5} =  \left(0,\frac{-1+\sqrt{5}}{2}\right).
\end{split}
\end{equation*}



Consider both the primes $p = 7$ and $p = 13$, which have different splitting behaviours in the three quadratic fields, 
as shown in the table below.
\[
\begin{array}{c|ccc}
p &  \chi_5(p)  &  \chi_{31}(p) &  \chi_{155}(p) \\ \hline
7 &  -1   &  +1      & -1        \\ 
13 & -1 & -1      &  +1         \\ 
\end{array}
\]

For $p = 7$ the   eigenvalues of $\sigma_p$ on $V_g$ 
 are $\alpha_g =1$ and $\beta_g = -1$,
 and so
$$(\alpha_g \cdot \alpha_g,\alpha_g \cdot \beta_g) = (1,-1), \qquad 
(\beta_g \cdot \alpha_g, \beta_g \cdot \beta_g ) = (-1,1).$$
The  $7$-adic iterated integrals attached to $(f_a,\gamma_{\pm 1},g)$ and
$(f_c,\gamma_{\pm 1},g)$  are
\begin{eqnarray*}
\int_{\gamma_{1}} f_a \cdot g   =   \int_{\gamma_{-1}}f_a \cdot g & = & 0 \bmod{7^{30}}\\
\int_{\gamma_{1}} f_c \cdot g   =   \int_{\gamma_{-1}} f _c\cdot g & = & 1416540068067896281523634 \times 7  \bmod{7^{30}}.
\end{eqnarray*}
For $p = 13$ the   eigenvalues of $\sigma_p$ on $V_g$ 
 are $\alpha_g = i$ and $\beta_g = -i$, and so
$$(\alpha_g \cdot \alpha_g,\alpha_g \cdot \beta_g) = (-1,1), \qquad 
 (\beta_g \cdot \alpha_g, \beta_g \cdot \beta_g ) = (1,-1).$$
 % (We choose $i \in \Z_{13}$ with $i \equiv 5 \bmod{13}$.)
The 
$13$-adic iterated integrals attached to $(f_a,\gamma_{\pm i}, g)$ and
$(f_c, \gamma_{\pm i}, g)$ are
\begin{eqnarray*}
\int_{\gamma_{i}} f_a \cdot g   =  -\int_{\gamma_{-i}}f_a \cdot g & = & 10546307292618612 \bmod{13^{15}}\\
\int_{\gamma_{i}} f_c \cdot g   =  \,\,\,\,\,\, \int_{\gamma_{-i}} f _c\cdot g & = & 21913581880269709 \bmod{13^{15}}.
\end{eqnarray*}
Following the notations in the rank $(1,1)$ setting of Conjecture $\shi$, the pair $(\varrho_1,\varrho_2)$ of
irreducible subrepresentations of $V_{gg}$
attached to   the curves $E_a$ and $E_c$ can be taken to be
  $(\chi_1,\chi_{31})$ and $(\chi_1,\chi_{5})$, respectively, where
  $\chi_1$ is the trivial character. 
    Observe that the eigenvalues of $\sigma_{13}$ 
are $(1,-1)$ in both cases. One checks to $15$-digits of $13$-adic precision
that
\[ \int_{\gamma_{i}} f_a \cdot g = \frac{80 \cdot \log_{E_a,13}(P_a) \log_{E_a,13}(P_{a,31})}{i \cdot 63 \cdot 13 \cdot \log_{13}(u^-)}\]
where $i $ is the unique square root of $-1$ in $\Z_{13}$ which is
$\equiv 5 \bmod{13}$,
and    $u^- = (1 + \sqrt{5})/2$ is a fundamental
unit for $\Q(\sqrt{5})$. 
Note that the same formula holds if one replaced $i$ by $-i$ on both sides.
Similarly one checks to the same amount of $13$-adic precision that
\[  \int_{\gamma_{i}} f_c \cdot g = -\frac{80 \cdot \log_{E_c,13}(P_c) \log_{E_c,13}(P_{c,5})}{21 \cdot 13 \cdot \log_{13}(u^-)}.\]

Turning now to $p = 7$ and the curve $E_a$,   observe that $(\chi_1(7), \chi_{31}(7))=(1,1)$, and hence  
the regulator term  in Conjecture $\shi$ vanishes.
This explains why
 the iterated integrals attached to $(f_a, \gamma_{\pm 1}, g)$ 
 were observed to
 vanish, at least  numerically
to $30$ digits of $7$-adic precision.
The situation for the  curve $E_c$ is different, since 
$(\chi_1(7),\chi_5(7)) = (1,-1)$. As predicted by Conjecture $\shi$,
the authors were able to verify, to $30$ digits of $7$-adic precision, that
\[ 
\int_{\gamma_{\pm 1}} f_c \cdot g  = - \frac{8 \cdot \log_{E_c,7}(P_c)  \log_{E_c,7} (P_{c,5})} {3 \cdot 7 \cdot \log_7(u^-)}.
\]
\end{example}
\medskip \medskip



\medskip\noindent
The final three chapters 
summarize  some of the numerical evidence that was obtained
 in support of Conjecture $\shi$  in settings where $g$ and $h$ are not both theta series.
%% induced from characters of a common quadratic field. 
 It is an apt moment to pause momentarily and thank again 
 Steve Donnelly: most of the interesting global points
in these chapters were found by him upon our request using his descent code.
We might also stress here that although the overconvergent projection
 algorithms used to compute $p$-adic iterated integrals are efficient,  such computations still remain something of a technological
{\em tour de force}: many of our examples pushed the available hardware and software (the algorithms of \cite{lauder} in particular) to their
current limits.

 

\section{Stark points over   $A_4$, $S_4$ or $A_5$ extensions and exotic weight one forms}
\label{sec:exotic-points}
This  chapter focusses on the case where $V_{gh}$ is reducible and at least one of $g$ or $h$ is an 
 exotic cuspform whose associated 
projective Galois representation  cuts out  a non-totally
real  extension of $\Q$ with Galois group $A_4$, $S_4$ or $A_5$. 
(The case in which $V_{gh}$ is irreducible is very difficult to test experimentally, due to the
apparent scarcity of such $V_{gh}$ of manageable size occurring with multiplicity two in the Mordell-Weil
group of elliptic curves of reasonable conductor. See \cite[Example 6.2]{DLR2} though for a
successful experiment in the analogous setting in which Mordell-Weil groups are replaced
by groups of units.)
\begin{lemma}
\label{lemma:from-DR2}
Assume that one of $g$ or $h$ is an exotic weight one form and that $g$ and 
$h$ satisfy the self-duality condition. Then 
the representation $V_{gh}$ is irreducible, unless 
%% $h$ is the twist of $g^*$ by a quadratic character $\iota$, 
 $g$ is the twist of $h^*$ by a quadratic character $\iota$,
in which case
$$V_{gh} = L(\iota) \oplus \mathrm{Ad}_g(\iota).$$
\end{lemma}
\begin{proof}
The discussion in Section 2
of \cite{DLR2} rules out the case where $V_{gh}$ contains an irreducible subrepresentation of 
dimension two. Hence $V_{gh}$ contains an irreducible one-dimensional subrepresentation,
and the result follows from Schur's Lemma (cf.~\cite[\S 2]{DLR2}).
\end{proof} 
So if $V_{gh}$ is reducible, Lemma \ref{lemma:from-DR2} implies that 
%% a decomposition as in Conjecture $\shi$
%% can only arise when 
%% $h$ is a quadratic twist of $g^*$.
 $g$ is a quadratic twist of $h^*$.
Since it is always possible to replace the form
$f$ by one of its quadratic twists,
there is no real loss in generality in assuming that 
%% $h=g^*$,
$g = h^*$
which is what shall be done in the rest of this chapter.
In this setting, 
Conjecture $\shi$ suggests a   new, 
% for curves of rank $1$ over $\Q$ and which satisfy a suitable hypothesis over the extension field 
 efficiently computable (conjectural) analytic formula for the
$p$-adic logarithm of  global points defined over the $A_4$, $S_4$ or $A_5$-extension cut out by the 
adjoint $\mathrm{Ad}_g$ (or, equivalently, by the projective representation attached to $g$).


 
\subsection{Theoretical preliminaries}
\label{sec:exotic-points-statement}
%% Let $g$ be an exotic weight one form,   let $h = g^*$ be its conjugate form, and 
 %% let $L$ be as before the field generated by the Fourier coefficients of $g$, or of $g^*$.
Let $h$ be an exotic weight one form,   let $g = h^*$ be its conjugate form, and 
 let $L$ be as before the field generated by the Fourier coefficients of $h$, or of $g = h^*$.
The representation $V_{gh} = L \oplus \mathrm{Ad}_g$,
  where $L$ refers to the trivial representation of $G_\Q$ with coefficients in $L$,
 factors through $\Gal(H/\Q)$, where
   $H$ is the field cut out by the 
   {\em projective } representation attached to $g$, so that
   $$ \Gal(H/\Q) \simeq A_4, S_4 \mbox{ or } A_5.$$
   Conjecture $\shi$ predicts that the trilinear form $I_p'$ in \eqref{eqn:trilinear-form-iterated-integral}
   arising from $p$-adic 
    iterated integrals  
    %%  is non-identically zero 
   %% precisely when either 
   may be non-identically zero only when
\begin{equation}
\label{ranks1}   
r(E,L) = r(E,\mathrm{Ad}_g) = 1,
\end{equation}
 or
 \begin{equation}
\label{ranks2}   
r(E,L) = 0, \qquad  r(E,\mathrm{Ad}_g) = 2.
\end{equation}
In the rank $(1,1)$ setting,
Conjecture $\shi$ asserts   that, for a suitable choice of    elements
   $$P_\Q\in E(\Q), \qquad 
   P_{g_\alpha} \in (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = \alpha_g/\beta_g}, \qquad \qquad
   u_{g_\alpha} \in ((\cO_H^\times)_L^{\mathrm{Ad}_g})^{\sigma_p = \alpha_g/\beta_g},$$ 
 which, since $\alpha_g\ne \pm \beta_g$, are {\em unique }
  up to scaling by $L^\times$,  and for a suitable choice  
  $\testf$, $\testgamma_{g_\alpha}$ and $\testh$ of test vectors attached to $(f,g_\alpha,h)$,
   \begin{equation}\nonumber
  %% \label{eqn:exotic}
    \int_{\testgamma_{g_\alpha}} \!\!\!\! \testf \cdot \testh \ \  \stackrel{?}{=} \ \ 
    \frac{ \log_{E,p}(P_\Q) \log_{E,p}(P_{g_\alpha}) }
   { \log(u_{g_\alpha})}. 
  \end{equation}
  In the rank $2$ setting, 
  Conjecture $\shi$ asserts that
    \begin{equation}\nonumber
  %% \label{eqn:exotic}
    \int_{\testgamma_{g_\alpha}} \!\!\!\! \testf \cdot \testh \ \  \stackrel{?}{=} \ \ 
    \frac{ \log_{E,p}(P_1) \log_{E,p}(Q_{g_\alpha})   -   \log_{E,p}(Q_1) \log_{E,p}(P_{g_\alpha}) 
    }
   { \log(u_{g_\alpha})},
  \end{equation}
  where $(P,Q)$ is a basis of $\Hom_{G_\Q}(\mathrm{Ad}_g, E(H)_L)$,
  and 
  $P_1$, $Q_1$ (resp.\,$P_{g_\alpha}$, $Q_{g_\alpha}$) denote the images of a vector $v_1\in \mathrm{Ad}_g^{\sigma_p=1}$ (resp.\,$v_2\in \mathrm{Ad}_g^{\sigma_p=\alpha_g/\beta_g}$)
  under these homomorphisms.
  

  
In the rank $(1,1)$ setting, the conjectural formula above gives an explicit, albeit conjectural, $p$-adic analytic expression for a global point---namely $P_{g_\alpha}$---defined over the field $H$ cut out by $\mathrm{Ad}_g$, depending on quantities which are a priori accessible computationally:
\begin{equation}
P_{g_\alpha} \stackrel{?}{=} \exp_{E,p} \left(  \frac{ \log(u_{g_\alpha})}{ \log_{E,p}(P_\Q) }\cdot     \int_{\testgamma_{g_\alpha}} \!\!\!\! \testf \cdot \testh\right).
\end{equation}

The point  $P_{g_\alpha} $ can be viewed as an elliptic curve analogue of the 
 Stark unit $u_{g_\alpha}$.
 For the sake of numerical experiments, it
  is worth giving  a more explicit description of   
   $P_{g_\alpha}$, depending on the nature of $\Gal(H/\Q)$.
   
   \subsubsection{Cases where $g$ is of tetrahedral type}
   \label{sec:A4-analysis}
   The group $G=\Gal(H/\Q) \simeq A_4$ has three representations of dimension one which factor through the quotient
   $A_4/V_4 \simeq A_3$,  and a unique faithful irreducible representation of dimension three,
   which is therefore isomorphic to the adjoint representation $\mathrm{Ad}_g$.
  Let $R(x)$ be any degree four polynomial whose splitting field is $H$, and let $M$ be the quartic field which is generated 
  by a single root of $R$. Then $\mathrm{Ad}_g$ is related to a permutation representation by the formula 
  $$ \mathrm{Ad}_g = \Ind_{G_M}^{G_\Q}  L  - L.$$
  It follows that
  $$ r(E,\mathrm{Ad}_g)  \ =  \  r(E, \Ind_{M}^\Q L) - r(E,L) \ = \  \mbox{rank}(E(M)) - \mbox{rank}(E(\Q)).$$
  (The Artin formalism for $L$-series implies similar equalities for the analytic ranks.)
  In particular when $r(E,\mathrm{Ad}_g)=1$, there is (up to scaling)  a unique point $P\in E(M)$ whose trace to $E(\Q)$ is zero, and
  $E(H)_L^{\mathrm{Ad}_g}$ is generated, as an $L$-vector space, by the four Galois translates of $P$, whose sum is zero.
  Let $P=:P_1$,  $P_2$ , $P_3$ and $P_4$ denote these Galois translates, on which $\Gal(H/\Q)=A_4$
   acts via the natural permutation action.
The point  $P_{g_\alpha}$  can be expressed as a linear combination of 
the $P_j$,  depending on the conjugacy class of $\sigma_p$.
  
