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\begin{document}

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



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

\dedicatory{To Glenn Stevens on his 60th birthday}

\begin{abstract}
This article can be read as a companion and sequel to
\cite{DLR}, which proposes a conjectural expression for the so-called
{\em $p$-adic iterated integrals} attached to a triple $(f,g,h)$ of classical
 eigenforms of weights $(2,1,1)$. When $f$ is a cusp form,
 this  expression involves   the  $p$-adic logarithms 
 of so-called {\em Stark points}: distinguished points  on the 
   modular abelian variety 
  attached to $f$,    defined over the number field cut out by the 
 Artin representations attached to $g$ and $h$. 
The goal of this paper is to formulate an analogous conjecture when $f$ is a weight two Eisenstein series rather than a cusp form. The resulting  formula  involves the $p$-adic logarithms of units and $p$-units in suitable number fields, and can be seen as a new   variant of  Gross's $p$-adic analogue of     Stark's conjecture on Artin $L$-series at $s=0$.
\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}

This article can (and probably should) be read as a companion and sequel to \cite{DLR}, which describes
 an arithmetic formula
for the so-called {\em  $p$-adic iterated integrals} attached to a triple $(f,g,h)$ of classical 
modular eigenforms of weights $(2,1,1)$.
In the setting considered in \cite{DLR}, where
  $f$ is assumed to be 
  a cusp form, this formula involves the $p$-adic 
formal group logarithms of global points on the modular abelian variety $A_f$ 
attached to $f$ by the Eichler-Shimura construction. 
The points that arise   are defined over the number field cut out by the 
tensor product $\varrho_g\otimes \varrho_h$ of the two Artin representations 
associated to $g$ and $h$. The $p$-adic iterated integral 
of \cite{DLR} can be understood as a $p$-adic avatar of the
second derivative  
of the Hasse-Weil-Artin $L$-series $L(A_f, \varrho_g\otimes \varrho_h,s)$ at its central point $s=1$, and
the main conjecture of loc.cit.~can  therefore  be viewed as an
analogue (in the $p$-adic setting) of Stark's conjecture on units \cite{stark1},  with unit groups 
 replaced by Mordell-Weil groups of elliptic curves or  more general 
 modular abelian  varieties.

The   goal of the present work is to 
bring the analogy with Stark's conjecture 
into sharper focus
 by studying the iterated integrals of \cite{DLR} when $f$ is an Eisenstein series of weight $2$.
The main conjecture of  Section 
\ref{sec:conjecture} below
transposes the ``Elliptic Stark conjecture" of loc.cit.
  to this setting, essentially
  by replacing points on elliptic curves
with units or  $p$-units in number fields having similar Galois-theoretic properties.
When $g$ and $h$ are Eisenstein series or theta series attached to a common
imaginary quadratic field in which the prime $p$ splits, Sections \ref{sec:circular} and
\ref{sec:elliptic} explain how
the $p$-adic Stark  conjecture can frequently be reduced to classical relations between the Kubota-Leopoldt or Katz $p$-adic $L$-functions and $p$-adic logarithms of circular or
elliptic units. 
To make up  for the lack of theoretical evidence in other settings, several numerical verifications  of the remaining cases are 
described    in Sections \ref{sec:gross-stark}
and \ref{sec:exotic}.



This article owes a lot to the vision  and insights of Glenn Stevens, 
who in several works (notably,
\cite{stevens1}, \cite{stevens2}, and \cite{stevens3}) exploits the rich interplay between the arithmetic properties of cusp forms and
Eisenstein series via their connection with irreducible and reducible 
two-dimensional Galois representations.
Around thirteen years ago, Glenn encouraged the first author to revisit 
his 
construction  of 
 Stark-Heegner points
attached to ``real quadratic cycles on ${\mathcal H}_p\times {\mathcal H}$"
\cite{darmon-hpxh} by
   replacing the 
 cusp form that arises in it   with an Eisenstein series. This led to a refinement 
\cite{DD}  of  the  $p$-adic Gross-Stark units in ring class fields of real quadratic fields\footnote{As another illustration of Glenn's
influence, let us mention the decisive impact,  described in
\cite{DP}, of his    theory of 
overconvergent modular symbols for the efficient 
polynomial time calculation of the
 Stark-Heegner points of \cite{darmon-hpxh}.}.
(These
$p$-units 
 appear in a new guise in
Section \ref{sec:gross-stark}, where they play the same role as the Stark-Heegner points of
\cite{darmon-hpxh} in Section 4 of \cite{DLR}.)
 Likewise, but in a purely archimedean setting, the construction
 \cite{DL}
 of Stark-Heegner points attached to ATR cycles on Hilbert modular varieties
 gives rise to the geometric construction \cite{CD} of 
  Stark units in abelian extensions of ATR extensions of a totally real field
 from  periods of 
Hilbert modular Eisenstein series.  To the extent that the present work   bears the same relation to
\cite{DLR} as do \cite{DD} and \cite{CD} to \cite{darmon-hpxh} and \cite{DL}
respectively, Glenn's suggestion continues to resonate through the years.
In the reverse direction, the strategy of substituting
elliptic cusp forms for the weight two
Eisenstein series that arise in the seminal work of Kato on the Birch and Swinnerton-Dyer conjecture
has
inspired the general program of
studying the arithmetic of elliptic curves based on 
  replacing   Beilinson elements attached to a pair
  of modular units  with
  Beilinson-Flach elements 
  \cite{BDR1}, \cite{BDR2}
 and generalised Gross-Schoen diagonal cycles \cite{DR1}, \cite{DR2}.
 (See \cite{BCDDPR} for an overview of this circle of ideas.)
 With profound gratitude for the influence he has exerted on the authors' 
 mathematical outlook, it
  is  a great pleasure to dedicate this article to Glenn
 Stevens on the occasion of his $60$th birthday.


\medskip\noindent
{\small
 \thanks{{\bf Acknowledgements.} The first author was supported by an NSERC Discovery grant,
  and the  third   by Grant MTM2012-34611.
}}

\medskip
\medskip


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

Following the notations of \cite{DLR} which will be adopted  
throughout, let $M_k(N,\chi)_L$ and $S_k(N,\chi)_L$ denote the spaces of modular forms
and cusp forms respectively of weight $k$, level $N$, and character $\chi$ with Fourier coefficients in a field $L$. When $L=\C$, it shall be  suppressed from the notation.

Let 
$$ g \in M_1(N_g, \chi_g), \qquad h \in M_1(N_h,\chi_h)$$
be classical eigenforms of weight one.
A crucial self-duality assumption made in 
\cite{DLR} required the even 
Dirichlet character $\chi_{gh} := \chi_g  \chi_h$
to be trivial, and thus the tensor product $$\varrho_{gh} := \varrho_g\otimes \varrho_h$$
of the Artin representations attached to $g$ and $h$ to be isomorphic to its contragredient. 

One of the novelties of
the Eisenstein series setting is that this
 self-duality assumption is not necessary anymore,
  and can therefore be relaxed: the character $\chi_{gh}$ can now
  be  an arbitrary Dirichlet character, of conductor dividing
   $N_{gh}:= \lcm(N_g,N_h)$. Fix a prime $p\nmid N_{gh}$ and let 
$$
\begin{matrix}
X^2-a_p(g)X+\chi_g(p) & =& (X-\alpha_g)(X-\beta_g), \\
X^2-a_p(h)X+\chi_h(p) &  = &  (X-\alpha_h)(X-\beta_h)
\end{matrix}
$$ 
be the $p$-th Hecke polynomials of $g$ and $h$ in $\C[X]$. 

It will be convenient to fix a finite extension $L\subset \C$ of $\Q$ over
which the representations  $\varrho_g$ and $\varrho_h$ (and hence also
$\varrho_{gh}$) can be realised. Such an $L$
contains the field generated by the traces of these representations, and can always be chosen
to be contained in a cyclotomic field. By enlarging $L$ if necessary, we also assume it contains the roots of unity $\alpha_g$, $\beta_g$, $\alpha_h$ and $\beta_h$.

Let $g_\alpha(z) = g(z)-\beta_gg(pz)$ and $g_\beta(z) = g(z)-\alpha_gg(pz) \in S_1(N_g p,\chi_g)_L$ denote the $p$-stabilizations of $g$ on which the Hecke operator $U_p$ acts with eigenvalue $\alpha_g$ (resp.\,$\beta_g$). 

 
 Let $$ f:=E_2(1,\chi_{gh}^{-1}) \in M_2(N_{gh},\chi_{gh}^{-1})$$
 be the weight two Eisenstein series with fourier expansion given by
 $$ f(q) := c_0 + \sum_{n=1}^\infty \left(\sum_{d|n} \chi_{gh}^{-1}(d) d \right) q^n.$$
 The product of the three nebentypus characters
 attached to the  triple $(f,g,h)$ of modular forms 
 of weights $(2,1,1)$ is trivial, and one can define the 
 $p$-adic iterated integral attached to this triple by letting
 $$ F := d^{-1} f = E_0^{[p]}(\chi_{gh}^{-1},1)$$
 be the overconvergent Eisenstein series of weight zero attached to the pair
  $(\chi_{gh}^{-1},1)$ of Dirichlet characters, with Fourier expansion
  given by
  $$ F(q) :=  \sum_{p\nmid n}
   \left(\sum_{d|n} \chi_{gh}^{-1}(n/d) d^{-1} \right) q^n,$$
 and considering the modular form
 $$ \Xi(g_\alpha,h) = e_{g_\alpha^*} e_{\ord} (F h),$$
 where 
  $$e_{\ord}: M_1^{\oc}(N_{gh},\chi_g^{-1})\lra
 M_1^{\oc}(N_{gh},
 \chi_{g}^{-1})$$
  denotes Hida's ordinary projection on the space $M_1(N_{gh},\chi_g^{-1})$ of overconvergent modular forms of weight one, and 
  $$ e_{g_\alpha^*}: M_1^{\oc}(N_{gh},\chi_g^{-1}) \lra 
  M_1^{\oc}(N_{gh},\chi_g^{-1})[[g_\alpha^*]]$$
  denotes the Hecke-equivariant projection to
  the generalised eigenspace attached to the system of 
  Hecke eigenvalues for the dual form  $g_\alpha^*$ of $g_\alpha$.
  The modular form $\Xi(g_\alpha,h)$ is overconvergent and ordinary, 
but need not be classical, since the conclusion of Coleman's classicality theorem
  fails in general in weight one. For this reason, it is convenient to
  assume
  that Hypothesis C of \cite{DLR}, referred to there as the
  ``classicality property", holds for $g_\alpha$. This hypothesis
  stipulates  that the natural inclusion
  $$ M_1(Np,\chi_{g}^{-1})_{\C_p}[g_\alpha^*] \hookrightarrow 
  M_1^{\oc,\ord}(N,\chi_g^{-1})[[g_\alpha^*]]$$
  of the eigenspace attached to $g_\alpha$
  (consisting entirely of classical forms) into the associated generalised eigenspace in the space of $p$-adic modular forms, maps $ M_1(Np,\chi_{g}^{-1})_{\C_p}[g_\alpha^*]$ into the generalized eigenspace $S_1^{\oc,\ord}(N,\chi_g^{-1})[[g_\alpha^*]]$ of $p$-adic {\em cusp forms} and gives rise to an isomorphism
$$ M_1(Np,\chi_{g}^{-1})_{\C_p}[g_\alpha^*] \simeq S_1^{\oc,\ord}(N,\chi_g^{-1})[[g_\alpha^*]]$$
 between these two $\C_p$-vector spaces.
 
When $g$ is cuspidal, it shall be assumed throughout 
that $\alpha_g \ne \beta_g$, i.e., that $g$
is {\em regular} at $p$, 
and that $g$ is not a theta series of a real quadratic field in which the prime $p$ is split. As explained in \cite[Proposition 1.2]{DLR}, the main result of \cite{BeDi}
then
 implies the classicality property for $g_\alpha$.

On the other hand, when $g = E_1(\chi^+,\chi^-)$ is the Eisenstein series associated to a pair of Dirichlet characters $\chi^+$ and $\chi^-$, the space $ M_1(Np,\chi_{g}^{-1})_{\C_p}[g_\alpha^*]$ maps into $S_1^{\oc,\ord}(N,\chi_g^{-1})[[g_\alpha^*]]$ if and only if  $\chi^+(p)=\chi^-(p)$, that is to say,  if and only if  $g$ is   {\em not }  regular at $p$ (cf.\,\cite[Proposition 1.3]{DLR}), and non-regularity is therefore  a necessary condition for the classicality property to be satisfied.
  
  Given an element $\gamma$ in the $L$-linear dual space
  $M_1(N_{gh},\chi_g^{-1})_L[g_\alpha^*]^\vee$, 
  the {\em $p$-adic iterated integral} attached to the pair 
  $(\gamma,h)$ is defined to be
\begin{equation}\label{ii}  
\int_\gamma f\cdot h := \gamma(e_{g_\alpha^*} e_{\ord}(Fh)) = \gamma(\Xi(g_\alpha,h)).
\end{equation}


  Our goal is to study  this $p$-adic iterated integral: more precisely, 
  to understand when  the classical modular form $\Xi(g_\alpha,h)$  with $\C_p$ coefficients
  is non-trivial, and, when it is, give an arithmetic interpretation
  for its  position   in $M_1(N_{gh},\chi_g^{-1})_{\C_p}[g_\alpha^*]$
  relative to the natural $L$-rational structure on this $\C_p$-vector space
  given by $M_1(N_{gh},\chi_g^{-1})_{L}[g_\alpha^*]$.
  
  
  As in \cite{DLR}, the  four-dimensional tensor product representation 
  $$\varrho_{gh} :=  \varrho_g \otimes \varrho_h$$
  plays a key role in this arithmetic interpretation.
  Let $H$ be the smallest number field through which 
  $\varrho_{gh}$ factors, and set
  $$ U_{gh} := L\otimes \cO_H^\times, \qquad  U^{(p)}_{gh} :=  L\otimes (\cO_H[1/p]^\times/p^\Z),$$
  viewed as finite-dimensional $L$-linear representations of $G_{\Q}$.
 
 
 \begin{lemma}
 \label{lemma:dirichlet} 

 Let  $s$ denote the multiplicity of the trivial representation in $\varrho_{gh}$. Set 
$$ d_{gh} := \dim_L \hom_{G_\Q}(\varrho_{gh},U_{gh}), \qquad d_{gh}^{(p)} :=  \dim_L \hom_{G_\Q}(\varrho_{gh},U^{(p)}_{gh}).
 $$
 Then
 \begin{enumerate}
 \item $d_{gh} = 2 - s$,
 \item $d_{gh}^{(p)} = 2  + \dim_L (\varrho_{gh})^{G_{\Q_p}}-  2s.$
 \end{enumerate}
 \end{lemma}
\begin{proof}
This follows  from  Herbrand's Galois-equivariant 
strengthening of Dirichlet's $S$-unit theorem, combined with 
Frobenius reciprocity. See for instance
\cite[Ch.I, \S3, \S4]{tate-book} or
\cite[Prop.~3.2.2]{DasMsc}.
\end{proof}
In particular, when $s=0$, Lemma \ref{lemma:dirichlet} shows that $d_{gh}= 2$ and that the same is true for 
$d_{gh}^{(p)}$ when $\varrho_{gh}$ admits no $G_{\Q_p}$-invariants.
Furthermore, it is not hard to see that $d_{gh}^{(p)} \ge 2$ when at
least one of $g$ or $h$ is a cusp form.
This is in marked contrast with the setting  explored in 
\cite{DLR} where $U_{gh}$ and $U^{(p)}_{gh}$ were replaced by 
the Mordell-Weil group $E(H)$ of an elliptic curve $E$, tensored with $L$: in this case
  no
simple elementary formula was available for the analogues of 
$d_{gh}$ and $d_{gh}^{(p)}$, and these quantities appear to be $0$ quite often. 
This is why the necessity of the ``vanishing hypothesis" (Hypothesis B)
arose in \cite{DLR}, whereas here it is  unecessary.