  \medskip\noindent
  {\em Case 1}: $\sigma_p =1$.
  In the case $\alpha_g/\beta_g = \beta_g/\alpha_g =1$, the representations 
  $\varrho_g$ and  $\mathrm{Ad}_g$ 
  are not regular at $p$, and Hypothesis
  C is not   satisfied. 
% Conjecture  $\shi$ then only asserts, when the $p$-adic iterated integrals attached to  $g_\alpha$ are defined, that $P_{g_\alpha}$ is some    undetermined linear combination of $P_1$, $P_2$, $P_3$ and $P_4$,  which all  belong to $E(\Q_p)$ after a $p$-adic embedding of $H$ has been chosen.
  
  \medskip\noindent
  {\em Case 2}: $\sigma_p$ is of order $2$. 
  In that case it may be assumed that
  the points $P_1$ and $P_2$ are defined over $\Q_{p^2}$ and are interchanged by $\sigma_p$, and 
  likewise for $P_3$ and $P_4$. It follows that  $\alpha_g/\beta_g = -1$ and that
  \begin{eqnarray*}
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = 1} &=&  L\cdot (P_1+P_2) = L\cdot (P_3+P_4) \qquad \mbox{ is one-dimensional}, \\
    (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = -1} &=&  L\cdot (P_1-P_2) \oplus L\cdot (P_3-P_4) \qquad \mbox{ is two-dimensional}.
    \end{eqnarray*}
   In this scenario, in which $V_g$ is regular but $\mathrm{Ad}_g$ is not,
    the point  $P_{g_\alpha}$ is a specific $L$-linear combination of
    the points $P_1-P_2$ and $P_3-P_4$, which lies in the well-defined one-dimensional subspace
    $\hom(V_g^{g_\beta},V_g^{g_\alpha})$ of $\mathrm{Ad}_g$.
    
    
    \medskip\noindent
  {\em Case 3}: $\sigma_p$ is of order $3$. 
  In that case, it  may be assumed that $\sigma_p$ fixes the point  $P_1$, which is thus defined over $\Q_p$,
  and cyclically  permutes the points $P_2$, $P_3$ and $P_4$. 
  The eigenvalues of $\sigma_p$ on $\mathrm{Ad}_g$ are then equal to $1$, $\alpha_g/\beta_g = \zeta_3$ and
  $\beta_g/\alpha_g = \zeta_3^{-1}$,  for some primitive cube root $\zeta_3$ of $1$,
   and $P_{g_\alpha}$ is determined, up to scaling, by the rule
 \begin{equation}
 \label{eqn:Pgalpha-A4-3}
  P_{g_\alpha} = P_2 + \zeta_3^{-1} P_3 + \zeta_3 P_4.
  \end{equation}
  In particular, 
  \begin{eqnarray*}
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = \alpha_g/\beta_g} &=&  L\cdot P_{g_\alpha}   \qquad \mbox{ is one-dimensional}.
    \end{eqnarray*}
    This is the only regular case, in which the point $P_{g_\alpha}$ is completely pinned down by its specified behaviour under the action of $\sigma_p$. 
   
   
   \begin{remark}
   There are   two distinct conjugacy classes of elements of order $3$ in $A_4$, 
    which are rational over $\Q(\zeta_3)$ and interchanged by the action of $\Gal(\Q(\zeta_3)/\Q)$. 
 Note  that $8$ of the $12$ elements in $A_4$ are of order $3$, and hence, by the 
   Chebotarev density theorem, the frobenius element $\sigma_p$ is regular for $\mathrm{Ad}_g$
    for a set of primes $p$
   of Dirichlet density $2/3$.
   \end{remark}
    
    
    
   \subsubsection{Cases where $g$ is of octahedral type}
   \label{sec:S4-analysis}
   The dimensions of the irreducible representations of the group $G=\Gal(H/\Q) \simeq S_4$ 
   are $ 1, 1, 2, 3, $ and $3$. The first two representations are the trivial and sign representations,
   and the two-dimensional representation 
   factors through the quotient $S_4/V_4 = S_3$. 
   In particular, the only faithful irreducible
    representations of $G$ are three-dimensional, and there are two such, whose characters are given in the 
   following table:
   $$
   \begin{array}{r|c|r|c|c|r|}
  \mbox{ size}      &  1    & 6       & 3           &   8      & 6  \\ 
\mbox{class}      &  1   & (12)   & (12)(34) & (123) & (1234)  \\ \hline 
\varrho_1                 &  3    & 1        & -1     &          0     &   -1   \\
\varrho_2                  & 3    & -1       & -1      &         0     &   1 
\end{array}
$$
The representation $\varrho_1$ occurs in the natural permutation representation of $G$ on $4$ letters,
while $\varrho_2$ is just the twist of $\varrho_1$ by the sign character.
Among the two representations   it is $\varrho_2$ which is the adjoint representation.
This can be seen by noting that the eigenvalues of $\varrho_1((1234))$ are visibly $-1$, $i$ and $-i$,
and that those of $\varrho_2((1234))$ are $1$, $i$, and $-i$, and that only the latter set of 
eigenvalues is consistent with the property of the  adjoint representation  that all elements in $G$
admit $1$ as eigenvalue.
  
  Let $R(x)$ be any degree four polynomial whose splitting field is $H$, and let $M$ be the quartic field which is generated 
  by a single root of $R$. 
  The field $H$ is an $S_3$-extension of $M$ and hence  there is  a unique quadratic  
  extension $\tilde M$ of $M$  contained in $H$.  The field $H$ also contains a unique quadratic field $K/\Q$,
  which is cut out by the sign character of $S_4$ and satisfies ${\tilde M } = M K$. 
 The  representation $\mathrm{Ad}_g$ is described by the formula
 $$ \mathrm{Ad}_g = \Ind_{G_M}^{G_\Q}  \sgn   - \sgn,$$
  where $\sgn$ is  viewed both as the quadratic character of $G_M$ and $G_\Q$ 
  induced by the sign characters on the finite quotients
   $\Gal(H/M) \simeq S_3$ and $\Gal(H/\Q) \simeq S_4$.
  It follows that
  $$ r(E,\mathrm{Ad}_g) = r(E, \Ind_{M}^H \sgn) - r(E,\sgn) = \dim_L E({\tilde M})^{-}_L - \dim_L E(K)^{-},$$
  where the superscript $-$ refers to the $-1$ eigenspace for the involutions in $\Gal(\tilde M/M)$ and 
  $\Gal(K/\Q)$ respectively.
 When $r(E,\mathrm{Ad}_g)=1$, there is, by Frobenius reciprocity, a unique point $P\in E({\tilde M})^{-}$, 
 up to scaling. Let 
  $P_1$, $P_2$, $P_3$ and $P_4$ denote the $A_4$ translates of $P=P_1$, and observe that
   $\sigma\in  G=S_4$ acts  on these points by the rule
  $$ \sigma P_j = \left\{ \begin{array} {rl} 
  P_{\sigma j}  & \qquad  \mbox{ if } \sigma \in A_4; \\
  -  P_{\sigma j} & \qquad  \mbox{ if } \sigma \in S_4-A_4.
  \end{array}\right.
  $$
  The  resulting explicit description of the action of $G$ on these four points, which together generate 
  $E(H)_L^{\mathrm{Ad}_g}$, make it possible  to describe the
  point $P_{g_\alpha}$ explicitly, depending on the conjugacy class  of $\sigma_p$ in $G$.
  
    
    \medskip\noindent
  {\em Case 1}: $\sigma_p =1$.
  In that case,   $\alpha_g = \beta_g$,
  and hence Hypothesis C' is not satisfied.
     
  \medskip\noindent
  {\em Case 2}: $\sigma_p$ is of order $2$ and odd, i.e., it is conjugate to the transposition
  $(12)$. 
 In that case, after relabelling the $P_j$,
 it can be  assumed that 
 $$ \sigma_p  P_1 = -P_2, \quad \sigma_p P_2 = -P_1, \qquad \sigma_p P_3 = -P_3, \quad \sigma_p P_4 = -P_4.$$
 In particular, the $x$-coordinates of the points $P_1$ and $P_2$ lie in $\Q_{p^2}$ and
 are permuted by $\Gal(\Q_{p^2}/\Q_p)$, while the $x$ coordinates of $P_3$ and $P_4$ are defined over $\Q_p$, even though all 
 four points are defined over $\Q_{p^2}$. 
 The eigenvalues of $\sigma_p$ are equal to 
 $1$ and $-1 = \alpha_g/\beta_g = \beta_g/\alpha_g$, and
  \begin{eqnarray*}
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = 1} &=&  L\cdot (P_1-P_2) \qquad \mbox{ is one-dimensional}, \\
    (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = -1} &=&  L\cdot  P_3  \oplus L\cdot P_4 \qquad \mbox{ is two-dimensional}.
    \end{eqnarray*}
    This is another irregular case, in which  the point 
    $P_{g_\alpha}$ is a specific  $L$-linear combination of
    $P_3$ and $P_4$ lying in the line $\hom(V_g^{g_\beta},V_g^{g_\alpha})$ of
    $\mathrm{Ad}_g$. 
    
   
    \medskip\noindent
  {\em Case 3}: $\sigma_p$ is of order $2$ and even, i.e., it is conjugate
   to the permutation $(12)(34)$.
  After eventually relabelling the $P_j$, it can be  assumed that
  $$ \sigma_p P_1 = P_2, \qquad \sigma_p P_2 = P_1, \qquad \sigma_p P_3=P_4, \qquad \sigma_p P_4 = P_3,$$
  so that
   \begin{eqnarray*}
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = 1} &=&  L\cdot (P_1+P_2) = L \cdot (P_3+P_4) \qquad \mbox{ is one-dimensional}, \\
    (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = -1} &=&  L\cdot  (P_1-P_2)  \oplus L\cdot (P_3-P_4) \qquad \mbox{ is two-dimensional}.
    \end{eqnarray*}
This is yet another irregular case,  and $P_{g_\alpha}$ is some unspecified 
 $L$-linear combination of
  $P_1-P_2$ and   $P_3-P_4$. 

  
    
    
    \medskip\noindent
  {\em Case 4}: $\sigma_p$ is of order $3$. 
  In that case,  it can be assumed that
  $\sigma_p$ fixes the point  $P_1$, which is defined over $\Q_p$,
  and permutes the points $P_2$, $P_3$ and $P_4$ cyclically. 
  The eigenvalues of $\sigma_p$ on $\mathrm{Ad}_g$ are then equal to $1$, $\alpha_g/\beta_g = \zeta_3$ and
  $\beta_g/\alpha_g = \zeta_3^{-1}$,  and $P_{g_\alpha}$ is determined, up to scaling, by the rule
  $$ P_{g_\alpha} = P_2 + \zeta_3^{-1} P_3 + \zeta_3 P_4.$$
  In particular,  
  \begin{eqnarray*}
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = \alpha_g/\beta_g} &=&  L\cdot P_{g_\alpha}   \qquad \mbox{ is one-dimensional}.
    \end{eqnarray*}
    
        
  \medskip\noindent
    {\em Case 5}: $\sigma_p$ is of order $4$, and hence is conjugate to $(1234)$.
    In that case, it can be  assumed that 
    $$ \sigma_p P_1 = -P_2, \qquad \sigma_p  P_2 = -P_3, \qquad  P_3 = -P_4, \qquad \sigma P_4 = -P_1.$$
    The eigenvalues of $\sigma_p$ are   equal to $1$, $i$ and $-i$, and  
    \begin{eqnarray}
    \nonumber
   (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = 1} &=&  L\cdot (P_1-P_2+P_3-P_4),   \\
   \label{eqn:S4-case5}
    (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = i} &=&  L\cdot  (P_1 +i P_2 -P_3 - i P_4), \\
     \label{eqn:S4-case5b}
     (E(H)_L^{\mathrm{Ad}_g})^{\sigma_p = -i} &=&  L\cdot  (P_1 -i P_2 -P_3+ i P_4). 
    \end{eqnarray} 
  In particular, the $\alpha_g/\beta_g$-eigenspace for $\sigma_p$ is  one-dimensional if and only if
    $\sigma_p$ has order $>2$, which occurs for $14$ of the $24$ permutations in $S_4$. 
   This regularity condition is therefore satisfied for a set of primes $p$ of density $7/12$.
    
    
    
     
        
   \subsubsection{Cases where $g$ is of icosahedral type}
   The dimensions of the irreducible representations of the group $G=\Gal(H/\Q) \simeq A_5$ 
   are $ 1, 3, 3, 4$  and $5$, and the characters of the two distinct three-dimensional representations
     are described in the following table:
    $$
   \begin{array}{r|c|c|c|c|c|}
  \mbox{ size}      &  1     & 15           &  20      & 12  &  12  \\ 
\mbox{class}      &  1      & (12)(34) & (123)  & (12345)  &  (12354) \\ \hline 
\varrho_1       &  3    & -1        &   0    &   \frac{1+\sqrt{5}}{2}           &   \frac{1-\sqrt{5}}{2}    \\
\varrho_2        & 3    & -1       &   0     &     \frac{1-\sqrt{5}}{2}     &    \frac{1+\sqrt{5}}{2} 
\end{array}
$$
In this case, both the  representations $\varrho_1$ and $\varrho_2$ arise as the adjoint of the 
Artin representation $\varrho_g$ and $\varrho_g'$ respectively, where $\varrho_g'$ denotes the
Artin representation of the conjugate form of $g$, whose Fourier coefficients lie in the field $\Q(\sqrt{5})$.