Fix an embedding of $H$ into $\C_p$. This equips $H$, and 
hence 
also 
$U_{gh}$ and $U^{(p)}_{gh}$, with natural actions of $G_{\Q_p}$.
Since $p$ is assumed to be prime to $N_{gh}$, the action of $G_{\Q_p}$ on $H$  is unramified; let $\Frob_p$ denote
the arithmetic Frobenius operator, inducing the automorphism $x \mapsto x^{p}$ on the residue fields. 

The galois element $\Frob_p$ 
 acts on $\varrho_{g}$ (resp.\,on $\varrho_{h}$) with
eigenvalues $\alpha_g$ and $\beta_g$  (resp.\,$\alpha_h$ and $\beta_h$).
Write
$$ 
\varrho_{g} := \varrho^{\alpha}_{g} \oplus \varrho_{g}^{\beta}, \qquad \varrho_h := \varrho_h^{\alpha} \oplus \varrho_h^{\beta}
$$
for the decomposition of $\varrho_{g}$ and $\varrho_h$ into one-dimensional
frobenius eigenspaces, when these are defined, i.e., when $\Frob_p$ acts with distinct eigenvalues.







When $g$ is cuspidal, so that $\alpha_g \ne \beta_g$ by our running assumption,
we attach to $g_\alpha$  a two-dimensional subspace of the representation
$\varrho_{gh}=\varrho_{g}\otimes \varrho_{h}$ by setting
\begin{equation}\label{ghcus}
 \varrho_{g h, \alpha} := \varrho^{\beta}_{g}\otimes \varrho_{h}.
\end{equation}

When $g$ is Eisenstein,  we have $\alpha_g = \beta_g$ and we 
 simply define 
\begin{equation}\label{gheis}
\varrho_{gh,\alpha}:= W\otimes \varrho_{h},
\end{equation}
where $W$ is any line in $\varrho_{g}$ which is {\em  not} stable under $G_{\Q}$. 


When $d_{gh}^{(p)}=2$, it is possible to associate to the pair $(g_\alpha, h)$ a 
$p$-adic regulator by choosing an $L$-basis $(\Phi_1,\Phi_2)$ for 
$\hom_{G_\Q}(\varrho_{gh},U^{(p)}_{gh})$ and an $L$-basis $(v_1,v_2)$ 
for $\varrho_{gh,\alpha}$ and setting
\begin{equation}\label{matrix}
R_{g_\alpha}(\varrho_{gh}) :=
 \left(\begin{array}{ll}
 \log_p(\Phi_1(v_1)) & \log_p(\Phi_1(v_2)) \\
 \log_p(\Phi_2(v_1)) & \log_p(\Phi_2(v_2)) 
 \end{array} \right).
\end{equation}
 This matrix is well-defined up to left and right multiplication by
 invertible 
 matrices with entries in $L$, and hence its determinant is well-defined modulo
 $L^\times$. This determinant belongs to $\Q_p^{\ur} \otimes L$, and $\Frob_p$ acts on it with the
 eigenvalue $\beta_g \alpha_h \beta_g \beta_h = \frac{\beta_g}{\alpha_g} \cdot \chi_{gh}(p)$.

Recall the unit $u_{g_\alpha}$ in $U_{gh}$ (when $g$ is a cusp form)
or in $U^{(p)}_{gh}$ (when $g$ is an Eisenstein series) attached to the 
$p$-stabilised eigenform $g_\alpha$ in \cite[\S 1.2]{DLR}.
It belongs to the $\Ad^0(g)$-isotypic part of $U_{gh}$ (resp.\,of $U^{(p)}_{gh}$),
and is an eigenvector for\footnote{This is indeed the same unit as in \cite{DLR}, where we rather employed the geometric frobenius $\sigma_p$ with acts on the unit with eigenvalue $\alpha_g/\beta_g$.} $\Frob_p$ with eigenvalue $\beta_g/\alpha_g$.

Finally, if $\chi$ is a Dirichlet character of conductor $m$, let  
$$\gauss(\chi) := \sum_{a=1}^m \chi^{-1}(a) e^{2\pi i a/m}$$
be the usual Gauss sum, on which $G_{\Q}$ acts through $\chi$ and thus $\Frob_p$ acts with eigenvalue $\chi(p)$. Our main conjecture is:
\begin{conjecture}
\label{conj:main}
If $d_{gh}^{(p)} >2$, then the modular form $\Xi(g_\alpha,h)$ is $0$.
Otherwise,
$$ \Xi(g_\alpha,h) \mbox{ belongs to }  \Omega_{gh} \cdot M_1(N_{gh},\chi_g^{-1})[g_\alpha^*]_L,$$
where the period $\Omega_{gh}$ is given by the formula
$$ \Omega_{gh} := \frac{ \det(R_{g_\alpha}(\varrho_{gh})) }{\gauss(\chi_{gh}) \log_p (u_{g_\alpha})}.$$
\end{conjecture}

As discussed in the introduction, this conjecture is directly inspired  by the 
``elliptic Stark conjectures"
 of  \cite{DLR} which
 apply to the 
 setting where   $f$ is replaced with a weight two cusp form attached to
  an elliptic curve. Conjecture \ref{conj:main} is closer in spirit to the $p$-adic
  Gross-Stark conjectures, since its regulator term involved units and $p$-units of number fields rather than global points on elliptic curves, but it does not appear to follow
from  any of the variants of Stark's conjecture previously
available in the literature.
  
The remainder of this paper shall be devoted to providing both theoretical
and experimental support for Conjecture \ref{conj:main}.


\section{Group-theoretic preliminaries}
\label{sec:gt-prelims}

This section  collects some 
general representation-theoretic results concerning  the structure
of the  representation  $\varrho_{gh} = \varrho_g \otimes \varrho_h$ appearing in Conjecture \ref{conj:main}. We refer to \cite{DS} and \cite{serre}
 for generalities on Artin representations and their connection to modular forms of weight one.
 
Let $V_g$ and $V_h$ denote two-dimensional $L$-vector spaces which 
realise $\varrho_g$ and $\varrho_h$ respectively, and denote by
$V_{gh} = V_g\otimes_L V_h$ the $L$-vector space realising $\varrho_{gh}$.

The image of $\varrho_g$ is a  finite subgroup  $G\subset\GL_2(\C)$. 
The well-known classification of such  groups implies that the natural 
image $\Gamma$ of $G$  in $\PGL_2(\C)$ is  isomorphic to 
\begin{enumerate}
\item a cyclic group, if $g$ is an Eisenstein series;
\item a dihedral group, if $g$ is the theta series attached to a finite order character
of a (real or imaginary) quadratic field;
 \item the group $ A_4$, $S_4$ or $A_5$. In this last case, the form $g$ is 
  said to be 
 {\em exotic}.
 \end{enumerate}




The group  $G$ is then isomorphic to
 \begin{equation}
 \label{eqn:decomp-central}
  G= (Z \times \tilde\Gamma)/\langle \pm 1\rangle,
  \end{equation}
 where $Z = G\cap \C^\times$ is   the center of $G$, and
 $\tilde \Gamma = G\cap \SL_2(\C)$ is the {\em universal  central extension} of $\Gamma$.
 It is a non-split central extension of $\Gamma$ by the group $ \langle\pm 1 \rangle\subset Z$,
and is called the  binary tetrahedral, octahedral, or icosahedral group (of
 order $24$, $48$, and $120$) when $\Gamma=A_4,$ $S_4$ and $A_5$ respectively.
 The irreducible representations of $G$ are obtained in a simple way from those of $\tilde\Gamma$: more precisely, if $\varrho_1,\ldots\varrho_t$ is a complete
 list of the irreducible representations of $\tilde\Gamma$, the irreducible representations 
 of $G$ are simply of the form
 $\chi_Z\otimes \varrho_j$, where $\chi_Z$ is a character of $Z$ satisfying $\chi_Z(-1) = \varrho_j(-1)$. 
 
 When $\varrho_g$ has cyclic projective image, it decomposes as a direct sum of two one-dimensional representations attached to Dirichlet characters $\chi_1$ and $\chi_2$, and
 $g=E_1(\chi_1,\chi_2)$ is the weight one Eisenstein series attached to this pair of characters.
 
 When $\varrho_g$ (or $\varrho_h$) has dihedral projective image, it is induced from a one-dimensional character
 $\psi_g:G_K\lra \C^\times$ of a quadratic field $K$, and $g$ is the theta series attached to
 this character. 
  The representation
$$V_{\psi_g}:= \Ind_K^\Q(\psi_g): G_\Q\lra \GL_2(\C) $$
becomes reducible when restricted to 
 $K$, where it is  isomorphic to $L(\psi_g) \oplus L(\psi'_g)$, where 
 $\psi'_g$ denotes the character obtained by 
 composing $\psi_g$ with the
 automorphism of $K/\Q$.
 Furthermore, $V_{\psi_g}$ is odd if and only if either
 \begin{itemize}
 \item[(a)]
 $K$  is an imaginary quadratic field, or
 \item[(b)]
 $K$ is a real quadratic field, and $\psi_g$ is of {\em mixed signature}, i.e.,
 it is even at one of the archimedean places of $K$, and odd at the other.
 \end{itemize}
 Note that $V_{\psi_g}$ is irreducible (and hence, $g$ is a cusp form) 
 precisely when $\psi_g \ne \psi_g'$. 
 
 



The most basic invariant of the tensor product $V_{gh}$ is its decomposition 
$$ V_{gh} = V_1 \oplus \cdots \oplus V_t$$
into irreducible representations. 
Write $\dd := (d_1,\ldots,d_t)$ for the dimensions of the $V_j$ that arise 
in this decomposition,  arranged by convention in non-increasing order.
There are five possibilities for $\dd$, corresponding to the possible partitions of $4$. 

If $g$ and $h$ are both reducible, i.e., correspond to Eisenstein series, then clearly 
$\dd=(1,1,1,1)$. Assume from now on that at least one of $g$ or $h$ is a cusp form, 
hence is associated
to an irreducible two-dimensional representation. 
\begin{lemma} 
\label{lemma:dd31}
The representation $V_{gh}$ contains a subrepresentation   
of dimension $1$ if
and only if $h$ is a twist of ${\bar g}$, i.e.,
$ V_h = V_g^\vee(\iota) $,  for some 
 Dirichlet  character $\iota$.
 In that case,
 \begin{enumerate}
 \item $\dd = (3,1)$ if and only if $g$ is exotic;
 \item $\dd = (2,1,1)$ if and only if $g$ is dihedral and $\psi_g^- := \psi_g/\psi_g'$ is
 not a quadratic (genus) character;
 \item $\dd = (1,1,1,1)$ if and only if $g$ is dihedral and $\psi_g^-$ is a genus character.
 \end{enumerate}
\end{lemma}
\begin{proof}
The   one-dimensional subrepresentation of $V_{gh}$ is
isomorphic to 
$L(\iota)$ for some Dirichlet character $\iota$ (viewed as a Galois character). 
This implies that
$$ V_{gh} \otimes L(\iota^{-1}) = \hom(V_g^\vee(\iota), V_h)  $$ 
has a $G_\Q$-invariant vector.   
Since at least one of $V_g$ or $V_h$ is irreducible, Schur's Lemma implies 
that  $V_h$ is isomorphic to $V_g^\vee(\iota)$.
 It follows that
 $$ V_{gh}  =  W \oplus L(\iota), \qquad \mbox{ where } W:= {\rm Ad}^0(V_g)(\iota) $$
is the twist by $\iota$ of the
 {\em adjoint representation} attached to $V_g$. 
 Lemma \ref{lemma:dd31} now follows by noting that:
\begin{enumerate}
\item[(i)]
if $\varrho_g$ has projective image isomorphic to $\Gamma=A_4$, $S_4$ or $A_5$,
then ${\rm Ad}^0(V_g)$ factors through $\Gamma$ and is a three-dimensional
irreducible representation of this group, as can be seen, for instance, by consulting the 
character tables of $\Gamma$ and $\tilde\Gamma$; 
\item[(ii)]
if $\varrho_g$  has dihedral projective image, and thus is
induced from a character 
$\psi_g$ of a quadratic field $K$, then
$$ \Ad^0(V_g) = L(\chi_K) \oplus V_{\psi_g^-},$$
where $\chi_K$ is the quadratic Dirichlet character attached to $K$.
\end{enumerate}
\end{proof}
The next lemma takes up the   case where $\dd=(2,2)$. 
\begin{lemma}
\label{lemma:dd22}
The representation $V_{gh}$ decomposes as a sum of two irreducible representations
of dimension two if and only if:
\begin{enumerate}
\item Exactly one of $g$ or $h$ is a weight one 
 Eisenstein series, say $h=E_1(\chi_1,\chi_2)$, in which case
$$V_{gh} = V_g(\chi_1) \oplus V_g(\chi_2).$$
\item Both
 $g$ and $h$
 are theta series attached to  characters $\psi_g$ and $\psi_h$ of the same 
quadratic field $K$  
for which neither of the characters
$$\psi_1 := \psi_g \psi_h, \qquad \psi_2 := \psi_g\psi_h'$$
  is equal to its conjugate over $K$.
  In this case,  
$$ V_{gh} = V_{\psi_1} \oplus V_{\psi_2}.$$
The representations $V_{\psi_1}$ and $V_{\psi_2}$ are non-isomorphic, 
and if $K$ is real quadratic, they are   of opposite (pure) signatures. 
\end{enumerate}
\end{lemma}
\begin{proof}
It is clear that $\dd=(2,2)$ when exactly one of $g$ or $h$ is an Eisenstein series. Assume
for the rest of the proof that both $g$ and $h$ are cuspidal.
Let $H_g/\Q$ and $H_h/\Q$ be the extensions of 
$\Q$ cut out by the   representations  $\varrho_g$ and $\varrho_h$,
let $H:= H_g\cap H_h$, and let $H_{gh}$ be the compositum of the extensions 
$H_g$ and $H_h$, which are linearly disjoint over $H$.
Write
$$ G_{gh} := \Gal(H_{gh}/\Q), \qquad 
G_g := \Gal(H_g/\Q), \quad G_h := \Gal(H_h/\Q), $$
$$  \cA_g := \Gal(H_g/H), \quad \cA_h := \Gal(H_h/H). \qquad G_0=\Gal(H/\Q).$$
The resulting field diagram is described below
$$\xymatrix{
   & H_{hg}   &    \\
 H_g \ar@{-}[ur]^{\cA_h} & & H_h \ar@{-}[ul]_{\cA_g} \\
 & H \ar@{-}[ul]_{\cA_g} \ar@{-}[ur]^{\cA_h}  \\
 &  \Q \ar@/^1.5pc/@{-}[uul]^{G_g} \ar@{-}[u]^{G_0}  \ar@/_1.5pc/@{-}[uur]_{G_h}   &    }
 $$
By hypothesis, the group $G_g$ (resp.~$G_h$) acts faithfully on $V_g$ (resp.~$V_h)$. 

Observe first that the groups $\cA_g$ and $\cA_h$ are necessarily abelian. 
For if $\cA_g$ (say) were non-abelian,   the two-dimensional vector space 
$V_g$ on which $\cA_g$ acts faithfully would be irreducible as an $\cA_g$-module.
Since $\cA_g$ acts trivially on $V_h$, the irreducible  
 $\cA_g$-submodules of 
$V_{gh}$ would have to all be of the form $V_g \otimes L_h$ for some one-dimensional
$L$-vector subspace $L_h \subset V_h$. This would force any two-dimensional
 irreducible sub-representation 
of $V_{gh}$  to also be of this form. But $V_h$ contains no $G_h$-stable lines
 and the vector space $V_g\otimes L_h$
  therefore cannot be stable under
the full action of $G_{gh}$. 