It turns out to be  easier to describe the 
direct sum $\varrho_1 \oplus \varrho_2$ in Galois-theoretic terms rather than each $\varrho_i$ individually. 
Let $D_{10}$ be the dihedral group of order $10$,
embedded naturally in $A_5$, and let $M$ be the non-Galois sextic field fixed by $D_{10}$.  
The field $M$ is generated by a root of  a sextic polynomial 
   $R(x)$, the so-called {\em Lagrange sextic resolvent}
   attached to the $A_5$ extension.
  Since $\Gal(H/M) = D_{10}$, there is  a unique quadratic  
  extension $\tilde M$ of $M$  contained in $H$, which is cut out by the unique
  non-trivial quadratic character $\chi^-$  of $D_{10}$.
 It is not hard to check the identity of virtual representations: 
  $$ \varrho_1 \oplus \varrho_2   = \Ind_{G_M}^{G_\Q}  \chi^-, $$
  which implies that the six-dimensional representation
  $ E(H)_L^{\mathrm{Ad}_g} \oplus E(H)_L^{W_{g'}}  $ is  generated as an $L$-vector space, by the unique
  point $P\in E({\tilde M})^-$ together with its $12$ Galois translates
  $$ P=:P_1, P_2, P_3, P_4, P_5, P_6, \quad -P_1, -P_2, -P_3, -P_4, -P_5, -P_6.$$
  
  
  The Stark point $P_{g_\alpha}$ is a $\Q(\sqrt{5})$-linear combination   of the
 $6$ linearly independent points $P_j$ with $1\le j \le 6$.  The coefficients that enter in this linear combination are completely determined by the requirement that $\sigma_p P_{g_\alpha} = \alpha_g/\beta_g P_{g_\alpha}$, provided the frobenius element $\sigma_p$ is not of order $\le 2$.
 Since $A_5$ has precisely $44$ elements of order $>2$,  this condition holds for a set of primes of
 Dirichlet density $11/15$. 

        
    
\subsection{Numerical evidence}
\label{sec:exotic-points-evidence}

In contrast to the situation in Sections \ref{sec:stark-heegner-evidence} and \ref{sec:abelian-extensions-evidence}, numerical evidence
is somewhat more bountiful in the setting of exotic modular forms. In particular, despite the appellation ``exotic'' such forms seem to be  more plentiful in small 
level than those studied in Section \ref{sec:stark-heegner-evidence}. (Note that in all our exotic examples we checked that hypotheses A, B and C hold.)


%% In all the exotic examples that were considered
%% (other than Example \ref{ex:283}) there is a factorisation of $L$-functions
%% $$
%% L(E,\varrho_{gh},s) = L(E,s) \cdot L(E,\mathrm{Ad}_g,s) \quad \mbox{with} \quad \ord_{s=1}  L(E,s) = \ord_{s=1} L(E,\mathrm{Ad}_g,s) = 1
%% $$
%% which mirrors the decomposition appearing in Lemma \ref{lemma:from-DR2} (with $\iota$ trivial). Thus the global signs of $L(E,s)$ and $L(E,%% \mathrm{Ad}_g,s)$ are
%% the same (and equal to $-1$), and we verified in each example that the local signs of both $L$-functions are always the same at every place $v$ of $\Q$. This implies that Hypothesis A holds for these examples. (Hypothesis B certainly holds too, and Hypothesis C by Proposition \ref{BeDi-criterion}.)

As in Example \ref{ex:83} it is helpful to consider algebraic factors similar to those defined in
\cite[Theorem 3.1]{DR1}, given by
%\[
%\begin{array}{rcl}
%{\mathcal E}(g_\alpha,f,h) & := & (1 - \alpha_g \alpha_f \alpha_h p^{-1}) \times (1 - \alpha_g \alpha_f \beta_h p^{-1}) \\
%& & \times \,(1 - \alpha_g \beta_f \alpha_h p^{-1}) \times (1 - \alpha_g \beta_f \beta_h p^{-1}) \\
%{\mathcal E}_0(g_\alpha) & := & (1 - \alpha_g^2 \chi^{-1}(p)) \\
%{\mathcal E}_1(g_\alpha) & := & (1 - \alpha_g^2 \chi^{-1}(p)p^{-1}).
%\end{array}
%\]
{\color{blue} 
\[
\begin{array}{rcl}
{\mathcal E}(g_\alpha,f,h) & := & (1 - \beta_g \alpha_f \alpha_h p^{-1}) \times (1 - \beta_g \alpha_f \beta_h p^{-1}) \\
& & \times \,(1 - \beta_g \beta_f \alpha_h p^{-1}) \times (1 - \beta_g \beta_f \beta_h p^{-1}) \\
{\mathcal E}_0(g_\alpha) & := & 1 - \beta_g^2 \chi(p) \\
{\mathcal E}_1(g_\alpha) & := & 1 - \beta_g^2 \chi(p)p^{-1}.
\end{array}
\]
}

Here when $p$ divides the level of $f$ we set $\alpha_f = a_p(f)$ and $\beta_f = 0$, and
otherwise define these so that $x^2 - a_p(f) x + p = (x - \alpha_f)(x - \beta_f)$. Economising on notation,
we define ${\mathcal E}_\alpha$ as
\begin{equation}\label{eqn-Ea}
{\mathcal E}_\alpha := \frac{{\mathcal E}(g_\alpha,f,h)}{{\mathcal E}_0(g_\alpha){\mathcal E}_1(g_\alpha)}  \in L.
\end{equation}
We define ${{\mathcal E}}_\beta$ similarly, switching the roles of $\alpha_g$ and $\beta_g$ in the above.
\medskip

Our first exotic example involves the tetrahedral form of level $133$ first constructed by Tate.
%% and studied from the point of view of the original Stark conjectures in \cite{chinburg}.
   
\begin{example}
\label{ex:133}
Let $\chi$ be a Dirichlet character of level $133 = 7 \cdot 19$ of order $2$ at $7$ and order $3$ at $19$. (There are two
such characters which are Galois conjugate to each other.) 
Denote by $\zeta$ a primitive $12$-th root of unity and define $\zeta_3 := \zeta^{-4}$, a primitive cube root of unity.
Then $S_1(133,\chi)$ is defined over
$\Q(\zeta_3)$ and has dimension two. It contains two eigenforms defined over $\Q(\zeta)$, namely
\[ h = q + (\zeta^2 - 1) q^2 - \zeta  q^3 + \zeta  q^5 + (-\zeta^3 + \zeta) q^6 + \zeta^3 q^7 - 
        q^8 + (\zeta^3 - \zeta) q^{10} + (-\zeta^3 + \zeta)q^{13} - \cdots \]
and its Galois conjugate obtained by replacing $\zeta$ by $-\zeta$. 
% Let $g = \bar h$.
Let $g = h^*$, the twist of $h$ by $\chi^{-1}$ (this has Fourier coefficients
complex conjugate to those of $h$).
The $A_4$-extension $H$ cut out by the
projective representation attached to $g$ is the normal closure of
the quartic field 
$$ M:= \Q(w), \qquad w^4 +3 w^2 -7w +4 = 0, $$ 
which has discriminant $133^2$.



Consider the elliptic  curves
\[
\begin{array}{l}
E_a :  y^2 + y = x^3 + x \\ 
E_b : y^2 + y = x^3 + x^2 - 7x + 5 
\end{array}
\]
of conductor $91 = 7 \cdot 13$ labelled $91a$ and $91b$ in Cremona's tables and write $f_a$ and $f_b$ for the
associated weight $2$ newforms. 
These curves both satisfy the conditions in the rank $(1,1)$ setting of Conjecture $\shi$, namely
$$
r(E_a,1) =1, \qquad r(E_a, \mathrm{Ad}_g)=1, 
\qquad 
r(E_b,1) =1, \qquad r(E_b, \mathrm{Ad}_g)=1.
 $$ 
%% as expected, similar identities hold for  the analytic ranks.
Let $P_\Q^{(a)}$ and $P_\Q^{(b)}$ denote Mordell-Weil generators of $E_a(\Q)$ and 
$E_b(\Q)$ respectively (modulo the torsion for $91b$), and let $P^{(a)}$ and $P^{(b)}$ denote generators of the  one-dimensional subspaces
of $E_a(M)_\Q$ and $E_b(M)_\Q$ respectively, 
consisting of   points with trace $0$.
A combination of naive search and two-descent shows that one can take
$$
\begin{array}{ll}
 P_\Q^{(a)}=(0,0), &   P^{(a)} = \left(\frac{-156 w^3 - 64 w^2 - 598 w + 654}{289} , \frac{823 w^3 + 1464 w^2 + 5515 w - 7663}{ 4913} \right), \\
 & \\
 P_\Q^{(b)} = (-1,3), &   P^{(b)} =  (9 w^3 + 5 w^2 + 31 w - 45 , -w^3 + 16 w^2 + 16 w + 83).
 \end{array}
 $$
 

Letting $p = 13$,
 the Hecke polynomial $t^2 - a_{13}(h) t + \chi(13)$ 
 has roots 
 $\beta_h = \zeta$ and $\alpha_h = - \zeta^3$.  
Hence  
the  two $13$-stabilisations $g_\alpha$ and $g_\beta$ of $g$ 
with $U_p$-eigenvalue 
$\alpha_g = \overline{\beta_h} = \zeta^{-1}$ and $\beta_g   = \overline{\alpha_h} =  \zeta^3$ 
 are distinct.
(Note we are twisting by $\chi^{-1}$ here and $\chi^{-1}(p) \beta_h = (\alpha_h \beta_h)^{-1}
\beta_h = \overline{\alpha_h}$, and likewise $\chi^{-1}(p) \alpha_h = \overline{\beta_h}$.)
Let $\gamma_{g_\alpha}$ and $\gamma_{g_\beta}$ denote the corresponding dual elements. 

Choose the embedding of $\Q(\zeta)$ into $\Q_{13}$ which sends $\zeta$ to the unique $12$th root of unity in
$\Z_{13}$ which is congruent to $6$ modulo $13$.
The values of the $p$-adic iterated integrals, evaluated to $20$ digits of $13$-adic accuracy,
 are then given by the four  quantities
%\[ 
%\begin{array}{rcl}
%\int_{\gamma_{g_\alpha}} f_a \cdot h & = & -6603156733219298602528 \bmod{13^{20}}\\
%\int_{\gamma_{g_\beta}} f_a \cdot h & = & 2705941994573919260213 \bmod{13^{20}}\\
%\int_{\gamma_{g_\alpha}} f_b \cdot h & = & -679336918469999467881 \times 13  \bmod{13^{20}}\\
%\int_{\gamma_{g_\beta}} f_b \cdot h & = & -2983964554989928769557 \bmod{13^{20}}.
%\end{array}
%\]
{\color{blue}

\[
\begin{array}{rcl}
\int_{\gamma_{g_\alpha}} f_a \cdot h & = & 2705941994573919260213 \bmod{13^{20}}\\
\int_{\gamma_{g_\beta}} f_a \cdot h & = &  -6603156733219298602528 \bmod{13^{20}}\\
\int_{\gamma_{g_\alpha}} f_b \cdot h & = &   -2983964554989928769557 \bmod{13^{20}}\\
\int_{\gamma_{g_\beta}} f_b \cdot h & = &  -679336918469999467881 \times 13 \bmod{13^{20}}.
\end{array}
\]

}

The ratio $\alpha_g/\beta_g = \zeta^{-4} = \zeta_3$ is  a primitive cube root of unity, and hence 
the eigenvalues of $\sigma_{13}$ on $\mathrm{Ad}_g$ are $1$, $\zeta_3$ and $\zeta_3^2$.
The fact that $\sigma_{13}$ has order $3$ puts us 
 in {\it Case 3} in the analysis in Section 
\ref{sec:A4-analysis}. 
Let 
 $$
 P_{g_\alpha}^{(a)}, P_{g_\beta}^{(a)} \in E_a(H)_L^{\mathrm{Ad}_g}, \qquad
  P_{g_\alpha}^{(b)}, P_{g_\beta}^{(b)} \in E_b(H)_L^{\mathrm{Ad}_g}$$
be the points constructed from $P^{(a)}$ and $P^{(b)}$ using 
 \eqref{eqn:Pgalpha-A4-3}.
  The units $u_{g_\alpha}$ and $u_{g_\beta}$ are constructed using the same
formula  with the point $P$ replaced by the  unit $u_1 := -w + 1$  which
 is the root of $x^4 - 4 x^3 + 9 x^2 - 3 x + 1 = 0$ in $M$ and its Galois conjugates $u_2$,
 $u_3$ and $u_4$.
 
It then turns out that to $20$ digits of $13$-adic accuracy,
\begin{eqnarray*}
\int_{\gamma_{g_\alpha}} f_a \cdot g  & = &  -\zeta_3^2 \cdot {\mathcal E}_\alpha \cdot \frac{ \log_{E_a,13}(P_\Q^{(a)}) \log_{E_a,13}(P_{g_\alpha}^{(a)})}{\log_{13}(u_{g_\alpha})}, \\
 \int_{\gamma_{g_\beta}} f_a \cdot g  & = &  -\zeta_3 \cdot {\mathcal E}_\beta \cdot \frac{ \log_{E_a,13}(P_\Q^{(a)}) \log_{E_a,13}(P_{g_\beta}^{(a)})}{\log_{13}(u_{g_\beta})},
\end{eqnarray*}
where ${\mathcal E}_\alpha$ is defined as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$.

 A similar pattern recurs with the elliptic curve $E_b$, for which  it
 was checked to $20$ digits of $13$-adic precision that
 \begin{eqnarray*}
 \int_{\gamma_{g_\alpha}} f_b \cdot h & = &  \frac{\zeta}{3} \cdot  {\mathcal E}_\alpha \cdot \frac{\log_{E_b,13}(P_\Q^{(b)}) \log_{E_b,13}(P_{g_\alpha}^{(b)})}{\log_{13}(u_{g_\alpha})}, \\
  \int_{\gamma_{g_\beta}} f_b \cdot h & =  & - \frac{\zeta^{-1}}{3} \cdot {\mathcal E}_\beta \cdot \frac{\log_{E_b,13}(P_\Q^{(b)}) \log_{E_b,13}(P_{g_\beta}^{(b)})}{\log_{13}(u_{g_\beta})},
\end{eqnarray*}
with ${\mathcal E}_\alpha$ as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$. 
(Here $\zeta \cdot {\mathcal E}_\alpha / 3 = \zeta^{-1} \cdot {\mathcal E}_\beta/ 3 = 4/(13 \sqrt{3})$.)

The choice of the prime $p = 13$ in the above calculations was dictated largely by convenience.
This prime has the virtue of splitting completely in $L=\Q(\zeta)$, so that $L$ embeds into $\Q_{13}$,
and it divides  the level of $f_a$ and $f_b$, which 
can thus be viewed as overconvergent $13$-adic  modular forms of 
tame  level $133$.
\end{example}


\medskip \medskip
The weight one tetrahedral form  
$g$ of level $133$ was  considered by Chinburg 
\cite{chinburg}   in connection with Stark's  original conjectures, 
motivated by the  final sentence in \cite{stark2},
\begin{quote}
{\em A meaningful verification for $N = 133$ would be of some interest.}
\end{quote}
Stark's concluding comment follows the verification of his conjecture for the weight one
modular form  considered in Example \ref{ex:145}.

\medskip
The next example involves the tetrahedral  modular form of minimal level $124 = 4 \cdot 31$ \cite[\S 2.2]{Buzzard}
which is also considered %% albeit in somewhat lesser detail, 
in \cite{chinburg}. It illustrates our conjecture in both rank $(1,1)$ and rank $2$ exotic settings.

\begin{example}\label{Ex-124}
%% See Sept10_124_p13_m15_again.m

Let $\chi$ be a sextic  Dirichlet character of conductor $124 = 4\cdot 31$, having order $2$ at $4$ and order $3$ at $31$. 
Denote by $\zeta$ a primitive $12$-th root of unity and define $\zeta_3 := \zeta^2 - 1$, a primitive cube root of unity.
The space 
$S_1(124,\chi)$ is defined over
$\Q(\zeta_3)$ and has dimension two. Just as in the previous example,
it contains two eigenforms, which are defined over $\Q(\zeta)$
and conjugate over
$\Q(\zeta_3)$, namely
\begin{equation*}
\begin{split}
h = q + \zeta^3 q^2 + \zeta q^3 - q^4 - & \zeta^2 q^5 + (\zeta^2 - 1) q^6 - \zeta q^7 - \\ 
   \zeta^3 q^8 + (-\zeta^3 + \zeta) q^{10} & 
         + (\zeta^3 - \zeta) q^{11} - \zeta q^{12} + 
        \zeta^2 q^{13} + \cdots
 \end{split}
 \end{equation*}
and its conjugate obtained by replacing $\zeta$ by $-\zeta$. Let $g = h^*$.
The field cut out by the projective representation attached to $g$ is 
the normal closure $H$ of the quartic field
$$M  = \Q(w), \qquad 
w^4 + 7 w^2 - 2 w + 14 = 0,$$
which  has discriminant $124^2$ and Galois group isomorphic to  $A_4$. 






Consider the elliptic  curve
\[ E: y^2 = x^3 + x^2 - 2x + 1 \]
of conductor $124$ labelled $124a$ in Cremona's tables and write $f$ for the associated weight $2$ newform.
The curve $E$ has rank one  over $\Q$ (with a  torsion subgroup of order $3$)
and rank two over $M$---this places us in the rank $(1,1)$ setting. Following the same notations of the previous example, 
generators for $E(\Q)$ and for the subgroup of points in $E(M)$ of trace zero 
are given by
$$ P_\Q = (-2,1), \qquad P  = \left(\frac{w^3 - w^2 + 3 w - 7}{2} ,   \frac{w^3 + 4 w^2 + 9 w + 10}{4}\right).$$
These points were found using a naive search.
  
Choosing the prime 
 $p = 13$ as in the previous example,  the Hecke polynomial
  $t^2 - a_{13}(h) t + \chi(13)$ has roots $\beta_h =1$ and $\alpha_h = \zeta_3$.
Hence $\alpha_g =   1$ and $\beta_g = \zeta_3^2$ and therefore the 
$13$-stabilisations $g_\alpha $ and $g_\beta $ of $g$ are distinct.


Choose the embedding of $\Q(\zeta)$ into $\Q_{13}$ which sends $\zeta$ to the unique $12$-th root of unity
in $\Z_{13}$ which is congruent to $7$ modulo $13$.
The values of the
$13$-adic iterated integrals were computed to be
%\[ 
%\begin{array}{rcl}
%\int_{\gamma_{g_\alpha}} f  \cdot h & = & 684158979097768450057 \times 13 \bmod{13^{20}}\\
%\int_{\gamma_{g_\beta}} f \cdot h & = & 5953034774844704381281 \bmod{13^{20}}.
%\end{array}
%\]
%% Note that taking the conjugate of g get negatives of these values:
%% not sure why.
{\color{blue}
\[ 
\begin{array}{rcl}
\int_{\gamma_{g_\alpha}} f  \cdot h & = &  5953034774844704381281 \bmod{13^{20}}\\
\int_{\gamma_{g_\beta}} f \cdot h & = & 684158979097768450057 \times 13 \bmod{13^{20}}.
\end{array}
\]

}

Note that $\alpha_g/\beta_g$ and $\beta_g/\alpha_g$ are both primitive cube roots of unity, and 
hence, just as in the previous example,  the setting is the one discussed 
 in {\it Case 3} in the analysis in Section 
\ref{sec:A4-analysis}. Just as before, the points $P_{g_\alpha}$ and 
$P_{g_\beta}$ in $E(H)_L^{\mathrm{Ad}_g}$ can be defined  as suitable linear combinations of the point $P$
and its Galois conjugates, following the formula in 
 \eqref{eqn:Pgalpha-A4-3}.
  The units $u_{g_\alpha}$ and $u_{g_\beta}$ are likewise  constructed using the same
formula  with the point $P$ replaced by the  unit $u_1 :=  w^2 - w + 3$, 
the unique  root of $x^4 + 2 x^3 + 18 x^2 - 6 x + 1$ in $M$.

With these conventions in place, it has been  checked to $20$
digits of $13$-adic precision that
\begin{eqnarray*}
\int_{\gamma_{g_\alpha}} f  \cdot h  & = & \frac{2}{\sqrt{3}} \cdot {\mathcal E}_\alpha \cdot \frac{\log_{E,13}(P_\Q) \log_{E,13}(P_{g_\alpha})}{\log_{13}(u_{g_\alpha})},\\ %% \qquad\quad
\int_{\gamma_{g_\beta}} f  \cdot h & = & \frac{2}{\sqrt{3}} \cdot {\mathcal E}_\beta \cdot \frac{\log_{E,13}(P_\Q) \log_{E,13}(P_{g_\beta})}{\log_{13}(u_{g_\beta})}, 
\end{eqnarray*}
where ${\mathcal E}_\alpha$ is as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$, 
with $\sqrt{3} = \zeta^3 - 2 \zeta$.
\medskip

We now turn to the curve 
\[ E: y^2 + x y + y = x^3 - x^2 - 3 x + 3\]
of conductor $26$ labelled $26b$. One checks
\begin{equation}\label{eqn:Adrank2}
r(E) = 0,\,r(E,\Ad_g) = 2,
\end{equation}
and likewise for analytic ranks,
and so here we are in the rank $2$ setting.
The required points in $E(H)^{\Ad_g}$ may be constructed from the generators
\[ 
\begin{array}{rcl}
P & = & \left(\frac{283 w^3 + 1295 w^2 + 1442 w + 9528}{4802}, \frac{29371 w^3 - 
    35875 w^2 + 146552 w - 319814}{235298} \right)\\
    & \\
Q & = & \left(\frac{-25 w^3 + 98 w^2 + 41 w  + 775}{121} , \frac{67 w^3 + 3832 w^2 + 2465 w + 17404}{1331}
\right)
\end{array}
\]
of the rank $2$ Mordell-Weil lattice in $E(M)$. We define $P_1$ and $P_{g_{\alpha}}$
as in the  Case 3 analysis for tetrahedral forms (and similarly $P_{g_\beta}$ switching
$\zeta_3$ for $\zeta_3^{-1}$). Likewise we define the points $Q_1$, $Q_{g_\alpha}$ and
$Q_{g_\beta}$.

Writing now $f$ for the form attached to $E$ we find, with the same embedding as before, that
%\[ 
%\begin{array}{rcl}
%\int_{\gamma_{g_\alpha}} f \cdot h & = & 6721481652857547558786 \bmod{13^{20}}\\
%\int_{\gamma_{g_\beta}} f \cdot h & = & -2549577176522983012568 \bmod{13^{20}}.
%\end{array}
%\]
{\color{blue} \[ 
\begin{array}{rcl}
\int_{\gamma_{g_\alpha}} f \cdot h & = &  -2549577176522983012568 \bmod{13^{20}} \\
\int_{\gamma_{g_\beta}} f \cdot h & = &  6721481652857547558786 \bmod{13^{20}}.
\end{array}
\]
}

Then to 20-digits of $13$-adic precision
\begin{align}
\int_{\gamma_{g_\alpha}} f \cdot h  & =  \frac{\zeta_3 \cdot {\mathcal E}_\alpha}{\sqrt{3}} \cdot \frac{\log_{E,13} (P_1) \log_{E,13} (Q_{g_{\alpha}}) - \log_{E,13} (Q_1) \log_{E,13} (P_{g_{\alpha}})}
{\log_p u_{g_{\alpha}}}  \label{eqn:26b_alpha} \\
\int_{\gamma_{g_\beta}} f \cdot h  & =  -\frac{\zeta_3^{-1} \cdot {\mathcal E}_\beta}{\sqrt{3}} \cdot \frac{\log_{E,13} (P_1) \log_{E,13} (Q_{g_{\beta}}) - \log_{E,13} (Q_1) \log_{E,13} (P_{g_{\beta}})}
{\log_p u_{g_{\beta}}}\label{eqn:26b_beta}
\end{align}
where ${\mathcal E}_\alpha$ is as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$, exactly as predicted by
Conjecture $\shi$.

Finally we consider the curve
\[ E: y^2 = x^3 + x - 10 \]
of conductor $52$ labelled $52b$ for which again \eqref{eqn:Adrank2} holds and so we are in the rank $2$ setting. Here we take
\[
\begin{array}{l}
P  =  \left(\frac{-225 w^3 - 1533 w^2 - 1458 w + 1078}{2809} , \frac{-247041 w^3 + 
    68295 w^2 - 1265094 w + 317530}{297754} \right),\\
     \\
Q  =  \left( \frac{-63391 w^3 - 51546 w^2
    - 698338 w - 942346}{390625} , \frac{84293978 w^3 - 33352782 w^2 + 
    603648754 w - 1348508882}{244140625} \right),
\end{array}
\]
as generators for the Mordell-Weil lattice in $E(M)$ and construct the points
$P_1,P_{g_\alpha},P_{g_\beta}$ and $Q_1,Q_{g_\alpha},Q_{g_\beta}$ from them.
Letting $f$ now be the newform attached to $E$ we have
%\[ 
%\begin{array}{rcl}
%\int_{\gamma_{g_\alpha}} f \cdot h & = & -1602512412162131808818 \bmod{13^{20}}\\
%\int_{\gamma_{g_\beta}} f \cdot h & = & 4989460414989865668627 \bmod{13^{20}}
%\end{array}
%\]
{\color{blue}
\[ 
\begin{array}{rcl}
\int_{\gamma_{g_\alpha}} f \cdot h & = & 4989460414989865668627 \bmod{13^{20}} \\
\int_{\gamma_{g_\beta}} f \cdot h & = &  -1602512412162131808818 \bmod{13^{20}}
\end{array}
\]
}
and one checks to 20-digits of $13$-adic precision that again \eqref{eqn:26b_alpha} and
\eqref{eqn:26b_beta} hold. (The algebraic constants which occur in equation \eqref{eqn:26b_alpha}
for $26b$ and $52a$ are in fact exactly the same, and likewise for \eqref{eqn:26b_beta}: both curves have non-split
multiplicative reduction at $13$.)

\end{example}
\medskip \medskip

The following example illustrates the vanishing of $p$-adic
iterated integrals predicted by Conjecture $\shi$ in a rank $2$ setting but where the regulator term vanishes 
for trivial reasons. 

\begin{example}\label{ex:283}
Consider the curve
\[ E : y^2 + x y = x^3 - 18 x + 31\]
of conductor $4811 = 17 \cdot 283$ labelled $4811a$ in Cremona's table and write $f$ for the associated
weight $2$ newform. 
Letting  $h$ be the octahedral weight one 
 form in level $283$ with quadratic character and $g = h^*$, it is readily checked that
 $$ r(E/\Q) = r_{\an}(E/\Q)= 2, \qquad  r_{\an}(E,\mathrm{Ad}_g) = 0.$$
 (The third equality was obtained by calculating $L(E,\mathrm{Ad}_g,1)$ 
numerically to a modest precision sufficient to verify its
non-vanishing.)
Taking $p$ a regular prime for $\rho_g$, 
it is clear that   hypotheses A,B,C are satisfied  but   that the matrix
$R_{g_\alpha}(E,\varrho_{gh})$ of Conjecture $\shi$ has determinant $0$.
The prediction that all associated $p$-adic iterated 
integrals should vanish
was verified  for $p = 17$ to $5$ digits of $17$-adic accuracy.
\end{example}
\medskip 

We note also here that we tested the vanishing prediction of Conjecture $\shi$ in several examples when $r(E,\varrho_{gh}) = 4$ and $6$
(involving CM rather than exotic forms).
\medskip
%% N = 131, p = 5, with Eisenstein series, ranks are 1,2,2,1 for summands of dimension 1,1,1,1
%% The CM rank 2 example: taking h = Eisenstein series, we get ranks 2,2 for dimensions 2,2.
%% Many examples involving Eisenstein series for ranks 1,1,1,1 and summand dimensions 1,1,1,1.