One can now distinguish two cases:

\noindent
{\em Case A}: At least one of  $G_g$ or $G_h$ has a projective image isomorphic to $A_4$, $S_4$ or $A_5$.
Assume without loss of generality that this is the case for $G_g$. The only abelian normal subgroups
of $\tilde\Gamma_g$ are contained in its center, and hence, the same is true for the abelian normal subgroups
of $G_g$.
It follows that $\cA_g$ acts on $V_g$ as scalar multiplications, 
and that $G_0$ has a quotient isomorphic to $\Gamma$. Hence the same assertions hold for  $G_h$:
in other words, both $G_g$ and $G_h$ are central extensions of a common $\Gamma\subset \PGL_2(\C)$ which arises as a quotient of $G_0$. 
  A study of the character table of $\tilde\Gamma$
  reveals that the tensor products of two irreducible two-dimensional representations of
  this group never decomposes as  a sum of two-dimensional representations. This rules out the
  scenario where $\dd=(2,2)$ in Case A. 
    
     \medskip\noindent
 {\em Case B}: The groups $G_g$ and $G_h$ have dihedral projective image. The representation
 $\varrho_g$ is then induced from a character  $\psi_g$ of a quadratic field $K$, which satisfies $\psi_g\ne \psi_g'$, since
 $V_g$ is irreducible. Hence
 $V_g$ is isomorphic to $L(\psi_g) \oplus L(\psi_g')$ as a $G_K$-module, and any element of 
 $G_\Q- G_K$ interchanges $L(\psi_g)$ and $L(\psi_g')$.  
In particular, the representation $V_{gh}$ is   isomorphic to  $V_h(\psi_g) \oplus V_h(\psi_g')$ as a $G_K$-module,
 and the following  cases arise:
 \begin{itemize}
 \item $V_h$ is reducible as a $G_K$-module, 
 and hence $\varrho_h$ is induced from a character of
 $K$.
 \item $V_h$ remains  irreducible as a $G_K$-module. The reducibility of $V_{gh}$ 
 then forces 
 \begin{equation}
 \label{eqn:VhequalsVhpsi}
 V_h(\psi_g)\simeq V_h(\psi_g'), \quad \mbox{ hence }  \ \ \  V_h \simeq V_h(\psi_g/\psi_g')  \quad \mbox{ as } G_K-\mbox{modules}. 
 \end{equation}
For otherwise, the only non-trivial $G_K$-stable subspaces of $V_{gh}$
 would be $V_h(\psi_g)$ and $V_h(\psi_g')$, which are not preserved by  elements of $G_\Q-G_K$.
Taking determinants on both sides  of 
\eqref{eqn:VhequalsVhpsi}
reveals that $\psi_g/\psi_g'$ is a quadratic (and hence a genus)
  character of 
 $K$, whose induced representation is a direct sum $\chi_1 \oplus \chi_2$ of two quadratic Dirichlet characters attached to quadratic
 extensions $K_1$ and $K_2$ respectively, 
 satisfying $\chi_1\chi_2=\chi_K$, where $\chi_K$ is the quadratic character attached to $K$.
  Inducing   \eqref{eqn:VhequalsVhpsi}
  from $K$ to $\Q$ shows that
 $$ V_h \oplus V_h(\chi_K) \simeq V_h(\chi_1) \oplus V_h(\chi_2) \qquad \mbox{ as $G_\Q$-modules}.$$
Since $V_h$ and $V_h(\chi_K)$ are assumed to be non-isomorphic, 
 it  may   be assumed, after eventually interchanging $\chi_1$ and $\chi_2$,
 that $V_h$ is isomorphic to $V_h(\chi_1)$, and therefore, that $V_h$ is induced from a character of 
$K_1$. But the same is also true of $V_g$, which is a twist  of a representation of the 
dihedral group of order $8$, and as such is simultaneously induced from characters of 
all three of
the quadratic fields $K$, $K_1$ and $K_2$.
\end{itemize}
 After replacing $K$ by $K_1$ in the second scenario, it follows that $V_g$ and $V_h$ are induced from 
  characters
 of the {\em same} quadratic field $K$.
 Writing 
  $\psi_h$ for the character    of $K$   satisfying
  $V_h = V_{\psi_h}$,  the representation $V_{gh}$ breaks up as
  $$V_{gh} = V_{\psi_1} \oplus V_{\psi_2}, \qquad \psi_1:= \psi_g\psi_h, \quad \psi_2 := \psi_g \psi_h'.$$  
 It follows from this that $\psi_1$ and $\psi_2$ have the same central character,
  and that $\dd=(2,2)$  precisely when
  $\psi_1 \ne \psi_1'$ and $\psi_2\ne \psi_2'$.
   (Recall that the central character of a character $\psi$ of $G_K$ 
   is the Dirichlet characters arising from the restriction to the id\'eles of $\Q$ of the Hecke character  of $K$ associated by class field theory to $\psi$.)
 To see that $V_{\psi_1}$ and $V_{\psi_2} $ are never isomorphic, note that  the irreducibility of 
 $V_h$ implies that $\psi_h \ne \psi_h'$, and hence that
$\psi_1\ne\psi_2$;  and that likewise, the irreducibility of
$V_g$ implies that 
 $\psi_1 \ne \psi_2'$.  When $K$ is real quadratic, it is readily checked that, since
 $\psi_g$ and $\psi_h$ are both of mixed signature, the ring class characters
 $\psi_1 = \psi_g\psi_h$ and $\psi_2 = \psi_g\psi_h'$ are
 of opposite pure signatures.
 The lemma follows.
\end{proof}


The ``generic" case where $\dd=(4)$ encompasses all scenarios where the 
representations $\varrho_g$ and $\varrho_h$ cut out linearly disjoint
 extensions of $\Q$. When 
it is irreducible,
the Artin representation $V_{gh}$ occurs with mutiplicity two in $L\otimes \cO_H^\times$
 and
is  undistinguished enough to  probably never have garnered explicit attention
 in connection with the Stark  or Gross-Stark  conjectures. 
 Nonetheless the  special
 cases  described in Lemmas \ref{lemma:interesting-S4} and 
 \ref{lemma:interesting-A5} below
 are worth singling out.
 
 
 Suppose that 
  $$\varrho_h: \Gal(H_h/\Q) \lra \tilde \Gamma \subset \GL_2(\C)$$
   has projective image $\Gamma= \tilde\Gamma/Z \simeq S_4$,
  and that  $\varrho_g$ is  the (non-faithful) two-dimensional
  representation of $\Gal(H_h/\Q)$ obtained from $ \varrho_h$ 
  by composing it with the natural 
  sequence of homomorphisms   
  $\tilde\Gamma \lra S_4 \lra S_3 \subset \GL_2(\C).$
  Then $\varrho_{gh}$ can be described concretely in terms of the   induced representation of a
  quadratic character of the quartic subfield $K$ of $H_h$ fixed by $S_3\subset S_4$.
  More precisely,
  the natural exact sequence
  $$ 1 \lra Z \lra \Gal(H_h/K) \lra S_3 \lra 1$$
  splits. Since $Z$  is a cyclic
  group of even cardinality, it follows that $\Gal(H_h/K)$  has three distinct surjective
  homomorphisms to $\pm 1$, of which only one is trivial on $Z$. Let $\eta$ be a 
   quadratic character of $\Gal(H_h/K)$ which is non-trivial on $Z$, and let $M$ be the octic extension of $\Q$
  cut out by $\eta$, as 
  summarised in the following field diagram:
  $$ \xymatrix{
 & H_h\ar@/_2pc/@{-}[dd]_{S_3\times Z} \ar@{-}[d] \ar@/^0.2pc/@{-}[drr]^Z \\
 & M \ar@{-}[d]_{\pm 1} & & \bar H_h  \ar@{-}[d]^{\Z/2\Z \times \Z/2\Z} \ar@/_0.2pc/@{-}[dll]_{ S_3} \ar@{-}[ddll]^{S_4} \\
 & K \ar@{-}[d]_4  & & H_g \ar@/^0.2pc/@{-}[dll]^{S_3} \\
 & \Q 
  }
 $$
 \begin{lemma}
 \label{lemma:interesting-S4}
For $\varrho_g$ and $\varrho_h$ as above, 
  $$ \varrho_{gh} = \Ind_K^\Q \eta.$$
  In particular, 
  $$ \hom(V_{gh}, \C\otimes \cO_H^\times) = \C\otimes  \cO_{M,1}^\times,$$
  where $\cO_{M,1}^\times$ denotes the group of units of $M$ whose norm to $K$ is equal to $1$.
  \end{lemma}
  \begin{proof}
  The first assertion follows from a direct calculation using the character table of ${\tilde S_4}$, and the second from Frobenius reciprocity.
  \end{proof}
  
  
  
 Another simple, concrete instance of an irreducible $V_{gh}$ 
 arises when $K$ is a non-totally real quintic extension of $\Q$ whose Galois closure
 $\tilde K$ has Galois group $A_5$ over $\Q$.
There are exactly two conjugacy classes of 
embeddings of   $ \Gal(\tilde K/\Q) = A_5$ into $ \PGL_2(\C)$;
choosing one
 gives rise to a  projective representation
$$\bar\varrho_g: G_\Q \twoheadrightarrow \Gal(\tilde K/\Q)\simeq A_5\subset \PGL_2(\C).$$
 Tate's lifting theorem produces a linear lift $\varrho_g: G_\Q\lra \GL_2(\C)$
of $\bar\varrho_g$, which is odd since $K$ is not totally real, and hence corresponds to a modular form $g$.
It can further be assumed that the   field $H$ cut out by $\varrho_g$
is a cyclic extension of $\tilde K$ of $2$-power  degree, so that
   $\varrho_g(Gal(H/\tilde K))$  
 consists of scalar matrices whose diagonal entries are $2$-power roots of unity.
 Let $\sigma$ be an automorphism of $\C$ which agrees with complex conjugation on the $2$-power roots of unity and sends $\sqrt{5}$ to $-\sqrt{5}$, and let   $h$ be
  the modular form obtained by applying $\sigma$  to $g$. 
  \begin{lemma}
 \label{lemma:interesting-A5}
 For $\varrho_g$ and $\varrho_h$ as above, 
 \begin{equation}
 \label{eqn:char-A5-tilde}
V_{gh}   \oplus \ \C  \ \ \  = \ \ \ \ V_5,
 \end{equation}
 where $V_5$ is the natural permutation representation of $A_5$ acting on $5$ letters.
 In particular, 
  $$ \hom(V_{gh}, \C\otimes \cO_H^\times) = \C\otimes \cO_{K}^\times, 
  \qquad 
   \hom(V_{gh}, \C\otimes \cO_H[1/p]^\times) = \C\otimes (\cO_{K}[1/p]^\times/p^\Z).
  $$
   \end{lemma}
   \begin{proof}
   See for instance the proof of Thm.~6.9 of \cite{DR2}.
   \end{proof}
   
   
   
   
\section{Circular units and Eisenstein series of weight one}
\label{sec:circular}
In this section we begin by considering the simplest case where 
$$ g = E_1(\chi_g^+,\chi_g^-), \qquad h = E_1(\chi_h^+,\chi_h^-)$$
are both Eisenstein series attached to pairs of Dirichlet characters.
It is assumed that $\chi_g^+$ is even and $\chi_g^-$ is odd, and likewise for
$\chi_h^+$ and $\chi_h^-$. The nebentype characters of $g$ and $h$ are equal to
$\chi_g := \chi_g^+\chi_g^-$ and $\chi_h:= \chi_h^+\chi_h^-$ respectively.

Fix a prime $p$ not dividing the conductors of $\chi_g$ and $\chi_h$, and recall we assume the classicality hypothesis on $g$, which in particular implies that $\chi_g^+(p)=\chi_g^-(p)$.
 

The representation $\varrho_{gh}$ decomposes as a direct sum of four one-dimensional
characters
$$ \varrho_{gh} = \chipp \oplus \chimm  \oplus \chipm \oplus \chimp := \chi_1 \oplus \chi_2 \oplus \chi_3 \oplus \chi_4,$$
where 
$$
\chipp := \chi_{g}^+ \chi_h^+, \qquad
\chimm := \chi_{g}^- \chi_h^-, \qquad
\chipm := \chi_{g}^+ \chi_h^-, \qquad
\chimp := \chi_{g}^- \chi_h^+.\qquad
$$
Note that the characters $\chipp$ and $\chimm$ are even, while
the characters $\chipm$ and $\chimp$ are odd (and in particular, non-trivial).

Given an even character $\chi$ factoring through a finite abelian extension
$H_\chi$ of $\Q$ and taking values in $L$, denote by $u(\chi)\in L\otimes (\cO_{H_\chi}^\times)$ the
 fundamental unit in the $\chi$-eigenspace for the natural $G_\Q$-action.
Given an odd character $\chi$ for which $\chi(p)=1$, let 
$u_p(\chi)\in L\otimes (\cO_{H_\chi}[1/p])^\times$
 denote the fundamental $p$-unit in the 
$\chi$-eigenspace for the Galois action.
 

In light of Lemma \ref{lemma:dirichlet}, the form taken on by Conjecture \ref{conj:main} depends   on the number of 
even characters among the $\chi_j$ which are non-trivial, and hence it is convenient to break up our discussion into three cases:

\medskip
\noindent
{\bf Case 1}: The even characters $\chipp$ and $\chimm$ are both non-trivial.
In that case, Conjecture \ref{conj:main} predicts that $\Xi(g_\alpha,h)=0$
if  $\chi_j(p) =1$  for some $1 \le j \le 4$, and that otherwise  
\begin{equation}
\label{eqn:cyc-case1}
\Xi(g_\alpha,h) \in \Omega_{gh} \cdot M_1(N_{gh},\chi_g^{-1})[g_\alpha^*]_L \quad \mbox{where} \quad \Omega_{gh} = 
\frac{\log_p(u(\chipp)) \log_p(u(\chimm)) } {\gauss(\chi_{gh}) \log_p(u_{g_\alpha})}.
 \end{equation}
 
 \medskip
 \noindent
 {\bf Case 2}: Exactly one of the even characters $\chipp$ and $\chimm$ is trivial. 
Our running assumption that $\chi_g^+(p) = \chi_g^-(p)$
 implies that $\chimp(p) = 1$ if $\chipp=1$ and that $\chipm(p) =1$
 if $\chimm=1$.  Conjecture \ref{conj:main} predicts that $\Xi(g_\alpha,h)=0$
  if  $\chi_j(p) =1$  for at least two non-trivial characters
  among the $\chi_j$.  Assuming that this is not the case, let
  $\chi_{gh}^+$ denote the unique  non-trivial character in $\{\chipp,\chimm\}$
  and let $\chi_{gh}^-$ denote the unique character in
  $\{\chipm,\chimp\}$ satisfying $\chi_{gh}^-(p) = 1$.
  Conjecture \ref{conj:main} then predicts that  
  \begin{equation}
  \label{eqn:cyc-case2}
 \Xi(g_\alpha,h) \in \Omega_{gh} \cdot M_1(N_{gh},\chi_g^{-1})[g_\alpha^*]_L \quad \mbox{where} \quad 
  \Omega_{gh} = 
 \frac{\log_p(u(\chi_{gh}^+)) \log_p(u_p(\chi_{gh}^{-})) } {\gauss(\chi_{gh}) \log_p(u_{g_\alpha})}.
  \end{equation}
  A notable special case arises when $\chi_g^+ = \chi_h^+ = 1$ and the odd characters
  $\chi_g^-$ and $\chi_h^-$ are both quadratic, with 
  $\chi_g^-(p) = -\chi_h^-(p) = 1$.
 In that scenario, the $p$-units 
 $u_p(\chi_{gh}^{-})$ and $u_{g_\alpha}$ that occur in the numerator and denominator of \eqref{eqn:cyc-case2} cancel out, and the conjectural formula for the period
 simplifies to 
  \begin{equation}
  \label{eqn:cyc-case2-bis}
   \Omega_{gh} = 
  \frac{\log_p(u(\chimm))}{\gauss(\chi_{gh})}.
   \end{equation}
One obtains an appealing expression for the $p$-adic
 logarithm of the fundamental  unit of a real quadratic field in terms of 
 $p$-adic iterated integrals. This leads to a (somewhat) 
  new method for solving Pell's equation, albeit one which is
   hardly an improvement over the classical method of continued fractions!
   
    \medskip
    \noindent
    {\bf Case 3}: Both of the even characters $\chipp$ and $\chimm$ are trivial.
    This implies that 
    $$ \chi_h^+ = (\chi_g^+)^{-1}, \qquad
    \chi_h^- = (\chi_g^-)^{-1}, $$
     so that, after setting $\chi_g^{\pm} := \chi_g^+/\chi_g^-$ and
     $\chi_{g}^{\mp} := \chi_g^-/\chi_g^+$,
     $$ \varrho_{gh}  = 1 \oplus 1 \oplus \chi_{g}^{\pm} \oplus \chi_g^{\mp}.$$
     Both of the odd characters that arise in this decomposition are trivial 
     at $p$. 
      Conjecture \ref{conj:main} therefore predicts that $\Xi(g_\alpha,h)\ne 0$
     and that  
    \begin{equation}
    \label{eqn:cyc-case3}
\Xi(g_\alpha,h) \in \Omega_{gh} \cdot M_1(N_{gh},\chi_g^{-1})[g_\alpha^*]_L \quad \mbox{where} \quad 
     \Omega_{gh} = 
    \frac{\log_p(u_p(\chi_{g}^{\pm})) \log_p(u_p(\chi_g^{\mp}))}
    { \log_p(u_p(\chi_g^{\pm})) + \log_p(u_p(\chi_g^{\mp}))}.
     \end{equation}
     A notable special case arises when the odd character
      $\chi_g^{\pm}$ is quadratic, and hence equal to $\chi_g^{\mp}$. 
      The expression in
      \eqref{eqn:cyc-case3} then simplifies to yield (modulo $L^\times$, as always)
     \begin{equation}
     \label{eqn:cyc-case3-bis}
      \Omega_{gh} = 
      \log_p(u_p(\chi_g^{\pm})).
      \end{equation}
   One thus obtains, somewhat like  in
   \eqref{eqn:cyc-case2-bis},
    an appealing expression for the $p$-adic
    logarithm of the fundamental  $p$-unit of an imaginary
     quadratic field in terms of 
    $p$-adic iterated integrals of Eisenstein series.
      
      The main result of this section is the unconditional proof of certain  special cases of Conjecture \ref{conj:main} when $g$ and $h$ 
      are both Eisenstein series:
\begin{theorem}
\label{thm:cyclotomic}
Assume that $g$ and $h$ are   Eisenstein series, and that  $\chi_g^+$ and $\chi_g^-$ are both quadratic
characters of relatively prime conductors.
 Then 
Conjecture \ref{conj:main} holds, i.e.,
equations \eqref{eqn:cyc-case1}, \eqref{eqn:cyc-case2} and
\eqref{eqn:cyc-case3} hold in cases 1, 2 and 3 respectively.
\end{theorem}
 \begin{proof}
 Recall that the 
classicality hypothesis imposed on $g$ implies that
 \begin{equation}
  \label{eqn:irreg-g}
  \chi_g^+(p) = \chi_g^-(p).
  \end{equation} 
 The pair $(\chi_g^+,\chi_g^-)$ corresponds to a genus character $\psi_g$ of the
 imaginary quadratic field $K$ cut out by the odd  Dirichlet
 character $\chi_g:= \chi_g^+ \chi_g^-$. Condition \eqref{eqn:irreg-g}
 implies that $\chi_g(p)=1$, and hence that the prime $p =: \fp \bar\fp$   splits in $K/\Q$. 
 The weight one Eisenstein series $g= E_1(\chi_g^+,\chi_g^-)$ 
 is equal to the theta series $\theta_K(\psi_g)$ attached to   $\psi_g$, and the modular form $g$ admits three natural ordinary deformations, arising from the fact that
  $E_1(\chi_g^+,\chi_g^-)= E_1(\chi_g^-,\chi_g^+)=\theta_K(\psi_g)$:
 \begin{itemize}
  \item The Hida families $\uE(\chi_g^+,\chi_g^-)$  and $\uE(\chi_g^-,\chi_g^+)$ of Eisenstein series, whose weight 
  $k$ specialisations are described by
  $$\uE_k(\chi_1,\chi_2) := \sum_{n=1}^\infty \sigma_{k-1,\chi_1,\chi_2}^{(p)}(n) q^n, \qquad \sigma_{k-1,\chi_1,\chi_2}^{(p)}(n) := \sum_{d|n, \ \ p\nmid d} \chi_1(n/d) \chi_2(d) d^{k-1};$$
  \item The cuspidal family $\utheta(\psi_g)$ of ordinary theta series whose weight $k$ specialisation is given by
  $$ \utheta_k(\psi_g) = \theta(\psi_g \psi_p^{k-1}) = 
  \sum_{\fa\subset \cO_K, \ \  \bar\fp \nmid \fa} \psi_g(\fa) \psi_p^{k-1}(\fa) q^{\fa\bar{\fa}}.$$
  In this equation, the character $\psi_p: I(K)\lra \bar{\Q_p}^\times$
  on the group $I(K)$ of fractional ideals of $K$
   is obtained from a canonical
  Hecke character $\psi_\infty: I(K) \lra \C^\times$  in the sense of
  Gross and Rohrlich,  sending a principal ideal $(\alpha)$ to
  $\pm \bar\alpha$ for all $\alpha$ which are relatively prime to the conductor
  of $\psi_\infty$. The character $\psi_\infty$  has infinity type $(0,1)$ and takes its values in a finite extension $L$ of $K$, and $\psi_p$ is obtained by composing it with a $p$-adic embedding of $L$ sending $K$ to its completion at $\fp$. 
 \end{itemize}

%{\blue{V to H: }}

Let $\theta_{3+2t}(\psi_g)$ denote the weight $3+2t$ specialisation 
of the Hida family of theta-series arising from the genus character $\psi_g$.
The classicality assumption on $\theta_1(\psi_g)$ implies that $\utheta(\psi_g)$ 
 is the unique cuspidal Hida family specialising to $\theta_1(\psi_g)$. 

 Because of this,
  the $p$-adic iterated integral \eqref{ii} attached
 to the pair  $(\theta_K(\psi_g), E_1(\chi_h^+,\chi_h^-))$ 
agrees, up to non-zero algebraic factors, with
  the $p$-adic limit   of the algebraic numbers (depending on $t\ge 0$)
  $$ \Omega_{gh}^{[p]}(t) := \frac{\langle \theta_{3+2t}(\psi_g), d^t E_2^{[p]}(1,\chi_{gh}^{-1}) \times E_1(\chi_h^+,\chi_h^-) \rangle}{\langle\theta_{3+2t}(\psi_g),\theta_{3+2t}(\psi_g)\rangle}$$
 as $t$ tends to $-1$ in weight space. (Note that this ratio does not depend on any choice of normalisation for the Petersson scalar product.)
  Applying Cor.~4.13 of \cite{DR1} after making the substitutions
 $$ (k,\ell,m) \leftarrow (3+2t,2,1), \quad f_k \leftarrow \theta_{3+2t}(\psi_g), \quad g \leftarrow E_2(1,\chi_{gh}^{-1}), \quad h  \leftarrow E_1(\chi_h^+, \chi_h^{-}),$$
 one obtains
   $$ \Omega_{gh}^{[p]}(t) = \frac{\cE_{gh}(t)}{\cE_g(t)} \times \Omega_{gh}(t),$$
 where
\begin{eqnarray*} 
 \cE_{gh}(t) &:=&  (1-\psi_p({\bar\fp})^{2+2t} \chi_h^+(p) p^{-2-t})  \times  (1-\psi_p({\bar\fp})^{2+2t} \chi_h^-(p)  p^{-2-t}) \\
  & &                            \times (1-\psi_p({\bar\fp})^{2+2t} \bar\chi_{h}^-(p) p^{-1-t}) \times (1-\psi_p({\bar\fp})^{2+2t} \bar\chi_{h}^+(p) p^{-1-t}), \\
 \cE_g(t) &:=&  (1-\psi_p(\bar\fp)^{4+4t} p^{-3-2t}),       \\
 \Omega_{gh}(t) &:=&  \frac{\langle \theta_{3+2t}(\psi_g), d^t E_2(1,\chi_{gh}^{-1}) \times E_1(\chi_h^+,\chi_h^-) \rangle}{\langle\theta_{3+2t}(\psi_g),\theta_{3+2t}(\psi_g)\rangle}.
\end{eqnarray*}
It will be convenient to express $\Omega_{gh}^{[p]}(t)$ as a ratio
$$ \Omega_{gh}^{[p]}(t) := \frac{\Omega_{gh}^{\num}(t)}{ \Omega_g^{\den}(t)}$$
of two quantities that can (as will be shown) be individually interpolated $p$-adically, namely
\begin{eqnarray*}
  \Omega_{gh}^{\num}(t)  &:=& \frac{
\cE_{gh}(t) \times \Omega_{K,p}^{4+4t} \times \langle \theta_{3+2t}(\psi_g), d^t E_2(1,\chi_{gh}^{-1}) \times E_1(\chi_h^+,\chi_h^-) \rangle }{\Omega_{K}^{4+4t}},  \\
\Omega_g^{\den}(t)&:=& \frac{\cE_g(t) \times \Omega_{K,p}^{4+4t} \times
 \langle \theta_{3+2t}(\psi_g), \theta_{3+2t}(\psi_g)\rangle}{ \Omega_{K}^{4+4t}},
\end{eqnarray*}
where $\Omega_{K}\in \C^\times$ is   Shimura's canonical period attached to the imaginary quadratic field $K$ and $\Omega_{K,p}\in \C_p^\times$ is the corresponding $p$-adic
period which appears in the interpolation formula for
the Katz $p$-adic $L$-function.
The strategy of proof is to show that both of these functions of $t\in \Z^{\ge 0}$, viewed as taking values in $\C_p$, extend  to an analytic function on weight
space  by relating them to   the Katz $p$-adic $L$-function, and  then to evaluate them at the point $t=-1$ by invoking    Katz's $p$-adic analogue of the
Kronecker limit formula.
 
 We begin with the expression $\Omega_{gh}^{\num}(t)$. To evaluate it, we invoke formula (18) of
 \cite{BD1}, after making the substitutions
 $$ (k,\ell,m) \leftarrow (3+2t,1,2), \quad c \leftarrow 2+t,  \quad f \leftarrow \theta_{3+2t}(\psi_g), \quad g \leftarrow E_1(\chi_h^+,\chi_h^-),$$
 to obtain
 $$   \langle \theta_{3+2t}(\psi_g), d^t E_2(1,\chi_{gh}^{-1}) \times E_1(\chi_h^+,\chi_h^-) \rangle = C(t) \times  L(\theta_{3+2t}(\psi_g)\otimes E_1(\chi_h^+,\chi_h^-), 2+t),$$
 where 
 $$ C(t) := 
 \frac{(1+t)!^2   \times \Omega_{K,p}^{4+4t} }{(-1)^{t-1} 2^{2+2t} (2\pi)^{4+2t}
  (-iN)^{-2}\gauss(\chi_{gh})}.$$
 Formula (19) of loc.cit.  implies that
\begin{eqnarray*}
 \Omega_{gh}^{\num}(t) &=& C(t) \cE_{gh}(t)  \times \frac{L(\theta_{3+2t}(\psi_g),\chi_h^+, 2+t)}{\Omega_K^{2+2t}} 
 \times \frac{L(\theta_{3+2t}(\psi_g),\chi_h^-,2+t)}{\Omega_K^{2+2t}} \\
 &=&  C(t)\cE_{gh}(t)  \times \frac{L(\psi_g \chi_h^+ \psi_t^{-1},0)}{\Omega_K^{2+2t}} \times
  \frac{L(\psi_g\chi_h^- \psi_t^{-1},0)}{\Omega_K^{2+2t}},
 \end{eqnarray*}
 where the $L$-functions in the second line are those attached to Hecke characters of $K$, and 
 $\psi_t$ is the Hecke character
 $$ \psi_t :=  \frac{ (\psi_p')^{2+t}}{\psi_p^{t}}.$$

In view of our definition of $\psi_\infty$, the character $\psi_t$ has infinity type $(2+t,-t)$ and therefore lies in the range of classical interpolation for the 
 Katz $p$-adic $L$-function precisely when $t\ge 0$. 

Let $L_p^{\katz}$ denote the two-variable Katz $p$-adic $L$-function attached to Hecke characters of $K$, and 
 $\Omega_{K,p}$ denote the $p$-adic counterpart of the complex period $\Omega_K$ that appears in the interpolation 
 property of $L_p^{\katz}$.

 The interpolation formula for the Katz $p$-adic $L$-function, as it is described for instance in 
 \cite[\S 3, (45)]{DLR}, shows that
 $$ \Omega_{gh}^{\num}(t)  \sim_t \gauss(\chi_{gh})^{-1} 
 \times L_p^{\katz}(\psi_g\bar\chi_h^+ \psi_t) \times  L_p^{\katz}(\psi_g \bar\chi_h^- \psi_t),$$
 where the symbol $\sim_t$ denotes an equality up to an elementary fudge factor which (as a function of the variable $t$) extends to an 
 analytic function on weight space and takes $L$-rational, non-zero values at $t=-1$.

 
 Evaluating this relation at $t=-1$, one finds
 $$ \Omega_{gh}^{\num}(-1) \sim \gauss(\chi_{gh})^{-1} \times L_p^{\katz}(\psi_g \bar\chi_h^+ {\bf N}) L_p^{\katz}(\psi_g \bar{\chi}_h^-{\bf N}),$$
 where ${\bf N}$ denotes the norm character (of infinity type $(1,1)$) and
 we have (in many of the equations above) identified the Dirichlet characters
 $\chi_h^+$ and  $\chi_h^-$ with their restrictions to $G_K$, viewed as (finite order) Hecke characters of $K$.
 Note that the  smallest extension $H$ of $K$ cut out by the characters $\psi_g \chi_h^+$ and $\psi_g\chi_h^-$ is Galois and abelian over $\Q$  as well.