Our final exotic example 
 involves the exotic octahedral modular form of minimal level $148 = 4\cdot 37$ 
 \cite[\S 2.3]{Buzzard} and rank $(1,1)$ settings of Conjecture $\shi$.

\begin{example}\label{ex:148}
Let $\chi$ be the Dirichlet character of level $148 = 2^2 \cdot 37$ which is trivial at $2$ and has order $4$ at
$37$ (there are two such characters and they are Galois conjugate). The space 
$S_1(148,\chi)$ is
defined over $L=\Q(i)$ and has dimension $1$. It contains the single eigenform
\[ h = q - i q^3 - q^7 + i q^{11} + (i - 1) q^{17} + (-i + 1) q^{19} + \cdots. \]
The field $H$ cut out by the projective representation attached to $g = h^*$ is the normal closure
of the quartic field
$$M : = \Q(a),  \qquad a^4 - a^3 + 5 a^2 - 7 a + 12 $$
of discriminant $2^2 \cdot 37^3$  and having Galois group $S_4$. 
The unique quadratic subfield of $H$ is $K = \Q(\sqrt{37})$, and hence 
$\tilde{M} = MK$ is the unique quadratic extension of $M$ contained in $H$.

Testing Conjecture $\shi$ in the rank $(1,1)$ setting requires 
elliptic curves $E$ of rank $1$ over $\Q$ for which
in addition $r(E,\mathrm{Ad}_g) = 1$.  
%% Under hypothesis A, the equality
%% $r_{\an}(E/\Q)= 1$ implies that $r(E,\mathrm{Ad}_g)$ is odd and (probably) equal to one 
%% in most of the small examples that lie in the range of numerical experiments.
We choose $p=17$ and consider elliptic curves $E$ of rank one and 
conductor dividing $148 p$ which satisfy the necessary
hypotheses. (Restricting the
conductor of $E$ in this way makes it possible to
 confine the $17$-adic  iterated integral
 computations  to spaces of overconvergent $p$-adic modular forms  tame level $148$.) 
Analytic rank computations suggest these are the curves $37a$, 
$629a$, $629c$ and $629d$ in Cremona's tables. 

Following the general notations introduced in Section \ref{sec:S4-analysis},
Conjecture $\shi$ was verified 
for  each of $E=37a$, $629a$ and $629d$ 
 using points $P\in E(\tilde{M})^{-}$ 
found by Steve Donnelly. (For $629c$  the authors were unable to find the point $P$. 
This leaves opens possibility of constructing it using Conjecture $\shi$,
a challenge that has not yet been addressed.) 


The elliptic curves of interest are
\[
\begin{array}{ll}
E_{37a}: & y^2 + y = x^3 - x\\
E_{629a}: & y^2 + xy = x^3 - x^2 + 11 x - 18\\
E_{629d}: & y^2 + xy + y = x^3 - x^2 - 171x + 1904
\end{array}
\]
with associated modular forms $f_{37a}$, $f_{629a}$ and $f_{629d}$. 

With $p = 17$ the Hecke eigenvalues
for the octahedral form 
$h$ are $\beta_h= i$ and $\alpha_h = -1$, and  therefore
$\alpha_g =-i$ and $\beta_g = -1$.
It follows that the frobenius eigenvalues at $17$ for   the adjoint representation 
$\mathrm{Ad}_g$ are $1,\alpha/\beta = i$ and $\beta/\alpha = -i$, which places this experiment
 in {\it Case 5} of the analysis  in Section \ref{sec:S4-analysis}. 

Choose the embedding of $L = \Q(i)$ into $\Q_{17}$ which sends $i$ to the unique square root of
$-1$ which is congruent to $4$ modulo $17$.
%% Embed $S_1(148 \cdot 17,\chi)$ into $\Z_{13}[[q]]$ by sending  $i $ to the unique square root of $-1$ which is congruent to $ 4 $ modulo $17$. 
A calculation of iterated integrals 
to $15$-digits of $17$-adic precision reveals that
%\[
%\begin{array}{lll}
%\int_{\gamma_{g_\alpha}} f_{37a} \cdot h & = & 795182232610993468 \bmod{17^{15}}\\
%\int_{\gamma_{g_\beta}} f_{37a} \cdot h & = & 591309544450892129 \bmod{17^{15}}\\
%\int_{\gamma_{g_\alpha}} f_{629a} \cdot h & = &  231921888434186915 \bmod{17^{15}}\\
%\int_{\gamma_{g_\beta}} f_{629a} \cdot h & = &  776162921415439615 \bmod{17^{15}}\\
%\int_{\gamma_{g_\alpha}} f_{629d} \cdot h & = & -612176557243786922 \times 17 \bmod{17^{15}}\\
%\int_{\gamma_{g_\beta}} f_{629d} \cdot h & = & 329220710071055453 \bmod{17^{15}}.
%\end{array}
%\]
{\color{blue}
\[
\begin{array}{lll}
\int_{\gamma_{g_\alpha}} f_{37a} \cdot h & = & 591309544450892129 \bmod{17^{15}} \\
\int_{\gamma_{g_\beta}} f_{37a} \cdot h & = & 795182232610993468 \bmod{17^{15}}\\
\int_{\gamma_{g_\alpha}} f_{629a} \cdot h & = & 776162921415439615 \bmod{17^{15}} \\
\int_{\gamma_{g_\beta}} f_{629a} \cdot h & = &  231921888434186915 \bmod{17^{15}}\\
\int_{\gamma_{g_\alpha}} f_{629d} \cdot h & = & 329220710071055453 \bmod{17^{15}} \\
\int_{\gamma_{g_\beta}} f_{629d} \cdot h & = & -612176557243786922 \times 17 \bmod{17^{15}}.
\end{array}
\]


}


Consider first the elliptic curve 
$E = E_{37a}$. The generator of $E(\Q)$ is chosen to be $P_\Q = (0,0)$. The
point
\begin{center}
\resizebox{13cm}{!}
{
\begin{tabular}{c}
 $\left(\frac{-152551 a^3 + 327117 a^2 - 384541 a + 1037221}{21316},
\frac{ -110958819 a^3 + 69702635 a^2 + 95157821 a + 314525497}{3112136}\right)$
\end{tabular}
}
\end{center}
on the quadratic twist $y^2 + y = x^3 - 1369 x + 12663$ of $E$ by $37$ 
gives rise  to the point
\begin{center}
\resizebox{13cm}{!}
{
\begin{tabular}{ccl}
$P$  & =  & $\left(\frac{-10817 b^7 - 15535 b^6 - 11412 b^5 - 43863 b^4 - 97162 b^3 + 30857 b^2
    + 39063 b - 42}{21316}, \right.$  \\
    && \\
     & & $\left. \frac{2553946 b^7 - 379967 b^6 - 1189613 b^5 + 
    10482748 b^4 + 12706439 b^3 - 35039416 b^2 + 19176309 b + 6012477}{15560680}\right)$
\end{tabular}
}
\end{center}
in $E(\tilde{M})^-$, where 
$$ \tilde{M} = \Q(b), \qquad  b^8 + b^7 + b^6 + 4 b^5 + 8 b^4 - 4 b^3 + b^2 - b + 1 = 0. $$
Writing $P_1,P_2,P_3$ and $P_4$ for the Galois translates of $P$ so that they are
 permuted cyclically
by $\sigma_{17}$,  define
\[ 
P_{g_\alpha} = P_1 - i P_2 - P_3 + i P_4, \qquad P_{g_\beta} = P_1 + i P_2 -P_3 -i P_4,
\]
following \eqref{eqn:S4-case5b} and \eqref{eqn:S4-case5}.
Likewise, taking $u := -b^2$ as one of the two roots of $x^8 - x^7 + 9 x^6 - 10 x^5 + 102 x^4 - 10 x^3 + 9 x^2 - x + 1$
in $\tilde{M}$
leads to a definition of the 
 Stark units $u_{g_\alpha }$ 
 and $u_{g_\beta} \in (\cO_H^\times)_L^{\mathrm{Ad}_g}$ 
  attached to  the two stabilisations of $g$.

The following identities were verified to 
 $15$-digits of $17$-adic precision:
\begin{eqnarray*}
\int_{\gamma_{g_\alpha}} f_{37a} \cdot h 
 &=& 
\frac{4}{3} \cdot {\mathcal E}_\alpha \cdot \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\alpha})}{\log_{17}(u_{g_\alpha})}, \\
 \int_{\gamma_{g_\beta}} f_{37a} \cdot h 
 &=& \frac{4}{3} \cdot {\mathcal E}_\beta \cdot
 \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\beta})}{\log_{17}(u_{g_\beta})},
 \end{eqnarray*} 
with ${\mathcal E}_\alpha$ as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$.
%% where $t_\alpha$ and $t_\beta \in L$ are the two roots of the polynomial
%% $$ x^2 -\left(\frac{ i }{2^2 \cdot 3}\right) x - \frac{5 \cdot 29}{2^{12} \cdot 3^2}. $$



Consider next the elliptic curve $E = E_{629a}$. Here   the 
generator  for $E(\Q)$  is chosen to be $P_\Q = (2,2)$, and, just as 
above, the point
\begin{center}
\resizebox{12cm}{!}
{
\begin{tabular}{ccl}
$P$ & = & $\left(\dfrac{-5 b^7 - 11 b^6 - 16 b^5 - 31 b^4 - 74 b^3 - 47 b^2 - 25 b + 57}{36},  \right.$ \\
& &  $\left.
\dfrac{1083 b^7 + 1784 b^6 + 2366 b^5 + 5569 b^4 + 11572 b^3 + 3647 b^2 + 
    1907 b - 3464}{1080} \right)$
\end{tabular}
}
\end{center}
belongs to $E(\tilde{M})^-$. 
The points $P_{g_\alpha}$ and $P_{g_\beta}$, and the units 
$u_{g_\alpha}$ and $u_{g_\beta}$, are defined just as they were before.
It was then checked to  $15$-digits of $17$-adic precision that
\begin{eqnarray*}
\int_{\gamma_{g_\alpha}} f_{629a} \cdot h  &=& \frac{-2i - 2}{3} \cdot {\mathcal E}_\alpha \cdot \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\alpha})}{\log_{17}(u_{g_\alpha})}, \\
\int_{\gamma_{g_\beta}} f_{629a} \cdot h  &=&  \frac{2i - 2}{3} \cdot {\mathcal E}_\beta \cdot \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\beta})}{\log_{17}(u_{g_\beta})}, 
\end{eqnarray*}
with ${\mathcal E}_\alpha$ as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$.

%% where $t_\alpha$ and $t_\beta \in L$ are the two roots of the polynomial
%% $$ x^2 + \left(\frac{p }{2 \cdot 3 \cdot 5 \cdot 29}\right) x +  \frac{1}{2^4 \cdot 3^2}. $$

Finally, for the curve
$E = E_{629d}$,   let  $P_\Q = (16,47)$ be the Mordell-Weil generator for $E(\Q)$ and 
let
\begin{center}
\resizebox{12cm}{!}
{
\begin{tabular}{ccl}
$P$  & = & $\left(\dfrac{-1369 b^7 + 259 b^6 - 518 b^5 - 4625 b^4 - 5994 b^3 + 12987 b^2 - 
    17871 b + 9656}{400} \right.,$ \\ 
    & &  $\left. 
    \dfrac{-1258148 b^7 + 64491 b^6 - 649461 b^5 - 
    5104594 b^4 - 5377617 b^3 + 14354298 b^2 - 11310937 b + 2516289}{40000}\right)$
\end{tabular}
}
\end{center}
be the generator of  $E(\tilde{M})^-_L$. 

With the same definitions for the  points $P_{g_\alpha}$ and $P_{g_\beta}$ in terms of $P$
 as before,   the following remarkably simple relation was checked to 
 $15$ digits of $17$-adic precision:
\begin{eqnarray*}
\int_{\gamma_{g_\alpha}} f_{629d} \cdot h &=& (-10 i + 10) \cdot {\mathcal E}_\alpha \cdot \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\alpha})}{\log_{17}(u_{g_\alpha})}, \\
\int_{\gamma_{g_\beta}} f_{629d} \cdot h &=&  (-10 i - 10) \cdot {\mathcal E}_\beta \cdot \frac{\log_{E,17}(P_\Q) \log_{E,17}(P_{g_\beta})}{\log_{17}(u_{g_\beta})}, 
\end{eqnarray*}
with ${\mathcal E}_\alpha$ as in (\ref{eqn-Ea}), and likewise for ${\mathcal E}_\beta$. Here 
$(-10 i + 10) \cdot {\mathcal E}_\alpha = 160/17$ and $(-10 i - 10) \cdot {\mathcal E}_\beta  = -160/17$.
\end{example}

\section{Stark points  attached to odd two-dimensional Artin representations}
\label{sec:beilinson-flach-points}

This chapter considers the case where $g\in S_1(N,\bar\chi)$ is a cuspidal newform and 
$h = E(1,\chi)$ is the Eisenstein series attached to the pair $(1,\chi)$. 
In that case,
$$ V_{gh} = V_{g} \oplus V_{g^*}, \qquad 
L(E,\varrho_{gh},s) = L(E,\varrho_g,s) L(E,\bar\varrho_g,s).$$
The factorisation of $L$-functions guarantees that Hypothesis A of the introduction is automatically satisfied in this setting
but Hypothesis B is only satisfied  when $L(E,\varrho_g,1)=0$. Furthermore,
 the conditions formulated in Conjecture $\shi$  are fulfilled precisely when $L(E,\varrho_{gh},s)$ has
a {\em double zero} at $s=1$, so that
$$ r_{\an}(E,V_g) = r_{\an}(E,\bar{V_g}) = 1.$$
%The eigenvalues of $\sigma_p$ acting on $V_g$ and on $V_h$ respectively  are given by
We have
%$$ \alpha_g, \qquad \beta_g, \qquad\qquad \alpha_h =1, \qquad \beta_h = \chi(p) = (\alpha_g %\beta_g)^{-1},$$
$${\color{blue}\alpha_h =1, \qquad \beta_h = \chi(p) = (\alpha_g \beta_g)^{-1},}$$
and therefore 
$$ \alpha_g \alpha_h = \alpha_g, \qquad \quad \alpha_g\beta_h = \bar\beta_g.$$
Let $H$ be the Galois extension of $\Q$ through which $\varrho_g$ (or, equivalently, $\bar\varrho_g$) 
factors. 
Assuming that $V_g$ is not isomorphic to its contragredient, its underlying field $L$ of coefficients
is a CM subfield of a cyclotomic field, and the spaces $E(H)_L^{V_g}$ and $E(H)_L^{V_{g^*}}$ 
decompose as $\sigma_p$-modules as
$$ 
E(H)_L^{V_g} = L\cdot P_{g,\alpha} \oplus L \cdot P_{g,\beta}, \qquad\quad
E(H)_L^{\bar V_g} = L\cdot P_{g^*,\bar \alpha} \oplus L \cdot P_{g^*,\bar\beta},$$
where $P_{g,\alpha}$ is an eigenvector for  $\sigma_p$  with  eigenvalue $\alpha_g$, and similar conventions are adopted for
 the other points.

When $g$ is regular at $p$ and \eqref{psplit-red} is satisfied---and 
so Hypothesis C holds by Proposition \ref{BeDi-criterion}---Conjecture $\shi$ of the introduction predicts that, for  a suitable choice of test vectors
$$(\testf, \testgamma_{g_\alpha}, h) \in S_2(N)_L \times S_1(N,\chi)_L^\vee[g_\alpha] \times M_1(N,\chi)_L, $$
(and likewise with $\alpha_g $ replaced by $\beta_g$) 
the following identities hold
\begin{eqnarray}
\label{eqn:gcusp-heis}
 \int_{\testgamma_{g_\alpha}} \testf \cdot \testh & =& 
\frac{ \log_{E,p}(P_{g,\alpha}) \cdot \log_{E,p}(P_{g^*,\bar\beta})}{\log_p(u_{g_\alpha})},  \\
 \int_{\testgamma_{g_\beta}} \testf \cdot \testh & =& 
\frac{ \log_{E,p}(P_{g,\beta}) \cdot\log_{E,p}(P_{g^*,\bar\alpha})}{\log_p(u_{g_\beta})}, 
\end{eqnarray}
where $u_{g_\alpha}$ is precisely the same Stark unit 
as arose in Chapter \ref{sec:exotic-points}.

\medskip
The identities above turned out to be harder to check experimentally, because  
\begin{itemize}
\item
 instances where $L(E,\varrho_g,s)$ acquires a simple zero at $s=1$
are  ostensibly sporadic  in nature.
When $\varrho_g$ is not isomorphic to its contragredient representation, this $L$-function does not
admit a sign in its functional equation which might be used to systematically 
predict the vanishing of
$L(E,\varrho_g,1)$, and it is expected that cases
where $L(E,\varrho_g,1)=0$ are exceedingly rare
 (although very little data has been gathered to be 
able to make serious pronouncements in this direction);
\item 
 the field cut out by $\varrho_g$   is  larger than the field cut out by its adjoint $\mathrm{Ad}_g$. 
More precisely, it is an abelian extension of it, of degree $t\ge 2$, where $t$ is the order of the non-trivial
 central
character attached to $\varrho_g$. In particular, the point $P_{g,\alpha}$ that arises in this section
is entirely independent of the point denoted $P_{g_\alpha}$ in Chapter \ref{sec:exotic-points},
is defined over a   larger degree extension, and is therefore  harder to produce numerically 
in practice, in addition to not arising in as systematic a manner. 
\end{itemize}

The problem of producing numerical evidence becomes more tractable when $g=g^*$, which occurs precisely 
when $g$ is induced from a ring class character of an imaginary quadratic field. Such examples 
are arguably less interesting since they also
fall under the scope of   Chapter
\ref{sec:heegner}. See the second half of Example \ref{ex:79}
of Section \ref{p-inert}
for a numerical verification of Conjecture $\shi$ in the case where $g$ is 
further assumed to be self-dual. (See \cite[Example 5.2]{DLR2} though for numerical evidence when
$g \ne g^*$ in the analogous setting in which $f$ is replaced by a weight two Eisenstein series, and  Mordell-Weil groups are replaced by unit groups.)


\section{Stark points over cyclotomic fields  and Eisenstein series}
 \label{sec:abelian-extensions}
 
 \subsection{Theoretical preliminaries}
 \label{sec:abelian-extensions-statement}
  
This section turns to the case where $g$ is an Eisenstein series satisfying condition (2) of Hypothesis C'. Recall that in this case,
 $V_{gh}^{g_\alpha} := W \otimes V_h$ where $W$ is any line in $V_g$ which is not an eigenspace for
the global Galois action; that is, not one of the two one-dimensional irreducible summands of $V_g$.
Conjecture $\shi$ of the introduction  predicts that, for  a suitable choice of test vectors
$$(\testf, \testgamma_{g_\alpha}, \testh) \in S_2(N)_L \times S_1(N,\chi)_L^\vee[g_\alpha] \times M_1(N,\chi)_L, $$
(and likewise with $\alpha_g $ replaced by $\beta_g$) 
the following identities hold
\begin{eqnarray*}
 \int_{\testgamma_{g_\alpha}} \testf \cdot \testh & =& 
\frac{ \log_{E,p}(P_{g,\alpha}) \cdot \log_{E,p}(P_{g^*,\bar\beta})}{\log_p(u_{g_\alpha})},  \\
 \int_{\testgamma_{g_\beta}} \testf \cdot \testh & =& 
\frac{ \log_{E,p}(P_{g,\beta}) \cdot\log_{E,p}(P_{g^*,\bar\alpha})}{\log_p(u_{g_\beta})}, 
\end{eqnarray*}
where $u_{g_\alpha}$  and $u_{g_\beta}$ are   the Gross-Stark $p$-units rather than genuine units. Otherwise, the statement of Conjecture $\shi$  is identical to the 
case  \eqref{eqn:gcusp-heis}
 where the roles of $g$ and $h$ are interchanged.
 
 For the same reasons as were discussed in the previous section, we have been unable to gather much  experimental evidence for it, 
 except in the simplest (and 
 particularly appealing) special case
 where $g$ and $h$ are {\em both } 
 weight one Eisenstein series of the form
 $$ g = E_1(1,\chi^{-1}), \qquad h = E_1(\chi_1,\chi_2), \quad \mbox{ where } \chi_1 \chi_2 = \chi.$$
It is natural to restrict attention to the case
where  at least one of $\chi_1$ or $\chi_2$ is not quadratic, to exclude scenarios
that were  already treated in Chapter \ref{sec:heegner}, where   both
 $V_g$ and $V_h$ are   induced from genus characters of the imaginary quadratic field attached to the quadratic Dirichlet character  $\chi$.
 Let $L$ be the field in which the characters $\chi_1$ and $\chi_2$ take values. The representations $V_g$, $V_h$, and $V_{gh}$ decompose as direct sums
 of one-dimensional representations:
 $$ V_g = L  \oplus L(\chi^{-1}), \quad
 V_h = L(\chi_1) \oplus L(\chi_2), \quad V_{gh} = L(\chi_1) \oplus L(\chi_2) \oplus L(\bar\chi_1) \oplus L(\bar\chi_2).$$
 It is worth noting that the representation $V_{gh}$ is isomorphic to its dual, but that its irreducible constituents are not all self-dual when $\chi_1$ or $\chi_2$ is non-quadratic. This phenomenon, which is specific to the case where at least one of the forms $g$ or $h$ is
 an Eisenstein series, makes the occurence of situations where $L$-functions vanish 
 somewhat more rare and sporadic, just as in the setting described in 
 Chapter \ref{sec:beilinson-flach-points}.

 
  The adjoint representation $\mathrm{Ad}_g$ attached to $V_g$ breaks up as a direct sum
 $$ \mathrm{Ad}_g = L \oplus L(\chi) \oplus L(\chi^{-1}).$$
Let $H_\chi$ be the abelian extension of $\Q$ cut out by the character $\chi$.
Recall that Hypothesis C' requires that
 \begin{equation} \nonumber
%% \label{eqn:g-is-not-regular}
 \mbox{ $\chi(p) =1$,    i.e.,  $V_g$ is not regular and $\chi_1(p) = \chi_2(p)^{-1}$}.
 \end{equation}
Since   $\chi$ is non-quadratic the  $L$-vector spaces
$$ 
 (\cO_{H_\chi}[1/p]^\times)_L^\chi, \qquad
   (\cO_{H_\chi}[1/p]^\times)_L^{\chi^{-1}}, \qquad
   (\cO_{H_\chi}[1/p]^\times)_L^{\chi+\chi^{-1}}$$
  have dimensions $1$, $1$ and $2$ respectively. 
       Let $u_{g_\alpha}\in\cO_{H_\chi}[1/p]^\times)_L^{\chi+\chi^{-1}}$ 
       be the Gross-Stark unit associated to $g_\alpha$ following the 
    recipe of Section \ref{sec:stark}.

 
 Let $f$ be a weight two cusp form attached to an elliptic curve. 
 Note that Hypothesis A is automatically satisfied in this setting. When $r_{\an}(E,\chi_1 \oplus \chi_2) > 0$ (and so
 Hypothesis B holds)
 %% Assuming Hypothesis A and B are satisfied (which is the case of the examples considered below), 
 Conjecture $\shi$ 
  predicts that the iterated integrals attached to $(f,g,h)$ is non-vanishing if and only if
  $r_{\an}(E,\chi_1 \oplus \chi_2) = 1$. 
    Assume (after possibly re-ordering $\chi_1$ and $\chi_2$)
  that 
  $$ r_{\an}(E,\chi_1) = 1, \qquad r_{\an}(E,\chi_2) = 0.$$
   Under this assumption, Conjecture $\shi$
 predicts the existence of    points  
 $P_{\chi_1}  \in E(H_{\chi_1})_L^{\chi_1}$ 
 and 
  $P_{\bar\chi_1}  \in E(H_{\chi_1})_L^{\bar\chi_1}$ 
 for which
 $$ \int_{\gamma} f\cdot h \stackrel{?}{=} 
 \frac{\log_{E,p}(P_{\chi_1} )\log_{E,p}({P}_{\bar\chi_1})}{ \log_p(u_{g_\alpha})}.
 $$
  A notable feature of this conjecture is that it involves global points over abelian
   extensions of $\Q$, whose occurrence seems
   sporadic, in contrast with the self-dual setting that is germane to
    the theory of Heegner and Stark-Heegner points.
  
  
\subsection{Numerical evidence}
\label{sec:abelian-extensions-evidence}


Although Eisenstein series are plentiful,  it is the very
scarcity of points over odd degree abelian extensions, 
and the necessity of working at a  prime $p$
which splits completely
in this abelian extension, 
which makes the numerical verification of Conjecture $\shi$
 so challenging in this situation. Frequently,  when such a point does exist, the common level of the Eisenstein series and cusp 
form $f$, and size
of the prime $p$, pushes the example beyond the current reach of experimentation.  

The authors  were rather fortunate to encounter the
following two examples (and in addition another cubic field example in which 
$r_{\an}(E,\chi_1 \oplus \chi_2) = 2$ and the integral vanished, as predicted by Conjecture $\shi$).
In the first one we give for variety numerical details of the logarithms computed --- these were suppressed  elsewhere to lighten the burden of unilluminating long numbers in the paper.

\begin{example}\label{ex:42a}
  Let  $$E : y^2 +xy+y = x^3+x^2-4x+5$$
be the elliptic curve of conductor $2\cdot 3 \cdot 7$ denoted $42a$
   in the tables of Cremona.
  Let $\chi_1$ be the (even) cubic character of conductor $19$, and let
  $\chi_2$ be the (odd) quadratic character of conductor $3$,
  so that $\chi= \chi_1\chi_2$ is an odd sextic character of conductor 
  $57$. As before, write $L=\Q(\zeta_3)$ for the field of coefficients generated by the values of $\chi_1$.
  A direct calculation reveals that
    $$E(\Q)_L=0, \qquad r(E,\chi_1) = r(E,\bar\chi_1)=1, \qquad r(E,\chi_2) =  r(E,\bar\chi_2) = 0,$$
and likewise for the analytic ranks.
    More precisely, the
 cyclic cubic field cut out by $\chi_1$ is given concretely by
  $$ H_{\chi_1} = \Q(\alpha), \quad \mbox{ where }  
  \mbox{  $\alpha$  satisfies } \alpha^3 - \alpha^2 - 6\alpha + 7 = 0.$$
  The Mordell-Weil group $E(H_{\chi_1})$  has rank two, and is generated as a 
  $\Gal(H_{\chi_1}/\Q)$-module by the point
%% $$ P := (64\alpha^2 + 80\alpha  - 195, 1104\alpha^2 + 1424\alpha - 3391 )$$
%% The next point looks a bit smaller - individual logs of points are different
%% but product stays the same.
\[ P := (2 \alpha^2 + 3 \alpha - 8 , 4 \alpha^2 + 6 \alpha - 10) \]
  with coordinates in $H_{\chi_1}$. 