 If $\psi$ is any non-trivial character of $\Gal(H/K)$, denote by $u_K(\psi)$ any element of
 $L\otimes \cO_H^\times $ which generates the $\psi$-eigenspace for the 
 action of $\Gal(H/K)$, and when $\chi$ is trivial, write
 $u_K(1)$ for the fundamental $p$-unit of the imaginary quadratic field $K$.
 Katz's  $p$-adic analogue of the Kronecker limit formula (Cf.~for instance
 \cite[\S 10.4, 10.5]{Katz1}, \cite[p.~90]{gross}, \cite[Ch.~II,\S 5.2]{de-shalit}) shows that
 \begin{equation}
 L_p^{\katz}(\psi_g\bar{\chi}_h^+{\bf N}) = 
 \left\{ 
 \begin{array}{ll}
 0 & \mbox{ if } \psi_g \chi_h^+\ne 1, \ \ \psi_g\chi_h^+(p) =1; \\
 \log_p(u_K(\psi_g\chi_h^+)) & \mbox{ otherwise. }
 \end{array} \right.
 \end{equation}
 Because the character $\psi_g \chi_h^+$ arises
 from  the restriction of a character of $\Q$ and $\psi_g$ 
 is the genus character attached to the pair $(\chi_g^+,\chi_g^-)$ of
 quadratic Dirichlet characters,  it can be seen that:
 $$ 
 u_K(\psi_g \chi_h^+) = \left\{ 
 \begin{array}{ll}
 u(\chipp) & \mbox{ if } \psi_g\chi_h^+ \ne 1; \\
 u_p(\chimp) & \mbox{ if } \psi_g\chi_h^+ =1.
 \end{array} \right.
 $$
 Therefore, it follows that
 \begin{equation}
  L_p^{\katz}(\psi_g\bar{\chi}_h^+{\bf N}) \sim 
  \left\{ 
  \begin{array}{ll}
  0 & \mbox{ if } \chipp \ne 1, \mbox{  and } \chi_h^+(p)=\chi_g^{+}(p) = \chi_g^-(p);  
 \\
  \log_p(u(\chipp)) & \mbox{if  } \chipp\ne 1,\mbox{ and }
    \chi_h^+(p) \ne \chi_g^+(p); \\
  \log_p(u_p(\chimp)) & \mbox{ if } \chipp=1.
  \end{array} \right.
  \end{equation}
 Similar considerations applied to the character
 $\psi_g\chi_h^-$ 
show that
 \begin{equation}
  L_p^{\katz}(\psi_g\bar{\chi}_h^-{\bf N}) \sim 
  \left\{ 
  \begin{array}{ll}
  0 & \mbox{ if } \chimm \ne 1, \mbox{  and } \chi_h^-(p)=\chi_g^{+}(p) = \chi_g^-(p);  
 \\
  \log_p(u(\chimm)) & \mbox{if  } \chimm\ne 1,\mbox{ and }
    \chi_h^-(p) \ne \chi_g^-(p); \\
  \log_p(u_p(\chipm)) & \mbox{ if } \chimm=1.
  \end{array} \right.
  \end{equation}
 It follows that 
\begin{equation}
\label{eqn:Omega-gh}
 \Omega_{gh}^{\num}(-1) \sim 
 \left\{
 \begin{array}{ll}
 \frac{\log_p(\chipp)\log_p(\chimm)}{\gauss(\chi_{gh})} & \mbox { in case 1}; \\
 \frac{\log_p(u(\chi_{gh}^+)) \log_p(u_p(\chi_{gh}^-))}{\gauss(\chi_{gh})}, &
 \mbox{ in case 2};  \\
 \log_p(u_p(\chi_g^{\pm})) \log_p(u_p(\chi_g^{\mp})) & \mbox{ in case 3}.
 \end{array}
 \right.
\end{equation}

In addition, it follows from \cite[Lemma 3.8]{DLR}, where $t=k-2$, that
$\Omega_g^{\den}(t)$ likewise extends to a $p$-adic analytic function of $t$, and
\begin{equation}
\label{eqn:Omega-g}
\Omega_g^{\den}(-1) \sim \log_p(u_p(\chi_g))
\end{equation}
by \cite[Theorem 3.9 (54)]{DLR}.

 The theorem follows directly from equations
 \eqref{eqn:Omega-gh} and \eqref{eqn:Omega-g} 
 in light of the fact that
  $$\Omega_{gh} \sim \frac{\Omega_{gh}^{\num}(-1)}{\Omega_g^{\den}(-1)}.$$
 \end{proof}

 
\begin{remark}
The analogue of Theorem \ref{thm:cyclotomic}   in the setting
explored in \cite[\S 7]{DLR}, 
  where the weight two
 Eisenstein series $E_2(1,\chi_{gh}^{-1})$ is replaced by a cusp form $f$  attached to an
 elliptic curve $E$ over $\Q$,  poses significant difficulties and remains unproved. This analogue 
 would involve   points on $E$ defined over  the cyclotomic fields 
 cut out by the characters $\chipp$, $\chipm$, $\chimp$, and $\chimm$, and no
 appropriate  variant of the  Heegner   construction leading to such points is available, at least when 
  the characters involved are non quadratic. Such a variant would occupy the same position vis-\`a-vis
  Heegner points as circular units vis-\`a-vis elliptic units.
  \end{remark}


\medskip\noindent
{\bf Numerical examples}.  




\begin{example}\label{ex:eisen}
The following experiments illustrate Theorem \ref{thm:cyclotomic}
in the special case where $g = E_1(1,\chi_g^-)$ and $h = E_1(1,\chi_h^-)$, 
where $\chi_g^-$ and $\chi_h^-$ are odd  quadratic Dirichlet characters satisfying
$(\chi_g^-(p),\chi_h^-(p)) = (1,-1)$. This corresponds to the special case of
Conjecture \ref{conj:main} described in equation \eqref{eqn:cyc-case2-bis} of
Case 2 treated above, involving the $p$-adic logarithm 
of the fundamental unit of the real quadratic field cut out by the even character $\chimm$. 

 
\medskip\noindent
{\bf \ref{ex:eisen}a.} For this experiment, let $\chi_g^-$ and $\chi_h^-$ be the odd quadratic 
Dirichlet characters of conductors $3$ and $11$  respectively, and choose $p := 7$.
The $7$-stabilised eigenform $g_\alpha := E_1^{(7)}(1,\chi_g^-)$ (of level $3 \cdot 7$) 
has two distinct $11$-stabilisations, denoted $g_{1}$ and $g_{-1}$, of 
level $3 \cdot 7 \cdot 11$, with $U_{11}$-eigenvalues $1$ and $-1$, respectively. 
Denote  
by $\gamma_{ 1}$ and $\gamma_{-1}$   the linear functionals on 
$S_1(3\cdot 7 \cdot 11)[g_\alpha] = \C_p g_1 \oplus \C_p g_{-1}$ sending 
$(g_1,g_{-1})$ to $(1,0)$ and $(0,1)$ respectively.
A computer calculation using the algorithms based on \cite{lauder}
shows
that
$$
\begin{array}{lll}
\int_{\gamma_{1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = & -9955228630224805563482516 \pmod{7^{30}}\\
\int_{\gamma_{-1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = & 0 \pmod{7^{30}}.
\end{array}
$$
A careful examination of the non-vanishing $7$-adic iterated integral above suggests that
$$ \int_{\gamma_1 } E_2(1,\chimm) \cdot E_1(1,\chi_h^-) = - \frac{2 \cdot \sqrt{33} \cdot \log_7{(u(\chimm))}}{7} $$
to $30$ digits of $7$-adic precision, where 
  $u(\chimm)$ is the fundamental unit in the real quadratic field cut out by the 
  even character $\chimm$, satisfying the polynomial
  $x^2 - 46 x + 1$.
 
\medskip\noindent
{\bf \ref{ex:eisen}b.} 
We place ourselves in exactly the same settting as in 
\ref{ex:eisen}a, but taking this time $\chi_h^-$ to be the odd quadratic 
character of conductor $23$ rather than $11$. Adopting the same notations as for the 
previous example, we find:
$$
\begin{array}{lll}
\int_{\gamma_{1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = & 10649075716435865817575346 \pmod{7^{30}}\\
\int_{\gamma_{-1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = & 0 \pmod{7^{30}}
\end{array}
$$
and 
$$
\int_{\gamma_{1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) = - \frac{3 \cdot \sqrt{69} \cdot \log_7(u(\chimm))}{7}
$$
to $30$ digits of $7$-adic precision. 

\medskip\noindent
{\bf \ref{ex:eisen}c.}  In the third experiment of this kind, we take $\chi_g^-$ to have conductor $7$ and
$\chi_h^-$ to have conductor $3$, with
$p := 11$. Then
$$
\begin{array}{lll}
\int_{\gamma_{1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = &  -11038147038719013039751993 \pmod{11^{25}}\\
\int_{\gamma_{-1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-) & = & 0 \pmod{11^{25}}
\end{array}
$$
and we find again that
$$
\int_{\gamma_{1}} E_2(1,\chimm) \cdot E_1(1,\chi_h^-)  = -\frac{\sqrt{21} \cdot \log_{11}(u(\chimm))}{11}
$$
to $25$ digits of $11$-adic precision. 
\end{example}

\begin{example}
Let $\chi_3$ and $\chi_4$   be the odd quadratic characters of conductors $3$ and $4$, respectively, and let
$\chi_5$ be the even quadratic character of conductor $5$.
Let $g := E_1(1,\chi_3)$ and  $h := E_1(\chi_5,\chi_4)$. We have found experimentally
that the $7$-adic iterated integrals attached to $g$ and $h$, for all the stabilisations of
$g$ in level $3 \cdot 7 \cdot 4 \cdot 5$,  are all zero, except for
%% see 2014_Aug23_chi3chi4chi4.m
$$ \int_{\gamma} E_2(1,\chi_3\cdot\chi_4\cdot\chi_5)  \cdot E_1(\chi_5,\chi_4) 
= -1510481263266793866295817 \pmod{7^{30}}, $$
where $\gamma$ is the unique element of the dual space of $S_1(7\cdot 60,\chi_3)[g_\alpha]$
sending the stabilisation of $g$ with 
  $ U_2$ and $U_5$ eigenvalues $ 1$ and $ -1$, respectively to $1$, and all other stabilisations to $0$.
Experimentally, one finds
\[
 \int_{\gamma } E_2(1,\chi_3 \cdot\chi_4\cdot \chi_5)  \cdot E_1(\chi_5,\chi_4) = 
 \frac{2^4 \cdot \sqrt{60} \cdot \log_7(u(\chi_5)) \cdot \log_7(u(\chi_{12}))}{3 \cdot 7 \cdot \log_7(u_7(\chi_3))}
\]
to $30$-digits of $7$-adic precision, in perfect agreement with the most ``generic'' instance of 
Conjecture \ref{conj:main} treated in Case 1 above.
\end{example}


\section{Elliptic units and theta series of imaginary quadratic fields}
 \label{sec:elliptic}
 

 \subsection{Theoretical evidence}

Let $K$ be an imaginary quadratic field of discriminant $-D_K$ and let $\chi_K$ denote the quadratic Dirichlet character associated to it. The aim of this chapter is to study
Conjecture \ref{conj:main}  in the case where both $g$ and $h$ are theta series 
 associated to $K$. 



Fix two arbitrary finite order characters 
$\psi_g$, $\psi_h$ of $K$ of conductors $\mathfrak c_g$, $\mathfrak c_h$. Let $\chi_g$, $\chi_h$ denote the central characters of $\psi_g$ and $\psi_h$. Let 
 $$g := \theta_{\psi_g} \in M_1(N_g,\chi_K\chi_g), \qquad h:= \theta_{\psi_h}\in M_1(N_h,\chi_K \chi_h)$$
be the theta series associated to them. Here $N_g=D_K\cdot N_{K/\Q}({\mathfrak c}_g)$ and $N_h=D_K\cdot N_{K/\Q}({\mathfrak c}_h)$.

Define $\psi_1= \psi_g \psi_h$ and $\psi_2 = \psi_g \psi'_h$ where, for any character $\psi$ of $G_K$, we denote $\psi'$ the character given by the rule $\psi'(\sigma) = \psi(\sigma_0 \sigma \sigma_0^{-1})$ for any choice of $\sigma_0\in G_\Q - G_K$.  Unlike in  \cite[\S 3]{DLR}, here we are not imposing the restriction $\chi_g=\chi_h^{-1}$, and thus $\psi_1$ and $\psi_2$ are not necessarily ring class characters.
%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. 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 $\psi_2$. 

Let $V_g= \Ind_K^\Q (\psi_g)$ and $V_h= \Ind_K^\Q (\psi_h)$ denote the
 two-dimensional induced representations of $\psi_g$ and $\psi_h$ from $G_K$ to $G_\Q$, with coefficients in a finite extension $L$ of $\Q$ which we may take to contain the fourier coefficients of $g$ and $h$ and the Gauss sum $\gauss(\chi_{gh})$. By Lemma \ref{lemma:dd22}, there is a decomposition of Artin representations
\begin{equation}
\label{eqn:dihedral-decomp}
 V_{gh} = V_{\psi_g}\otimes V_{\psi_h} \simeq V_{\psi_1} \oplus V_{\psi_2},
 \end{equation}
 where for $i=1,2$, $V_{\psi_i}$ is irreducible if and only if $\psi_i\ne \psi_i'$. If $\psi_i = \psi_i'$, let $\chi_i$ denote the central character of $\psi_i$. Then $V_{\psi_i} \simeq \chi_i \oplus \chi_K \chi_i$ decomposes as the sum of an odd and even Dirichlet character. The odd one can never be trivial, while the even one is trivial precisely when $\psi_i=1$. 


Let $N=\mathrm{gcd}(N_g,N_h)$ and let $p\nmid N$ be a prime that splits in $K$. By Lemma \ref{lemma:dirichlet}, it follows that
\begin{equation}\label{2}
 d_{gh}^{(p)} = 2 \quad \mbox{ if and only if } \quad \psi_g \ne \psi_h^{-1}, (\psi'_h)^{-1},
 \end{equation}
which we assume for the remainder of this chapter. In that case we in fact have $d_{gh} = d_{gh}^{(p)} = 2$ and, more precisely,
$$
 \dim_L(V_{\psi_i},U_{gh})=1 \quad \mbox{ for } i=1,2.
$$

Write $U_{gh}^{\psi_i}$ for the image of $V_{\psi_i}$ in $U_{gh}$.
The theory of elliptic units allows us to pin down distinguished choices of elements in $U_{gh}^{\psi_i}$ as follows.


Let $H_{g}/K$ and $H_{h}/K$ denote the abelian extensions cut out by the characters $\psi_g$ and $\psi_h$, respectively, and let $H_{gh}=H_g H_h$ denote their composition.  Since $p$  splits in $K$, the frobenius  element $\Frob_p$ belongs to $\Gal(H_{gh}/K)$,
and the choice of a prime $\wp$ of $K$ above $p$ is sufficient to
determine $\Frob_p$ 
as an element  of this group, satisfying $\psi(\wp) = \psi(\Frob_p)$ for all characters $\psi$ of $\Gal(H_{gh}/K)$. 

The choices of $\alpha_g, \beta_g, \alpha_h$ and $\beta_h$ can then be made so that
$$ \alpha_g = \psi_g(\wp), \qquad \beta_g = \psi_g(\bar\wp), \qquad \alpha_h=\psi_h(\wp), \qquad \beta_h = \psi_h(\bar\wp). $$

Let $K_{\mathfrak f}$ denote the smallest ray class field containing $H_{gh}$ and let $u\in K_{\mathfrak{f}}^\times$ be an elliptic unit constructed as in \cite[\S 2.6]{de-shalit}. For any character $\psi$ of $G=\Gal(K_{\mathfrak f}/K)$, define
\begin{equation}\label{uaa}
u_{\psi}  = \sum_{\sigma \in G} \psi^{-1}(\sigma) u^\sigma.
%\begin{matrix}
%u_{\alpha \alpha}  = \sum_{\sigma \in \Gal(K_{\mathfrak f}/K)} \psi_1(\sigma^{-1}) u^\sigma, & u_{\beta \beta}  = \sum_{\sigma \in %\Gal(K_{\mathfrak f}/K)} \psi'_1(\sigma^{-1}) u^\sigma, \\
%u_{\alpha \beta}  = \sum_{\sigma \in \Gal(K_{\mathfrak f}/K)} \psi_2(\sigma^{-1}) u^\sigma, &  u_{\beta \alpha}  = \sum_{\sigma \in %\Gal(K_{\mathfrak f}/K)} \psi'_2(\sigma^{-1}) u^\sigma,
%\end{matrix}
\end{equation}

If $\psi\ne 1$, the element $\sum \psi^{-1}(\sigma) \cdot\sigma$ belongs to the augmentation 
ideal of $L[G]$ and \cite[\S 2.6]{de-shalit} shows that $u_{\psi}$ lies in $L\otimes \cO_{K_{\mathfrak f}}^\times$. Since, by \eqref{2}, the characters $\psi_i$, $i=1,2$, are not trivial, the four units $u_{\psi_1}$, $u_{\psi'_1}$, $u_{\psi_2}$, $u_{\psi'_2}$ 
lie in $U_{gh}$. In fact, by construction $u_{\psi_1}$ belongs
to  $U_{gh}^{\psi_1}$ and $\Frob_p$ acts on it with eigenvalue  $\psi_1(\Frob_p) = \alpha_g\alpha_h$. Analogous 
properties hold for $u_{\psi_1'} \in U_{gh}^{\psi_1}$ and  $u_{\psi_2} $, $u_{\psi_2'} \in U_{gh}^{\psi_2}$.

Both when $g$ is cuspidal or Eisenstein, the matrix in \eqref{matrix} is 
$$
R_{g_\alpha}(\varrho_{gh}) =
 \left(\begin{array}{ll}
 \log_p(u_{\psi'_2}) & 0 \\
 0 & \log_p(u_{\psi'_1}) 
 \end{array} \right).
$$

In addition to the above units, let us introduce also a canonical choice for the unit $u_{g_{\alpha}}$ entering in  
Conjecture \ref{conj:main}. 

If $g$ is cuspidal,  let $\psi_{\mathrm{ad}}:=  \psi'_g/\psi_g$ denote the character associated to the adjoint of $g$ and define $u_{g_\alpha} = u_{\psi_{\mathrm{ad}}}$.
Since $\psi_g \ne \psi_g'$, it follows as above that $u_{g_\alpha}$ belongs to $L\otimes \cO_{H_g}^\times$ and that $\Frob_p$ acts on it with eigenvalue $ \beta_g/\alpha_g$. 

If $g$ is Eisenstein, i.e.\,$\psi_g = \psi_g'$, define 
\begin{equation}\label{ubb}
u_{g_\alpha}:= u_{\wp} \in  L\otimes\cO_{H_g}[1/p]^\times 
\end{equation}
 to be any element such that $\wp^{h_K} = (u_{\wp})$, where $h_K$ denotes the class number of $K$.


Conjecture \ref{conj:main} then predicts that  
 \begin{equation}
 \label{eqn:caseK}
\Xi(g_\alpha,h) \in \Omega_{gh} \cdot M_1(N_{gh},\chi_g^{-1})[g_\alpha^*]_L \quad \mbox{where} \quad 
  \Omega_{gh} = \frac{\log_p(u_{\psi'_1}) \log_p(u_{\psi'_2}) } { \gauss(\chi_{gh})\log_p(u_{g_\alpha})}.
  \end{equation}



The main result  of this section is:
\begin{theorem}
\label{main-theta} 
Conjecture \ref{conj:main} holds in the setting described above, i.e., \eqref{eqn:caseK} holds true.
\end{theorem}

The remainder of this section is devoted to the proof of Theorem \ref{main-theta}. Similarly as in the proof of the theorem of the previous section, fix a Hecke character $\lambda$ of  infinity type $(0,1)$ and conductor $\bar\wp$, with values in a number field $E$. Write $\cO$ for the ring of integers of the $p$-adic closure of $E$ in $\bar\Q_p$ and decompose its group of units
   as $\cO^\times=\mu\times W$, where $\mu$ is finite and $W$ is free over $\Z_p$. Let $\langle \cdot\rangle:\cO^\times \rightarrow W$ denote the natural projection.


For every integer $k\geq 1$ define $\psi^{(p)}_{g,k-1}=\psi_g \langle \lambda \rangle^{k-1}$ and let $\psi_{g,k-1}$ be the Hecke character of infinity type $(k-1,0)$ given by
\begin{equation}\label{he}
\psi_{g,k-1}(\mathfrak q) = \begin{cases} \psi^{(p)}_{g,k-1}(\mathfrak q) & \mbox{ if } \mathfrak q \ne \bar\wp \\ \chi_g(p) p^{k-1}/\psi^{(p)}_{g,k-1}(\wp) & \mbox{ if } \mathfrak q = \bar\wp.
\end{cases}
\end{equation}

Let $g_k = \theta(\psi_{g,k-1})$ denote the theta series associated to $\psi_{g,k-1}$, an eigenform of weight $k$. These modular forms vary $p$-adically, in the sense that there is a Hida family $\hg$ of tame level $N_g$ and character $\chi_g$ over an open subset $\mathcal U_g \subset 1+p\Z_p$ of weight space such that its classical specializations are the ordinary $p$-stabilizations of $g_k$. Note that $\psi_{g,1}=\psi_g$ and thus $g=g_1$; the specialization of $\hg$ in weight one is $g_\alpha$. 

For the remainder of this section, we replace the Hida family $\hg$ with a $\Lambda$-adic modular form $\htg(q) = \sum_{d\mid \frac{N}{N_g}} \lambda_d\hg(q^d)$ of tame level $N$ satisfying that all its classical specializations are not only eigenvectors for the Hecke operators $T_\ell$ for $\ell \nmid Np$ but also for the operators $U_q$ for all $q\mid Np$. Likewise, replace $h$ with an oldform $\testh\in M_1(N,\chi_h)$ that is an eigenvector for all good and bad Hecke operators.



Together with \eqref{he}, define also the Hecke characters
\begin{equation}\label{Hecke-gh}
\Psi_{gh}(k)=\psi_{g,k-1}^{-1} \cdot \psi_h^{-1} \cdot \mathbf{N}^{\frac{k+1}{2}} \quad \mbox{and} \quad \Psi_{gh'}(k)=\psi_{g,k-1}^{-1} \cdot  (\psi'_h)^{-1} \cdot \mathbf{N}^{\frac{k+1}{2}},
\end{equation}
where $\mathbf{N}$ is the Hecke character induced by the norm from $K$ to $\Q$.

%For $k\in U_g := \Z_{\geq 2} \cap \mathcal U_g$ 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 $\kappa = \frac{k-3}{2} \geq 0$. Hence for $k\in U_g$ the %values $L_p(f,K)(\Psi_{gh}(k))$, $L_p(f,K)(\Psi_{gh'}(k))$ and $\Lp^g(\hg,k)(k)$ all lie in the region of interpolation. 

It follows immediately from the definitions that, for all $k\geq 1$, the Rankin-Selberg $L$-function $L(\testg_k,\testh,s)$ factors as the product of the two $L$-series associated to the (inverse of the) Hecke characters appearing in \eqref{Hecke-gh}, that is to say:
\begin{equation}\label{eq}
L\left(\testg_k, \testh,\frac{k+1}{2}\right) =  \mathfrak f_0(k)\cdot L(\Psi_{gh}^{-1}(k),0) \cdot L(\Psi_{gh'}^{-1}(k),0).
\end{equation}

Here $ \mathfrak f_0(k)$ is a fudge factor which is a rational function on $q^k$ as $q$ ranges over the divisors of $N$, with coefficients in $L$. Moreover, $\mathfrak f_0(k) \in L^\times$ for all $k\in \Z_{\geq 1}$. In the terminology coined in \cite[Definition 4.3]{DLR}, functions satisfying these properties are called $L$-admissible, and we identify them with their unique analytic extension to $\mathcal U_g$.

The factorisation formula \eqref{eq} admits a $p$-adic analogue, of which Theorem \ref{main-theta} is a consequence, as we show later. Since the reader may find it of independent interest, we state it below as a separate result. For every $k\geq 1$, define the Hecke character
$$
\Psi_g(k) := \psi_{g,k-1}^{-2} \,\chi_g\,\mathbf{N}^k.
$$

Let $\Lp(\htg,\testh)$ denote Hida's $p$-adic Rankin $L$-function on $\mathcal U_g$ associated to the pair $(\htg,\testh)$. We refer e.g.\,to \cite{BDR1} for its precise definition, and we shall use the normalizations adopted therein. 

As in the previous chapter, let also $L_p^{\katz}$ denote Katz's $p$-adic $L$-function, defined and normalized as in \cite{DLR}. 

%The domain of this function is the $p$-adic completion of a suitable set of Hecke characters, which %contains $\mathcal U_g$ in a natural way via the identification of a Hecke character with its associated %theta series.

\begin{theorem}\label{factor} There is an $L$-admissible function $\mathfrak{f}$ such that the following factorisation of analytic functions on $\mathcal U_g$ is valid:
\begin{equation}\label{fact0}
\Lp(\htg,\testh)(k)\,\cdot  \, L_p^{\katz}(\Psi_{g}(k))  =  \mathfrak{f}(k) \,\cdot \, L_p^{\katz}(\Psi_{gh}(k)) \, \cdot \,L_p^{\katz}(\Psi_{gh'}(k)).
\end{equation}
\end{theorem}

\begin{proof} It suffices to show that \eqref{fact0} holds true for all $k\in U_g := \mathcal U_g \cap \Z_{\geq 2}$, as this is a dense subset of $ \mathcal U_g$ for the $p$-adic topology. Since points in $U_g$ belong to the region of interpolation of the four $p$-adic $L$-functions appearing in \eqref{fact0}, their values are non-zero multiples of critical classical $L$-values. Namely, there are explicit functions $\Omega_i(k)$, $i=1,\dots,4$, such that
\begin{equation*}
\begin{matrix}
\Lp(\htg,\testh)(k) = \Omega_1(k)\cdot L\left(\testg_k, \testh,\frac{k+1}{2}\right) , & L_p^{\katz}(\Psi_{g}(k)) = \Omega_2(k)\cdot  L(\Psi_{g}^{-1}(k),0),\\
 L_p^{\katz}(\Psi_{gh}(k)) = \Omega_3(k)\cdot L(\Psi_{gh}^{-1}(k),0),   & L_p^{\katz}(\Psi_{gh'}(k)) = \Omega_4(k)\cdot  L(\Psi_{gh'}^{-1}(k),0).
\end{matrix}
\end{equation*}

Functions $\Omega_i(k)$ involve powers of suitable complex and $p$-adic periods and products of Euler-like factors at primes dividing $pN$. The reader may find completely explicit formulas for them in \cite[\S 2]{BDR1} and \cite[\S 3.2]{DLR}, respectively. In particular, it is apparent from these expressions that each $\Omega_i(k)$ individually does not interpolate $p$-adically to a continuous function on $\mathcal U_g$, thus a fortiori they are not $L$-admissible at all. However, \cite[Lemma 3.8]{DLR} together with a rather tedious but elementary computation show that the ratio 
$$
\mathfrak f(k) := \frac{\Omega_1(k) \Omega_2(k)}{  \Omega_3(k) \Omega_4(k)} \cdot  L(\Psi_{g}^{-1}(k),0)
$$ 
does vary $p$-adically continuously, and gives rise to an $L$-admissible function. We omit the details, since the proof of this fact proceeds along similar lines as in \cite[Theorem 3.9]{DLR}, although the calculations are much simpler than in loc.\,cit.\,because here we avoid the subtleties arising in \cite{DLR}  from the choice of test vectors for Garrett's triple-product $L$-function. 

The theorem now follows from \eqref{fact0}.
\end{proof}


In order to prove Theorem \ref{main-theta}, we proceed by relating the values at $k=1$ of the functions appearing in \eqref{fact0} to a $p$-adic iterated integral and the $p$-adic logarithm of suitable units. Note that 
\begin{equation*}
\Psi_{gh}(1)=\psi_{g}^{-1} \cdot \psi_h^{-1} \cdot \mathbf{N}, \quad  \Psi_{gh'}(1)=\psi_{g}^{-1} \cdot  (\psi'_h)^{-1} \cdot \mathbf{N}, \quad \Psi_g(1) = \psi_{g}^{-2} \,\chi_g\,\mathbf{N}  = \frac{\psi'_{g}}{\psi_g}\,\mathbf{N}.
\end{equation*}

Since for any {\em finite order}
character $\psi$  of $G_K$, Katz's $p$-adic $L$-function satisfies the functional equation
$$
L_p^{\katz}(\psi)  = L_p^{\katz}((\psi')^{-1} \mathbf{N}_{K}),
$$
Theorem \ref{factor} specializes to the following formula at $k=1$:
\begin{equation}\label{fact1}
\Lp(\htg,\testh)(k)\,\cdot  \,L_p^{\katz}(\frac{\psi'_{g}}{\psi_g})  =  \mathfrak{f}(1) \,\cdot \, L_p^{\katz}(\psi'_2) \, \cdot \,L_p^{\katz}(\psi'_1).
\end{equation}

\begin{lemma}\label{le} Let $g^*_\alpha = g_\alpha \otimes \chi_g^{-1}$ denote the twist of $g_\alpha$ under the inverse of its nebentypus. There exists a linear functional
$$
\testgamma_\alpha: M_1(N_{gh} p,\chi^{-1}_g)_L[g^*_{\alpha}] \,\lra \, L
$$
such that
\begin{equation}\label{le-2}
 \gauss(\chi_{gh})\times  \Lp(\htg,\testh)(1)  =\int_{\testgamma_{\alpha}}  f \cdot \testh.
\end{equation}
\end{lemma}

\begin{proof}
Since the $p$-adic $L$-function $\Lp(\htg,\testh)$ of \cite[\S 2.2.2]{BDR1} is constructed using the method of \cite{DR1}, the proof of this statement follows verbatim as in \cite[Proposition 2.6]{DLR}.
\end{proof}

In addition to the above lemma, recall Katz's $p$-adic analogue of Kronecker's limit formula, which asserts that for any {\em finite order}
character $\psi$   of $G_K$, the following equality holds up to scalars in $L^\times$:
\begin{equation}
\label{Katz-GZ}
L_p^{\katz}(\psi) = \begin{cases}   \log_p(u_{\psi}) &  \mbox{ if } \psi\ne 1\\
\log_p(u_\wp)   & \mbox{ if } \psi=1,
\end{cases}
\end{equation}
where $u_{\psi}$ and $u_\wp$ are as in \eqref{uaa} and \eqref{ubb} respectively.

Since $\mathfrak f$ is $L$-admissible, $\mathfrak f(1)$ is a non-zero scalar in $L$. Theorem \ref{main-theta} follows, as the combination of \eqref{fact1}, \eqref{le} and \eqref{Katz-GZ} show the sought-after equality
$$
\testgamma_\alpha(\Xi(g_\alpha,h)) = \lambda \cdot 
 \frac{\log_p(u_{\psi'_1}) \log_p(u_{\psi'_2})}{\gauss(\chi_{gh})\log_p(u_{g_\alpha})}
$$
up to some $\lambda\in L^\times$.



%{\em Now it remains to add a line explaining why this proves the conjecture in the form we stated it.}  Currently the conjecture asks us to show $\Xi(g_\alpha,h)$ is not zero in $M_1(Np,\chi_g^{-1})%[g_\alpha^*]_{\C_p}$ and moreover belongs to  $\Omega \cdot M_1(Np, \chi_g^{-1})[g_\alpha^*]_L$.

%This amounts to saying that for any element $\gamma$ in the $L$-dual of $M_1(Np, \chi_g^{-1})[g_\alpha^*]_L$,
%\begin{equation}\label{proved}
%\gamma(\Xi(g_\alpha,h)) = \lambda \cdot \Omega, \quad \mbox{ for some } \lambda \in L,
%\end{equation}
%and $\lambda\ne 0$ for at least one such $\gamma$. This may be more than we can prove, and perhaps is actually false. What we have proved so far is that $\Xi(g_\alpha,h) \ne 0$ and there is an element  %$\gamma$ in the $L$-dual of $M_1(Np, \chi_g^{-1})[g_\alpha^*]_L$ such that \eqref{proved} holds. This weaker version is also in the spirit of \cite{DLR}.

   
 \begin{example}
 We revisit the setting treated in \cite[Example 4.3]{DLR}.
 Let $\chi_5$ and $\chi_{31}$ denote the (even and odd, respectively) quadratic Dirichlet
  characters of conductor $5$ and $31$, and
 let $\chi := \chi_5 \cdot \chi_{31}$ be the quadratic character attached to the   quadratic field $K:=\Q(\sqrt{-155})$.
 The class group of $K$ is cyclic of order $4$, and the two-dimensional representation induced from a quartic character $\psi$
 of this class group is attached to the 
 weight one cuspidal eigenform
 \[ g = \theta_\psi =  q + q^4 - q^5 - q^9 + q^{16} - 2 q^{19} - \cdots \in S_1(155,\chi).\]
The Galois representation $\varrho_g$ attached to $g$  has  image the dihedral group
 $D_4$ of order $8$, and projective image isomorphic to the Klein $4$-group.
 It   is also induced from  quadratic
characters of $\Q(\sqrt{5})$ and $\Q(\sqrt{-31})$ (ramified at  a prime above $31$ and $5$, respectively).  
In particular, the classicality hypothesis for $g$ is satisfied for the prime $p$
precisely when $\chi_5(p) = -1$.
Set $h:= g = g^*$ and let $f:= E_2^{(p)}(1,1)$   be the
Eisenstein series of weight $2$ and level $p$ attached to the trivial character.