Fix the prime $p=7$ and let $\zeta_3$  be the unique  cube root of unity 
in $\Q_7$ which is congruent to $2$ modulo $7$. Fix a primitive $6$th root of unity 
$\zeta_6 = - \bar\zeta_3$, so 
$ \zeta_3 = \zeta_6^2$. After letting $\sigma$ be a generator of the
Galois group $\Gal(H_{\chi_1}/\Q)$, let
\begin{eqnarray*}
 P_{\chi_1} &=& 
  P + \bar\chi_1 (\sigma) P^\sigma + \bar\chi_1(\sigma^2) P ^{\sigma^2},\\
  P_{\bar\chi_1} &=& 
  P + \chi_1 (\sigma) P^\sigma + \chi_1(\sigma^2) P ^{\sigma^2}.
\end{eqnarray*}
A numerical evaluation  of the $p$-adic logarithms shows that
\begin{align}
\label{eqn:logs-of-points-cyclic-cubic}
 %% \log_{E,7}(P_{\chi _1})  &=& -29110112917237581854090152551826\times 7 \pmod {7^{39}}, \\
 \log_{E,7}(P_{\chi _1}) & = & -199213220217977173722637405653124 \cdot 7 \bmod {7^{38}} \\
 \nonumber
%% \log_{E,7} (P_{\bar\chi_1}) &=&  -7230650666490160622790674115807\times 7^2 \pmod {7^{39}}.
 \log_{E,7} (P_{\bar\chi_1}) & = & -61436332416679587407622380051451 \cdot 7^2 \bmod {7^{38}}. %% actually mod 7^40
\end{align}
Turning now to the Gross-Stark unit attached to $g$,  the defining polynomial of the
  sextic field $H_\chi$  is chosen to be $x^6 + 2x^5 - 2x^4 - 14x^3 + 55x^2 +
19x + 343$, whose roots generate the group of $7$-units in $H_\chi$.
Let $u$ be   the unique $7$-adic  root of this polynomial  satisfying $\ord_7(u)=3$,
and set 
\begin{eqnarray*}
 u_\chi &:=& \sum_{\sigma\in \Gal(H_\chi/\Q)  } \bar\chi(\sigma) \otimes u^\sigma, \\  
  u_{\bar \chi} &:=& \sum_{\sigma\in \Gal(H_\chi/\Q)  }  \chi(\sigma) \otimes u^\sigma.
  \end{eqnarray*}
  An evaluation of the $7$-adic logarithms of these units yields
\begin{eqnarray*}
 \log_7(u_\chi)   &=&  793452521874901120340916121448 \times 7 \bmod {7^{38}},  \\
\log_7(u_{\bar \chi}) &=&  2536773629823487601064250408052 \times 7  \bmod {7^{38}},
\end{eqnarray*}
and
\begin{equation}
\label{eqn:product-of-logs-cyclic-cubic}
\log_7(u_\chi) + \log_7(u_{\bar\chi})   = 475746593099769817343595218500 \times 7^2
\bmod {7^{38}}.  
\end{equation}

Let 
%% $$ \breve{g} = E_1(\chi_1,\chi_2)(q) - E_1(\chi_1,\chi_2)(q^7) + E_1(\chi_1,\chi_2)(q^2)
 %%               - E_1(\chi_1,\chi_2)(q^{14}), \qquad h = E_1(1,\chi),$$
 \[ g := E_1(1,\chi^{-1}),\qquad h := E_1(\chi_1,\chi_2)\]
 and take the stabilisation 
\[ \breve{g} := \left(g(q) - g(q^7)\right) - (- \zeta_6^{-1}  + 1) \left( g(q^2) - g(q^{14}) \right) \]
in (classical) level $2 \cdot 7 \cdot 57$ with $U_2$-eigenvalue $1$. These are also $7$-adic
modular forms of level $114 = 2 \cdot 57$. 
Letting $\breve{\gamma}$ be the element in the dual space which sends $\breve{g}$ to $1$ 
 and the other stabilisation  of $E_1(1,\chi^{-1})$ to zero, a computation of the $p$-adic
  iterated integral (by far the most resource intensive part of the experiment) 
  reveals that
  \begin{equation}
\label{eqn:iterated-integral-cyclic-cubic-case}
\int_{\breve{\gamma}} f \cdot h =   -1264003828062411821439581214808 \bmod {7^{36}}.
\end{equation}
A direct comparison of \eqref{eqn:iterated-integral-cyclic-cubic-case}
with the invariants obtained in \eqref{eqn:logs-of-points-cyclic-cubic} and
\eqref{eqn:product-of-logs-cyclic-cubic} shows
$$
\int_{\breve{\gamma}}   f  \cdot h  
= \frac{64}{7 \cdot 9}  \frac{ \log_{E,7}(P_{\chi_1}) \log_{E,7}(P_{\bar{\chi}_1}) } {    (\log_7(u_\chi) + \log_7(u_{\bar\chi}) )} \bmod {7^{36}}.
$$
Replacing $\breve{g}$ by the other stabilisation of $E_1(1,\chi^{-1})$, with $U_2$-eigenvalue
$- \zeta_6^{-1} + 1$, and defining $\breve{\gamma}$ accordingly just multiplies the integral value by $\zeta_3^2$; and 
likewise, taking $\breve{g}$ to be a stabilisation of $E_1(\chi_1^{-1},\chi_2^{-1})$ and $h := E_1(1,\chi)$ in the integral changes its
value by at most a cube root of unity. This is as predicted by Conjecture $\shi$.
\end{example}
\medskip \medskip

\begin{example}\label{ex:21}
 Consider the case where 
  $$E: y^2 + x y + y = x^3 - 5 x - 8$$
  is the elliptic curve of conductor $2\cdot 13$ denoted $26a$
  in the tables of Cremona. Write $f$ for the newform of weight $2$ attached to $E$.
  Let $\eta_1$ be an (even) cubic character of conductor $7$, and let
  $\eta_2$ be the (odd) quadratic character of conductor $3$.
  Define $\chi_1 := \eta_1 \eta_2$, an odd sextic character of conductor 
  $21$, $\chi_2 := 1$, and $\chi :=\chi_1 \chi_2 = \chi_1$. (This rather roundabout notation is chosen just to align with
  that in Section \ref{sec:abelian-extensions-statement}.)
 Letting $g := E_1(1,\chi^{-1})$ and $h := E_1(\chi_1,\chi_2) = E_1(\chi,1) = E_1(1,\chi)$,
  the representation $V_{gh}$ is given by
  $$ V_{gh}=  L \oplus L \oplus L(\chi) \oplus L(\bar\chi)$$
   where $L=\Q(\zeta_3)$ is the field generated by the values of $\chi$.
A  descent calculation reveals that
    $$E(\Q)_L=0, \qquad r(E,\chi) = r(E,\bar\chi)=1$$
    and likewise for the analytic ranks.
   The
 cyclic sextic field cut out by $\chi$ is given by
  $$ H_{\chi} = \Q(\alpha), \quad \mbox{ where } \quad 
  \mbox{$\alpha$  satisfies } \alpha^6 - \alpha^5 + 3 \alpha^4 + 5 \alpha^2 - 2 \alpha + 1= 0.$$
  The Mordell-Weil group $E(H_{\chi})$  has rank two, and is generated as a 
  $\Gal(H_\chi/\Q)$-module by the point
  $$ P := (2 \alpha^5 - 6 \alpha^4 + 8 \alpha^3 - 10 \alpha^2 + 4 \alpha - 23 , -50 \alpha^5 + 38 \alpha^4 - 134 \alpha^3 - 36 \alpha^2 - 242 \alpha + 50)$$
  with coordinates in $H_{\chi}$. (This point is the image of one on the quadratic twist of $E$ by $-3$ over the cyclic cubic field
  $H_{\eta_1}$ cut out by $\eta_1$.) The points $P_{\chi}$ and $P_{\bar \chi}$,
 and the units $u_{\chi}$ and $u_{\bar \chi}$,
 are now constructed in
 an analogous manner to
Example \ref{ex:42a}. 

We take $p = 13$. Noting here $a_2(g) = 1 + \zeta_6^{-1}$ with $\zeta_6$ a primitive sixth root of unity, we replace $g$ by its stabilisation
\[ \breve{g} := \left(g(q) - g(q^{13})\right) - \left(g(q^2) - g(q^{26}) \right) \]
in level $2 \cdot 13 \cdot 21$,
and let $\breve{\gamma}$ be the element in the dual space which sends $\breve{g}$ to $1$ 
and the other stabilisation to zero. Embedding $\Q(\zeta_6)$ in
$\Q_{13}$ so that $\zeta_6 \equiv 4 \bmod{13}$ we find
%% \int_\gamma f \cdot h = 1174849788806372600813 \cdot 13 \bmod{13^{20}}. Chose other stabilisation
\[
 \int_{\breve{\gamma}} f \cdot h = -1183909055963560023118938339892601 \times 13 \bmod{13^{30}}.
 \]
%% (With the other choice of stabilisation of $g$ the integral differs by a primitive third root of unity.)
It was then checked to $30$ digits of $13$-adic precision that
\[
\int_{\breve{\gamma}} f \cdot h  = \frac{4}{13 \cdot  \zeta_6} \frac{ \log_{E,13}(P_\chi) \log_{E,13}(P_{\bar\chi}) } {    (\log_{13}(u_\chi) + \log_{13}(u_{\bar\chi}) )}
\]
as predicted by Conjecture $\shi$.
\end{example}
\medskip \medskip

\begin{remark}
The experimental work in this paper fell into two phases---one of discovery and one of verification---between which  Conjecture  $\shi$ was  gradually 
formulated and refined. Formula \eqref{formula-precise} and Examples \ref{ex:83}, \ref{ex:145} and \ref{ex:42a} belong to the former phase, and the
remaining examples to the latter.  One notable feature   which eluded the authors at the outset was the possibilitly of non-vanishing of the
$p$-adic iterated integral in ``genuine rank two" settings. This discovery was made experimentally in the companion paper \cite[Example 6.1]{DLR2}, and 
it was this experiment which led to the definitive formulation of the conjecture.
\end{remark}

\medskip

\medskip

\medskip

\medskip

\begin{appendix}


\section{Digest of numerical examples}


{
The following table summarises some of the salient features of the numerical examples presented
in support of Conjecture $\shi$. It is intended as a navigational guide for the reader. Each numbered
example involves a choice of weight one modular forms $g$ and $h$ of some common level $N$.
This level, the dimensions of the irreducible summands of the tensor product representation
$V_{gh} := V_g \otimes V_h$, and the image of each of those irreducible representations as an abstract
group are given in the second, third and fourth columns of the table, respectively. (Here $D_n$ is the
dihedral group of order $n$.) Each numbered example then involves a further choice of one or
more elliptic curve $E$ and one or more primes $p$ with $\gcd(N,p) = 1$. The curve $E$ always has conductor dividing
$Np$ (and thus the attached form $f$ has $p$-adic level dividing $N$). The Cremona labels of the curves $E$ are shown in the fifth column, and
the ranks with which the irreducible
summands of $V_{gh}$ occur in the Mordell-Weil group of $E$ are given in the sixth. For reasons of space we do not
show the different primes $p$ considered; note however that whether $p$ was chosen to divide the conductor of $E$ or not can be readily deduced by comparing columns ``Level'' and ``Curves''. The examples in the table are grouped under headings indicating the nature of the points constructed.
}
\medskip
\medskip

\renewcommand{\arraystretch}{1.3}

\begin{center}
\resizebox{15.5cm}{!}
{
\begin{tabular}{|| c | c | c | c | c | c ||}\hline

Ex. & Level & Dimension of $V_{gh}$  &  Galois groups & Curves & Ranks  \\ \hline

\multicolumn{6}{| l |}{{\it Classical Heegner points over ring class fields of imaginary quadratic fields}} \\ \hline

\ref{ex:83} & $83$ &  $2 + 1 + 1$ & $D_6 \times 1 \times C_2$ & $83a$ & $1 + 1 + 0$  \\ \hline

\ref{ex:83} &  $83$ &  $2 + 2$ & $D_6  \times D_6$  & $83a$ & $1 + 1$ \\ \hline

\ref{ex:79} & $79$ &  $2 + 2$ & $D_{10}  \times D_{10}$ & $79a$ & $1 + 1$ \\ \hline

\ref{ex:57} &  $57$ &  $2 + 1 + 1$ & $D_6 \times 1 \times C_2$ & $57b$ & $2+ 0 + 0$  \\ \hline

\multicolumn{6}{| l |}{{\it Heegner points arising from a Shimura curve parametrisation}}\\ \hline

\ref{ex:35} &  $35$ &  $2 + 1 + 1$ & $D_6 \times C_2 \times 1$ & $35a$ & $1 + 1 + 0$ \\ \hline

\multicolumn{6}{| l |}{{\it Stark-Heegner points over ring class fields of real quadratic fields}} \\ \hline

\ref{ex:145} &  $145$ &  $2 + 1 + 1$ & $D_8 \times C_2 \times 1$  & $17a$ & $1 + 1 + 0$ \\ \hline

\multicolumn{6}{| l |}{{\it Points   arising   both as Heegner and Stark-Heegner points}} \\ \hline

\ref{ex:155} & 155 &  $1+1+1+1$ & $1 \times C_2 \times C_2 \times C_2$ & $155a$ & $1 + 0 + 1 + 0$ \\ \hline

\ref{ex:155} & 155 &  $1+1+1+1$ & $1 \times C_2 \times C_2 \times C_2$ & $155c$ & $1 + 1 + 0 + 0$ \\ \hline

\multicolumn{6}{| l |}{{\it Points over fields cut out by the projective representations attached to exotic forms}} \\ \hline

\ref{ex:133} & 133 & $1 + 3$ & $1  \times A_4$ & $91a,91b$ & $1 + 1$  \\ \hline 
 
\ref{Ex-124} &  124 & $1 + 3$ & $1  \times A_4$ & $124a$ & $1 + 1$ \\ \hline 

\ref{Ex-124} &  124 & $1 + 3$ & $1  \times A_4$ & $26b,52b$ & $0 + 2$ \\ \hline  
 
\ref{ex:283}& 283 & $1 + 3$ & $1 \times S_4$ & $4811a$ & $2 + 0$ \\ \hline 
 
\ref{ex:148} & 148 & $1+ 3$ & $1  \times S_4$ & $37a,629a,629d$ & $1 + 1$  \\ \hline 

\multicolumn{6}{| l |}{{\it Points over abelian extensions of $\Q$}} \\ \hline

\ref{ex:42a} & 57 & $1 + 1 + 1 + 1$ & $C_{3} \times C_{2} \times C_{3} \times C_{2}$ & $42a$ & $1 + 0 + 1 + 0$ \\ \hline 
 
\ref{ex:21} & 21 & $1 + 1 + 1 +1$ & $1 \times 1 \times C_{6} \times C_{6}$ & $26a$ & $0 + 0 + 1 + 1$ \\ \hline 

\end{tabular}
}
\end{center}
\medskip







\end{appendix}

\begin{thebibliography}{BCDDPR}

\bibitem[AL78]{atkin-li}
A.O.L.  Atkin and  W. Li.
{\em Twists of newforms and pseudo-eigenvalues of $W$-operators},
 Invent. Math. {\bf 48} (1978), no. 3, 221--243.