Although $g=h=\theta_\psi$  is  cuspidal, the tensor product $\varrho_{gh}$ decomposes as a direct sum
$$ \varrho_{gh} = 1 \oplus \chi_5 \oplus \chi_{31} \oplus \chi_{155},$$
while 
$$ \Ad(\varrho_g) = \chi_5 \oplus \chi_{31} \oplus \chi_{155}.$$
The period $\Omega_{gh}$ of Conjecture \ref{conj:main}  involves the $p$-adic logarithms of 
the fundamental unit $u(\chi_5)$ of the quadratic field $\Q(\sqrt{5})$ in both its numerator and 
 denominator. This quantity cancels out in the conjecture, which then
predicts that the iterated integrals attached to $(f,g,g)$ should agree with the logarithm of the fundamental
$p$-unit of the imaginary quadratic field $K=\Q(\sqrt{-155})$, if $\chi_{155}(p)=1$ 
(resp.~of $\Q(\sqrt{-31})$, if $\chi_{31}(p)=1$). 
The running assumption that $\chi_5(p)=-1$ ensures that one is always in precisely one of these two scenarios.

The prime $p := 7$ is inert in $K/\Q$, and the frobenius eigenvalues $\alpha_g$ and $\beta_g$ are equal to $1$ and $-1$ respectively.
Letting $\gamma_1$ and $\gamma_{-1}$ denote the associated elements in the dual space of 
$S_1(155,\chi)$, we calculate:
\[ \int_{\gamma_{\pm 1}} E_2^{(7)}(1,1)  \cdot \theta_\psi = 4365724287399866821223816 \pmod{7^{30}}, \]
and check to $30$ digits of $7$-adic precision that
\[ \int_{\gamma_{\pm 1}} E_2^{(7)}(1,1) \cdot \theta_\psi = - \frac{\log_7(u_7(\chi_{31}))}{7}.\]
The prime $p := 13$ splits in $K/\Q$,  and 
the associated frobenius eigenvalues are $i$ and $-i$. A numerical calculation reveals that
\[ \int_{\gamma_{\pm i}} E_2^{(13)}(1,1) \cdot \theta_\psi = 11525893437665284 \pmod{13^{15}}. \]
One observes, to $15$ digits of $13$-adic precision, that
\[ \int_{\gamma_{\pm i}} E_2^{(13)}(1,1) \cdot \theta_\psi = - \frac{3 \cdot \log_{13}(u_{13}(\chi_{155}))}{13}, \]
as predicted by Conjecture \ref{conj:main}. 
 \end{example}
 
 
 
 
 
 
 
\section{Gross-Stark units and theta series of real quadratic fields}
\label{sec:gross-stark}
The setting in which 
\begin{equation}
\label{eqn:theta-psig-psih}
g:= \theta_{\psi_g}, \qquad
h:= \theta_{\psi_h} 
\end{equation}
are induced
 from characters $\psi_g$ and $\psi_h$  of (the same) mixed signature of 
 a common real quadratic field $K$ 
is of special interest, since in that case $\varrho_{gh}$ factors through the Galois group 
 of an abelian extension of $K$. For instance, when
 \eqref{eqn:theta-psig-psih} holds, the representation
 $V_{gh}$ decomposes as a direct sum
 $$ V_{gh} = V_{\psi^+} \oplus V_{\psi^-}, \qquad \psi^+ := \psi_g \psi_h, \ \ \ \psi^- := \psi_g \psi_h'$$
 of two induced representations by characters $\psi^+$ and $\psi^-$ of $K$ which are
 totally even and totally odd respectively.
 
 If $V_{\psi^+}$ is irreducible, it occurs with multiplicity two in the unit group, and hence $d_{gh} = 2$.
 If $p$ is any prime satisfying the regularity hypothesis for $g$, then necessarily
 $p$ is inert in $K$ and therefore $\alpha_g = - \beta_g$, and likewise for $\alpha_h$ and $\beta_h$. 
 The period $\Omega_{gh}$ of Conjecture \ref{conj:main} is predicted to be non-zero precisely when
$ \beta_g\alpha_h \ne \pm 1$ and $\beta_g\beta_h\ne \pm 1$.
This period then involves, in its numerator, a $p$-adic regulator 
 formed from genuine units in the $\psi^+$ and $(\psi^+)'$ components
  of the unit group of an appropriate abelian extension of $K$.
  The unit $u_{g_\alpha}$ which appears in the denominator in the definition of
  $\Omega_{gh}$ is just the $p$-adic logarithm of a fundamental unit $u_K$ of $K$,
  in light of the regularity hypothesis which  implies that the prime $p$ is inert in $K$.
  (Cf.~case (b) in the proof of Lemma 1.5 in Sec.~1.2 of \cite{DLR}.)
  We have not tested any instance of this general conjecture numerically, and it might be interesting to
 do so. 
 
 
 In the special case where $\psi_h = \psi_g^{-1}$, the character $\psi^+$ is trivial and $\psi^- = \psi_g / \psi_g'$ is a ring class
 character of $K$. The abelian extension $H$  of $K$ cut out by $\psi^-$ is Galois over $\Q$ and 
 $\Gal(H/\Q)$ is a generalised dihedral group.
 The regularity hypothesis for $p$, which forces $p$ to be inert in $K/\Q$, implies that the frobenius element at $p$
 acts on $V_{\psi^-}$ with eigenvalues $1$ and $-1$, and hence the Artin representation
 $V_{\psi^-}$ occurs with multiplicity $1$ in the space $\C\otimes\cO_H[1/p]^\times$. 
 The $p$-units in this abelian extension are precisely the Gross-Stark units studied in 
 \cite{DD}, and the regulator of Conjecture \ref{conj:main} involves the $p$-adic logarithms of such units. 
 There is also the $p$-adic logarithm of a fundamental unit of $K$, which is in the minus eigenspace for the action of the frobenius automorphism at $p$,
 and which appears in both the numerator and the denominator and thus 
 cancels out.  The transcendental part of the period $\Omega_{gh}$ therefore involves only 
 the $p$-adic logarithms of a $p$-unit $u_p^+({\psi^-})$, which is in the $V_{\psi^-}$-component
 of $\cO_H[1/p]^\times$ and is fixed by the frobenius element at $p$.
  The next example illustrates this special instance of Conjecture \ref{conj:main}:
 
  



\begin{example}
\label{ex:145gg}
Let $K=\Q(\sqrt{5})$ be the real quadratic field attached to the even 
Dirichlet character $\chi_5$ of conductor $5$, 
and let $\psi$ be a quartic character of $K$ whose conductor is one of the primes of $K$ 
above $29$. The central character $\chi_{29} $ of $\psi$ is an odd quartic Dirichlet character of 
conductor $29$. 
Let  $\chi:= \chi_5\chi_{29}$,  and let
\[ g := \theta_\psi = q - i q^4 + i q^5 - i q^9 + (i - 1)q^{11} - q^{16} + 
        (i - 1) q^{19} + \cdots \in S_1(5 \cdot 29,\chi)\]
be the weight one modular form considered in \cite[Example 4.1]{DLR},  whose Galois 
representation agrees with the induced representation $V_\psi$. 
After setting $h := g^* = \theta_{\psi^{-1}}$, we have 
$$ V_{gh} =  1 \oplus \chi_{5} \oplus V_{\psi^-},$$
where  $\psi^- = \psi/\psi'$ is a quartic ring class character    of  $\Q(\sqrt{5})$.
The induced representation $V_{\psi^-}$ has image the dihedral group $D_4$ of order $8$, and is notable in that
it cannot be obtained as the induced representation of the character of any imaginary quadratic field.
   % (Level $145$ is the smallest in which such forms occur: indeed there is another in that level whose representation is induced from a character of  $\Q(\sqrt{29})$, but the next such form does not occur until level\footnote{As far as I remember. Double-check this claim!} $396$.) 
   
   
Set $p := 13$  
and choose the embedding of $\Q(i)$ into $\Q_{13}$ sending $i$ to the unique $4$th 
root
of $-1$ which is congruent to $5$ modulo $13$.
One finds  that $\chi(13) = 1$ and  $(\alpha_g,\beta_g) = (i,-i)$.
Letting $f:= E_2^{(13)}(1,1)$ , one computes
\[ \int_{\gamma_{\pm i}} E_2^{(13)}(1,1)  \cdot \theta_{\psi^{-1}} = -7724370085026350688358 \pmod{13^{20}}.\]
%% for precision 13^20 see 2014_Mar31_145_p13_m20_eis.m
Let 
\[ H := \Q(a) \mbox{ where } a^8 - 2 a^7 + 4 a^6 - 26 a^5 + 94 a^4 - 212 a^3 + 761 a^2 - 700 a + 980.\]
This is the quartic subfield of the 
ring class field  of $K$ of conductor $29$, which 
is  cut out by  $\psi^-$.
Let
\[ b := (-27 a^7 + 116 a^6 - 196 a^5 + 738 a^4 - 3602 a^3 + 11256 a^2
    - 22295 a + 53310)/5160, \]
denote the $13$-unit of $H$ satisfying  the quartic polynomial $x^4 - 11 x^3 + 20 x^2 + 89 x + 169$. 
Writing $\Gal(H/K) =  \langle \sigma
\rangle$,  and  defining   $u_{13}^+(\psi^-) := b/b^{\sigma^2}$, 
we have verified  that
\[  \int_{\gamma_{\pm i}} E_2^{(13)}(1,1) \cdot \theta_{\psi^{-1}}  = \frac{3}{2 \cdot 13} \cdot \log_{13}(u_{13}^+(\psi^-)), \]
to $20$-digits of $13$-adic precision.
%% in computation v = 2(b - \sigma^2 (b)) and the fraction is 3 / (4 x 13).
\end{example}


The setting in which 
\begin{equation}
\label{eqn:theta-psig-eis}
g:= \theta_{\psi_g}, \qquad
h:= E_1(1,\chi_g^{-1})  
\end{equation}
is also of great relevance for explicit class field theory for $K$, since in that 
case 
$$ V_{gh} = V_{\psi_g} \oplus V_{\psi_g^{-1}} = V_g \oplus V_{ g^*}.$$
 Let $u(\psi_g)$,  $u(\psi_g')$, $u(\psi_g^{-1})$,  and $u(\psi_g')$ denote the generators in
 $L\otimes\cO_{H_g}^\times$ for the relevant eigencomponent relative to the action of 
 $\Gal(H_g/K)$. Since the prime $p$ is inert in $K$, these four units can be chosen in such a way that
 $$ \Frob_p u(\psi_g) = u(\psi_g'), \qquad \Frob_p u(\psi_g^{-1}) = u(\psi_g^{\prime -1}).$$
 


\begin{example}\label{ex-145gh}
Let $g$ be the cusp form in level $145$ from Example
 \ref{ex:145gg} but now take 
\[ h := E_1(1,\chi) = (i + 1) + q + \cdots + 2 q^{13} + \cdots \] 
to be
the Eisenstein series  of weight $1$ attached to the pair of  characters $(1,\chi)$. 
The Artin representation $\varrho_{gh}$ is then given by
$$ V_{gh} = V_{\psi_g} \oplus V_{\psi_g^{-1}} = V_g \oplus V_{ g^*}.$$
The smallest abelian extension $H_g$ of $K$ cut out by the representation 
$V_{g}$ (and/or $V_{g^*}$) is larger than the ring class field $H$
of degree $8$ over $\Q$ that arose in Example \ref{ex:145gg}: it is the ray class field of $K$ of conductor
$$  (29) = (29,2 \sqrt{5}  + 17)(29,2 \sqrt{5} + 10) =: Q_1 Q_2, $$
and has degree $32$ over $\Q$.
The field $H_g = H_{g^*}$ is the compositum of the ray class fields $H_1$ and $H_2$ 
of $K$ of conductors $Q_1$ and $Q_2$, these ray class fields being the abelian extensions of $K$ cut
out by the characters $\psi_g$ and $\psi_g'$ respectively.

A calculation with Magma reveals that $H_g$ can be described as the splitting field of the polynomial
$$ x^8 - x^7 - 3 x^6 + 3 x^5 + 3 x^4 - 6 x^3 - 2 x^2 + 3 x + 1 =  P_1(x) P_2(x), $$
where
$$P_1(x) =  x^4 + \frac{(-\sqrt{5} - 1)}{2} x^3 - x^2 + x + \frac{(-\sqrt{5} + 3)}{2}, \qquad P_2(x) = \tau P_1(x),$$
and $\tau$ denotes the generator of $\Gal(K/\Q)$. 
The ray class fields 
 $H_1$ and $H_2$ are just  the splitting fields over $K$ of the  irreducible quartic factors $P_1(x)$ and $P_2(x)$ above with cyclic Galois group.
Hence, $H_1=K(u_1)$ and $H_2 = K(u_2)$, where $u_1$ and $u_2$ are roots of  $P_1(x)$ and $P_2(x)$ respectively. 
We define
$$  v_1 := \frac{1}{2} \left(-\sqrt{5} - 1\right) u_1^2, 
\qquad v_2 := \frac{1}{2} \left(\sqrt{5} - 1\right) u_2^2.$$
Letting $\sigma_1$ denote a generator of $\Gal(H_1/K)$, we have
$$  v_1 = (\sigma_1^2(v_1))^{-1}, \qquad \sigma_1(v_1) = (\sigma_1^3(v_1))^{-1}, $$
and the
 unit $v_1$ and its $\Gal(H_1/K)$-translates $\sigma_1 v_1$, $\sigma_1^2 v_1$, $\sigma_1^3 v_1$ generates the
 compositum of the $\psi_g$ and $\psi_g^{-1}$-isotypic parts in the unit group $\cO_{H_1}^\times$. 
 More precisely, using additive notation in $L\otimes \cO_{H_g}^\times$, the elements
 $$ v(\psi_g) :=  v_1  + i \otimes v_1^{\sigma_1}, \qquad
 v(\psi_g') := v_2 + i \otimes v_2^{\sigma_2}$$
 generate the $\psi_g$ and $\psi_g'$ eigenspaces respectively
 for the action of $\Gal(H_g/K)$,
 while the elements 
 $$ v(\psi_g^{-1}) :=  v_1  - i \otimes v_1^{\sigma_1}, \qquad
 v((\psi_g')^{-1}) := v_2 - i \otimes v_2^{\sigma_2}$$
 generate the $\psi_g^{-1}$ and $(\psi_g')^{-1}$-eigenspaces for this action.

  Assume that $p$ is inert in $K$ and that $\alpha_g = -\beta_g  = i$. 
 After fixing $v_1$ and $v_2$ in such a way that $\Frob_p(v_1) = v_2$, 
a direct calculation reveals that the regulator matrix $R_{g_i}(\varrho_{gh})$ is a diagonal matrix of the form
$$R_{g_i}(\varrho_{gh}) = \left(
\begin{array}{cc}
 \log_p(v(\psi_g))  - i \log_p(v(\psi_g')) & 0 \\
0 &  \log_p(v(\psi_g^{-1}))  -  i \log_p(v((\psi_g')^{-1})) \end{array} \right),$$
while 
$$ R_{g_{-i}}(\varrho_{gh}) = \left(
\begin{array}{cc}
 \log_p(v(\psi_g))  +  i \log_p(v(\psi_g')) & 0 \\
0 &  \log_p(v(\psi_g^{-1}))   +  i \log_p(v((\psi_g')^{-1})) \end{array} \right).
$$
 
%% Then $v_1$ is a unit in $H_1$ which is in the $i$-eigenspace for $\tau_1$, where here $\Gal(H_1/\Q(\sqrt{5})) = \langle
%% \tau_1 \rangle$. (To self: this is how it looks in the example file, but probably best to define
%% \[ H_1 := \Q(\sqrt{5})(v_1) \mbox{ where } v_1^4 + (2 \sqrt{5} + 3) v_1^3 + \frac{1}{2}(5 \sqrt{5} + 15) v_1^2 + (2 \sqrt{5} + 3) v_1 + 1 = 0.\]
%% One can check then $v_1^{-1}$ is also a root of this polynomial ...)

%% The ray class field $H_2$ for $Q_2$ is obtained by adjoining a root $u_1$ of the second factor
%% above and one constructs an analogous unit $v_2 \in H_2$ which is in the $i$-eigenspace for $\tau_2$,
%% where $\Gal(H_2/\Q(\sqrt{5})) = \langle \tau_2 \rangle$

 
Let $p := 13$ and embed $\Q(i)$ into $\Q_p$ so that $i \equiv 5 \bmod p$. 
A numerical calculation of the $p$-adic iterated integrals reveals that
\[
\begin{array}{rcl}
\int_{\gamma_{+i}} E_2^{(13)}(1,1)  \cdot E_1(1,\chi) & = & -8501755713365462060121  \pmod{13^{20}}\\
\int_{\gamma_{-i}} E_2^{(13)}(1,1) \cdot E_1(1,\chi) & = & 4274893846350140558653 \pmod{13^{20}}.
\end{array}
\]
%% This is the correct way around, assuming the alpha and beta in my code follow the same convention
%% as in the paper.

After further embedding $H_1$ and $H_2$ into $\Q_{p^4}$ in such a way that
 $\sigma_{p}(v_1)=v_2$,  one finds that
 \begin{eqnarray*}
\int_{\gamma_{i}} E_2^{(13)}(1,1) \cdot h &=& \frac{ (7 i + 6)^2  \cdot \det R_{g_i}
(\varrho_{gh})}{3 \cdot 8 \cdot 7 \cdot 13 \cdot  \log_{13}(u_K)}  \\
\int_{\gamma_{-i}} E_2^{(13)}(1,1) \cdot h &=& \frac{ (-7 i + 6)^2  \cdot \det R_{g_{-i}}
(\varrho_{gh})}{3 \cdot 8 \cdot 7 \cdot 13 \cdot  \log_{13}(u_K)},
\end{eqnarray*}
to $20$-digits of $13$-adic precision.
\end{example}

The analogous setting to that in Example \ref{ex-145gh} with Mordell-Weil groups replacing groups of units was discussed in \cite[Section 6]{DLR}. However, 
this case of the ``Elliptic Stark Conjecture'' was   difficult to check experimentally because of the apparent scarcity of elliptic curves $E$ for which $r(E,\varrho_g) >0$, and no evidence was given in \cite{DLR} (beyond
the case when $g = g^*$). Example \ref{ex-145gh} can be seen as providing some indirect evidence for the  main conjecture of \cite{DLR}, albeit with unit groups replacing
Mordell-Weil groups.


\section{Exotic forms}
\label{sec:exotic}
Cases where $g$ and/or $h$ are exotic forms often correspond   to settings 
that have been little studied 
previously from the point of view of either  the classical or the $p$-adic Stark conjectures.
 
 
One of the  simplest settings occurs  when $g$ is exotic   and $h=g^*$ is the dual form. 
The Artin representation 
 $\varrho_{gh}$ then factors through the extension $H$ cut out by the 
 {\em projective representation} attached to $g$, and it decomposes as
 $$  V_{gh} = 1 \oplus \Ad^0(V_g).$$
 The adjoint representation
 $\Ad^0(V_g)$ always occurs with multiplicity one in $\C\otimes \cO_H^\times$,  and with multiplicity at least two
 in $\C\otimes\cO_H[1/p]^\times$, since $1$ always arises as an eigenvalue for any linear transformation on the adjoint.
 More precisely, the frobenius automorphism $\Frob_p$ acts on $\Ad^0(V_g)$ with eigenvalues
 $1$, $\alpha_g/\beta_g$ and $\beta_g/\alpha_g$. 
 When $\Ad^0(V_g)$ is regular at $p$, there are therefore  
 (unique up to scaling) 
 elements
 \begin{equation}
 \label{eqn:units-ad}
 u_1, u_{g_\alpha}, u_{g_\beta} \in L\otimes \cO_H^\times 
 \end{equation}  
 generating a subrepresentation 
 of $L\otimes\cO_H^\times$ which is isomorphic to $\Ad^0(V_g)$, and such that
 $$ \Frob_p u_1 = u_1, \qquad
 \Frob_p u_{g_\alpha} =  (\beta_g/\alpha_g)  \otimes  u_{g_\alpha}, 
 \qquad
  \Frob_p u_{g_\beta} = ( \alpha_g/\beta_g) \otimes u_{g_\beta}.
  $$
  Likewise, the choice of a Galois-equivariant homomorphism $v:V_{gh} \lra  L\otimes (\cO_H[1/p])^\times$ whose image is not contained in $L\otimes\cO_H^\times$ produces three independent $p$-units
  \begin{equation}
  \label{eqn:p-units-ad}
  v_1, v_{g_\alpha}, v_{g_\beta} \in L\otimes(\cO_H[1/p])^\times
  \end{equation}
satisfying
 $$ \Frob_p v_1 = v_1, \qquad
 \Frob_p v_{g_\alpha} =  (\beta_g/\alpha_g) \otimes  v_{g_\alpha}, 
 \qquad
  \Frob_p v_{g_\beta} = ( \alpha_g/\beta_g) \otimes v_{g_\beta}.
  $$
 These units are the counterparts of the Stark points
 considered in  \cite[Example 5.3]{DLR}.  

  A simple calculation reveals that
  $$ R_{g_\alpha}(\varrho_{gh}) = \left(
  \begin{array}{cc}
  \log_p u_1 & \log_p u_{g_\alpha} \\
  \log_p v_1 & \log_p v_{g_\alpha} 
  \end{array} \right),$$
  and therefore that the period $\Omega_{gh}$ of Conjecture \ref{conj:main} can be described in this case
  by
  \begin{equation}
  \label{eqn:adjoint-reg}
  \Omega_{gh} = \frac{ \log_p u_1 \log_p v_{g_\alpha} - \log_p u_{g_\alpha} \log_p v_1}{\log_p u_{g_\alpha}}.
  \end{equation}
  
\begin{example}
This example examines this special case of Conjecture \ref{conj:main} when
  $g$ is 
 Tate's tetrahedral form in level $133$.
Recall the following from \cite[Example 5.3]{DLR}:
Let $\chi$ be a Dirichlet character of level $133 = 7 \cdot 19$ of order $2$ at $7$ and order $3$ at $19$.
Denote by $\zeta$ a primitive $12$-th root of unity and define $\zeta_3 := \zeta^{-4}$.
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 $\chi^{-1}$.
The  extension $H$ of $\Q$ 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 $133^2$ and Galois group $A_4$ over $\Q$.

As in \cite[Example 5.3]{DLR},
with $p := 13$ we find the
%% $\beta_h = \zeta$ and $\alpha_h = -\zeta^3$, and so the
two $13$-stabilisations $g_\alpha$ and $g_\beta$ of $g$ have
$U_p$-eigenvalues $\alpha_g = \zeta^{-1}$ and $\beta_g = \zeta^3$, and
choose the embedding of $\Q(\zeta)$ into $\Q_{13}$ which sends $\zeta$ to the $12$th root
of unity which is congruent to $6 $ modulo $13$. 

Letting $f:= E_2^{(13)}(1,1)$ be the Eisenstein series of weight $2$ and 
level $13$,  and letting $\gamma_\alpha$  be the element of the 
dual space  of  $S_2(133\cdot p,\chi)[g_\alpha]$  
sending $g_\alpha$ to $1$, (and likewise for $\gamma_\beta$, with $g_\alpha$ replaced by $g_\beta$), 
we find
\[
\begin{array}{rcl}
\int_{\gamma_{\alpha}} E_2^{(13)}(1,1) \cdot g  & = &  7956424094204877500900 \pmod {13^{20}}\\
\int_{\gamma_{\beta}} E_2^{(13)}(1,1)  \cdot g  & = & -5574982888910767132405 \pmod{13^{20}}.
\end{array}
\]
%% not completely sure these are correct way around.

Define the unit and $13$-unit
\[
\begin{array}{rcl}
u & := & -b + 1 \in {\mathcal O}_M^{\times}\\
v & := & 6 b^3 + 3 b^2 + 22 b - 21 \in {\mathcal O}_M[1/p]^\times, 
\end{array}
\]
respectively. One may choose an embedding of $H$ into $\Q_{13^3}$ so that 
$u_1 := u$ and $v_1 := v$ lie in $\Q_{13}$ and so are fixed by $\sigma_{13}$.
Let 
\[ u_{g_{\alpha}}, u_{g_{\beta}} \in \left({\mathcal O}_{H}^\times \right)_L^{{\rm Ad}_g}, \qquad 
v_{g_{\alpha}}, v_{g_{\beta}} \in \left({\mathcal O}_{H}[1/p]^\times \right)_L^{{\rm Ad}_g}\]
be the units and $p$-units    described in  terms of $u$ and $v$ in
\eqref{eqn:units-ad} and
\eqref{eqn:p-units-ad}.
The unit $u_{\alpha_g}$ (resp.~$u_{\beta_g}$) lies in the  $\beta_g/\alpha_g= \zeta_3^{-1}$ eigenspace (resp.~the  $\alpha_g/\beta_g = \zeta_3$
 eigenspace) for $\Frob_p$, and likewise for the
$p$-units $v_{\alpha_g}$ and $v_{\beta_g}$.

To
$20$-digits of $13$-adic precision we have found
\[ \int_{\gamma_{\alpha}} E_2^{(13)}(1,\chi^{-1}) \cdot g =  - \frac{\log_p(u_1) \log_p(v_{g_\alpha}) - \log_p(v_1) \log_p(u_{g_\alpha})}{13 \cdot \log_p(u_{g_\alpha})}\]
and likewise replacing $\alpha$ by $\beta$ throughout.  

This example is of a similar flavour to
\cite[Example 5.4, curves 26b, 52a]{DLR}. Indeed, it pre-dates the rank $2$ examples in \cite{DLR}
and was the experiment which led to the first serious evidence for (and correct formulation of) the 
 ``Elliptic Stark conjecture'' of \cite{DLR} in the more striking, but harder to test
scenarios  of   ``genuine rank two".
\end{example} 
 
 
Our final example involves a scenario where $V_{gh}$ is an
 irreducible $4$-dimensional representation, occuring with multiplicity two in the group of $p$-units. It is worth noting
  that no instances of
 this sort were ever successfuly tested numerically in the elliptic curve setting considered in \cite{DLR}, owing to
 the apparent scarcity of irreducible $V_{gh} $ of manageable size occuring with multiplicity two in the Mordell-Weil groups
 of elliptic curves of reasonable conductor. In the setting of units, such examples abound, since {\em any} irreducible 
 $V_{gh}$ has multiplicity two in the unit group.
The experiment below can thus also be viewed as providing some indirect
 support for the elliptic Stark conjecture of \cite{DLR} in the numerically
  less accessible setting where $\varrho_{gh}$ is irreducible.
\begin{example}
Let $\chi$ be the quadratic Dirichlet character of level $283$. The space $S_1(283,\chi)$ is $3$-dimensional and spanned by the
eigenforms
$$
\begin{array}{rcl}
g & = & q + q^4 - q^7 + q^9 - q^{11} - q^{13} + q^{16} - q^{23} + \cdots \\
h_{\pm} & = &  q \pm \sqrt{-2} q^2 \mp \sqrt{-2} q^3 - q^4 \mp \sqrt{-2} q^5 + 2 q^6 - q^7 - q^9 + 2 q^{10} + q^{11} \pm 
        \sqrt{-2} q^{12} + \cdots.
\end{array}
$$
 The pair $h_{\pm}$ are exotic forms whose
associated projective representations have image $S_4$, and 
$g$ 
 is the dihedral form who representation $\varrho_g$ is induced from an unramified character of the
imaginary quadratic field $\Q(\sqrt{-283})$ (which has class number $3$). 
The representation $\varrho_g$ is obtained by composing the projective 
representation attached to $\varrho_h$ with the exceptional homomorphism
$S_4 \lra S_3$. This   is precisely the set-up   of
Lemma \ref{lemma:interesting-S4}, which asserts that
$\varrho_{gh}$ is irreducible and induced from 
 a suitable quadratic character $\eta$ of a quartic extension $K$ of $\Q$. 
The quartic field $K$ occuring in this lemma,
whose Galois closure ${\bar H}_h$ is the $S_4$-extension cut out by the
projective representation attached to the exotic forms $h_{\pm}$,
 is equal in this case to
$$K = \Q(a), \qquad a^4  - a - 1=0. $$
 The quadratic extension $M$ of $K$
 cut out by $\eta$ is given by
\[ M := K(b) \mbox{ where } b^2 = -6 a^3 + a^2 - 3 a + 5.\]
The elements 
\[ 
\begin{array}{ll}
u := \frac{1}{2}(-a^3 + a^2 - a + 1) b + \frac{1}{2} (-4 a^3 + 3 a^2 - 2 a + 4)\\
 & \\
v := \frac{1}{2} (a^3 - 2 a) b + \frac{1}{2} (a^2 + 2 a - 3)
\end{array}
\]
are relative units in   $M/K$, and hence their Galois translates generate the 
$\varrho_{gh}$-isotypic components of the unit group of $H_h$.
In additive notation let
\[ u^{\zeta_8^{\pm 1}} := \sum_{j = 0}^7 \zeta_8^{ \pm } j u^{\Frob_p^j} \]
and likewise for $v^{\pm \zeta_8}$. Here $\zeta_8$ is a primitive $8$th root of
unity such that $\sqrt{-2} = \zeta_8^3 + \zeta_8$. 




Let $p := 5$, so that $\chi(5) = -1$ and $\alpha_g = -\beta_g = 1$. 
Taking $f = E_2^{(5)}(1,1)$,
 one computes
\[ \int_{\gamma_{+1}} E_2^{(5)}(1,1) \cdot h_{\pm} = \left(1365364418900524107116682 \pm 982832041040545110371696 \sqrt{-2} \right)/5 \]
to $35$-digits of $5$-adic precision (and likewise for the $-1$-stabilisation, replacing $\pm$ on the righthand side by $\mp$),
after choosing an
embedding of $\Q(\zeta_8)$ and $H$ into $\overline{\Q_p}$.
%% Note in the 283_irreducible the sextic polynomial is the square of this!
A direct calculation shows that in this example,
$$ \det R_{g_\alpha}(\varrho_{gh}) = \log_p(u^{\zeta_8}) \log_p(v^{\zeta_8^{-1}}) -
\log_p(u^{\zeta_8^{-1}}) \log_p(v^{\zeta_8}).$$

As for the Stark unit $u_{g_\alpha}$ occuring in the denominator of the period $\Omega_{gh}$ of
Conjecture \ref{conj:main} when $\beta_g/\alpha_g =-1$,
the Hilbert class field of $\Q(\sqrt{-283})$ is generated by an algebraic number
$w$ satisfying the relation $w^3 - 16 w^2 - 8 w - 1 = 0$.
The unit $u_{g_\alpha}$ is then described, using additive notation, by the formula
\[ u_{g_\alpha} := w^{\Frob_p} - w^{\Frob_p^2} \]
where  an embedding of the Hilbert class field into $\Q_{p^2}$  has been fixed so that
$w$ is defined over $\Q_p$.
It has been verified experimentally that
\[ \int_{\gamma_{+1}} E_2^{(5)}(1,1)  \cdot h_{-} = \frac{19 \zeta_8^3 + 34 \zeta_8^2 - 19 \zeta_8}{2^5 \cdot 3^2 \cdot 5} \cdot
\frac{\log_5(u^{\zeta_8}) \log_5(v^{\zeta_8^{-1}}) -
\log_5(u^{\zeta_8^{-1}}) \log_5(v^{\zeta_8})}{\log_5(u_{g_\alpha})}
\]
to $35$ digits of $5$-adic precision, in excellent agreement with Conjecture \ref{conj:main}.
\end{example}


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\end{thebibliography}

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