\bibitem[BeDi]{BeDi} 
J. Bella\"iche and  M. Dimitrov.
 {\em On the eigencurve at classical weight one points},
 to appear in Duke Math. J.


 \bibitem[BCDDPR]{BCDDPR}
 M. Bertolini, F.~Castella, H.~Darmon, S.~Dasgupta, K.~Prasanna, and V.~Rotger.
 {\em $p$-adic $L$-functions and Euler systems: a tale in two trilogies}. 
 in {Automorphic forms and Galois representations}, vol.~1, 
 LMS Lecture Notes {\bf 414}, 
  Cambridge University Press (2014) 52--102.
 
 

\bibitem[BD0]{bd-crelle}
M.~Bertolini and H.~Darmon. {\em 
Kolyvagin's descent and Mordell-Weil groups over ring class fields.} 
Journal f\"ur die reine und angewandte Mathematik {\bf 412} (1990) pp. 63-74.


\bibitem[BD1]{BD-genus}
M. Bertolini and H. Darmon. {\em  
The rationality of Stark-Heegner points over genus fields of real quadratic fields},
Annals of Mathematics {\bf 170} (2009), 343-369. 

\bibitem[BD2]{BD1}
M. Bertolini and  H. Darmon.
 {\em Kato's Euler system and rational points on elliptic curves I: A p-adic Beilinson formula}, Israel J. Math. {\bf 199} (2014), 163--188.


\bibitem[BDP1]{BDP1}
M. Bertolini, H. Darmon, and  K. Prasanna.
 {\em  Generalised Heegner cycles and $p$-adic
Rankin $L$-series}, Duke Math J. {\bf 162}, No. 6,  (2013) 1033--1148.

\bibitem[BDP2]{BDP2}
M. Bertolini, H. Darmon, and K. Prasanna.
 {\em  p-adic Rankin $L$-series and rational points on CM elliptic curves}, 
Pacific J. Math. {\bf 260} (2012), 261--303.


\bibitem[BDR1]{BDR1}
M.~Bertolini, H.~Darmon and V.~Rotger. 
{\em Beilinson-Flach elements and Euler Systems I: syntomic regulators and $p$-adic Rankin $L$-series}, {\em J. Algebraic Geometry} {\bf 24} (2015), 355--378.



\bibitem[BDR2]{BDR2} 
M.\ Bertolini, H.\ Darmon, and V.\ Rotger.
 {\em
Beilinson-Flach elements and Euler systems II: $p$-adic families
and the Birch and Swinnerton-Dyer conjecture}, 
 {\em J. Algebraic Geometry} {\bf 24} (2015), 569--604.


%% \bibitem[BDST]{bdst}
%% K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, R.~Taylor.
%% {\em  On icosahedral Artin representations}, 
%% Duke Math. J.  {\bf 109}  (2001),  no. 2, 283--318. 






\bibitem[Br]{Brooks}
E. H. Brooks.
 {\em Generalized Heegner cycles, Shimura curves, and special values of $p$-adic $L$-functions}, 
Ph.\,D dissertation, University of Michigan, 2013.

\bibitem[Bu]{Buzzard} 
K. Buzzard.
 {\em Computing weight one modular forms over $\C$ and $\bar  \F_p$}, 
in ``Computations with Modular Forms: Proceedings of a Summer School and
Conference, Heidelberg, August/September 2011'', G. B\"ockle and G. Wiese
(eds), Springer, 2014, 129-146.


% \bibitem[Ci]{Ci} C. Citro, IMRN.

% \bibitem[CS]{CoSc} J. Coates, C.-G. Schmidt, {\em Iwasawa theory for the symmetric square of an elliptic curve}, J. reine angew. Math. {\bf 375/376} (1987), 104-156.

% \bibitem[Co]{Col} P. Colmez, {\em Fonctions L $p$-adiques}, S\'eminaire Bourbaki {\bf 851}, 1998/99.

\bibitem[Ch]{chinburg}
T.~Chinburg. {\em Stark's conjecture for $L$-functions with first order zeroes at $s = 0$},
 Advances in Math. {\bf 48} (1983), p. 82-113.


\bibitem[Co]{cowan}
A.~Cowan. {\em Computational techniques for verifying the Birch and Swinnerton-Dyer conjecture
for curves over cubic number fields}, NSERC USRA (Undergraduate Summer Research Award) project,
available at {\tt http://www.math.mcgill.ca/darmon/theses/cowan.pdf}. 

\bibitem[CV]{cho-vatsal}
S.~Cho and V.~Vatsal. {\em Deformations of induced Galois representations},
J. Reine Angew. Math. {\bf 556 }  (2003), 79--98. 


\bibitem[Dar]{darmon-hpxh}
H.~Darmon, {\em  Integration on $\cH_p\times \cH$ and arithmetic applications}. 
Ann. of Math. (2) 154 (2001), no. 3, 589--639.

 
\bibitem[DDLR]{arizona}
H.~Darmon, M.~Daub, S.~Lichtenstein, and V.~Rotger. 
(With an appendix by W.~Stein) 
{\em Algorithms for Chow-Heegner points via iterated integrals}, {\em Mathematics of Computation} {\bf 84} (2015), 2505--2547.


\bibitem[DLR1]{DLR2}
H.~Darmon, A.~Lauder, and V.~Rotger.
{\em Gross-Stark units and $p$-adic iterated integrals attached to 
modular forms of weight one}, to appear in {\em Ann. Math. Qu\'ebec}, volume dedicated to Prof. Glenn Stevens on his 60th birthday. 

\bibitem[DLR2]{DLR3}
H.~Darmon, A.~Lauder, and V.~Rotger.
{\em Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields},  {\em Adv. Math.} {\bf 283} (2015) 130--142.


 \bibitem[DLR3]{DLR-irregular}
 H.~Darmon, A.~Lauder, and V.~Rotger.
 {\em Elliptic Stark conjectures and irregular weight one forms}, 
 in progress.


% \bibitem[DLR3]{DLR-irregular-cuspidal} H.~Darmon, A.~Lauder, and V.~Rotger. {\em Overconvergent generalised eigenforms of weight one: the irregular case}, in preparation.


\bibitem[DP]{DP}
H. Darmon and R. Pollack. {\em
The efficient calculation of Stark-Heegner points via overconvergent modular symbols},  
Israel Journal of Mathematics, {\bf 153} (2006), 319-354. 



\bibitem[DR1]{DR1}
H. Darmon and V. Rotger.
{\em Diagonal cycles and Euler 
systems I: a $p$-adic Gross-Zagier formula},  Annales 
Scientifiques de l'Ecole Normale Sup\'erieure {\bf 47}, n. 4, 47 p.

\bibitem[DR2]{DR2}
H.~Darmon and V.~Rotger. {\em 
Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-series},
submitted. 


\bibitem[DR3]{DR3}
H.~Darmon and V.~Rotger. {\em 
Elliptic curves of rank two and generalised Kato classes},
submitted. 

\bibitem[DRS]{DRS}
H.~Darmon, V.~Rotger, and I.~Sols. 
{\em Iterated integrals, diagonal cycles, and rational points on elliptic curves}. 
Publications Math\'ematiques de Besan\c{c}on, vol.2, 2012, pp. 19-46. 

\bibitem[deS]{dS} 
E. de Shalit,  Iwasawa theory of elliptic curves with complex multiplication. $p$-adic $L$-functions. 
{\em Perspectives in Mathematics} {\bf 3}, Academic Press, Inc., Boston, MA, 1987.


\bibitem[DFK]{DFK}
C.~David, J.~Fearnley, and H.~Kisilevsky.
{\em  Vanishing of $L$-functions of elliptic curves over number fields}. 
Ranks of elliptic curves and random matrix theory, 247--259, London Math. Soc. 
Lecture Note Ser. {\bf 341}, 
Cambridge Univ. Press, Cambridge, 2007. 


\bibitem[DuLi]{Duke-Li} 
W. Duke and Y. Li. {\em Harmonic Maass forms of weight one}, Duke Math. J. {\bf 164} (2015), 39--113.

\bibitem[Gh]{Gh} 
E. Ghate. {\em Ordinary forms and their local Galois representations},  
available for download at  {\tt{http://www.math.tifr.res.in/~eghate/math.html}}.




\bibitem[Gre]{Gre}
M.~Greenberg. {\em Stark-Heegner points and the cohomology of quaternionic Shimura varieties},
Duke Math. J. 147 (2009), no. 3, 541--575. 

\bibitem[Gro]{Gr} 
 B. Gross. {\em On the factorization of $p$-adic $L$-series}, 
 Invent. Math. {\bf 57}, No. 1, 83--95 (1980).

\bibitem[HaKu]{HaKu} 
M. Harris and  S. Kudla.
 {\em The central critical value of a triple product $L$-function},
Ann. Math. (2) {\bf 133} (1991), 605--672.

\bibitem[HaTi]{harris-tilouine}
M.~Harris and J.~Tilouine. {\em
$p$-adic measures and square roots of special values of triple product $L$-functions},
Math. Ann. 320 (2001), no. 1, 127--147. 

\bibitem[Hi1]{Hi81} 
H. Hida. {\em Congruences of cusp forms and special values of their zeta functions}, Invent. Math. {\bf 63}, 225-261 (1981).


\bibitem[Hi2]{hida-book}
H. Hida. Elementary theory of $L$-functions and Eisenstein series,
 {\em London Math. Soc. St. Texts} {\bf 26}, 1993.

\bibitem[Hi3]{hida-genuine}
 H. Hida. On the search of genuine $p$-adic modular $L$-functions for $\GL(n)$, 
M\'emoires de la Soci\'et\'e Math\'ematique de France, 2nd series, 67, (1996), 1-110.

\bibitem[Ich]{Ichino} 
A. Ichino.
 {\em Trilinear forms and the central values of triple product L-functions}. 
 Duke Math. J. {\bf 145} (2008), no. 2, 281--307.  
 
 
\bibitem[Ja]{Jac}
H. Jacquet. Automorphic forms on GL(2), 
Part II, {\em Lecture Notes Math.} {\bf 278} Springer,
Berlin-New York, 1972.
 
 
 \bibitem[Kato]{kato}
K.~Kato. 
{\em $p$-adic Hodge theory and values of zeta functions of modular forms},
Cohomologies $p$-adiques et applications arithm\'etiques. III. Ast\'erisque No. {\bf 295} (2004), ix, 117--290. 

\bibitem[Katz1]{Katz1}
N. M. Katz. {\em $p$-adic interpolation of real analytic Eisenstein series}, 
Annals Math. {\bf 104}, no.\,3 (1976), 459-571.

\bibitem[Katz2]{Katz2}
N. M. Katz. {\em $p$-adic $L$-functions for CM fields}, Invent. Math. {\bf 49} (1978), 199--297.


\bibitem[KW]{KW} 
C. Khare and J.-P. Wintenberger.
 {\em Serre's modularity conjecture (I)}, Invent. Math. {\bf 178} (2009), 485--504. 


\bibitem[La1]{lauder-alg}
A.~Lauder. {\em Computations with classical and p-adic modular forms}, 
LMS J. Comput. Math. {\bf 14} (2011) 214-231.

 
\bibitem[La2]{lauder}
A.~Lauder. {\em Efficient computation of Rankin $p$-adic $L$-functions}, 
in ``Computations with Modular Forms: Proceedings of a Summer School and
Conference, Heidelberg, August/September 2011'', G. B\"ockle and G. Wiese
(eds), Springer, 2014, 181-200.

 

\bibitem[Pr]{Pr} 
D. Prasad. {\em Trilinear forms for representations of $\GL(2)$ and local epsilon factors}, 
Compositio Math. {\bf 75} (1990), 1--46.

 
\bibitem[Si]{silverman}
J.H.~Silverman. {\em Computing rational points on rank $1$ elliptic curves via $L$-series 
and canonical heights}, Math. Comp. {\bf 68} (1999), no. 226, 835--858. 
 
\bibitem[St1]{stark1}
H.M. Stark. {\em $L$-functions at $s = 1$. II. Artin $L$-functions with Rational Characters}, Advances Math.
{\bf 17} (1975), 60--92.

\bibitem[St2]{stark2}
H.M. Stark. {\em Class fields and modular forms of weight one}. 
Modular functions of one variable, V 
(Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), 
pp. 277--287. Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977. 

\bibitem[Wa]{Wat} 
T.C.~Watson.  Rankin triple products and quantum chaos. 
Thesis (Ph.D.)--Princeton University. (2002) 81 pp. 

\bibitem[Wo1]{Wood}
 M. Woodbury. {\em Trilinear forms and subconvexity of the triple product $L$-function}, submitted.

\bibitem[Wo2]{Wood2} 
M. Woodbury. {\em On the triple product formula: real local calculations}, in preparation.

\bibitem[YZZ]{YZZ}
X. Yuan, S. Zhang, and W. Zhang.
 {\em The Gross--Zagier formula on Shimura curves}, 
 Annals of Mathematics Studies {\bf 184} (2013).

\bibitem[Zh]{zhang-msri}
S-W.~Zhang. {\em  Gross-Zagier formula for $GL(2)$. II}. in Heegner points and Rankin $L$-series, 191--214, Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, 2004.  


\end{thebibliography}

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