
\documentclass[a4paper,11pt]{amsart}

\newfont{\cyr}{wncyr10 scaled 1100}

\usepackage[left=2.7cm,right=2.7cm,top=3.5cm,bottom=3cm]{geometry}
\usepackage{amsthm,amssymb,amsmath,amsfonts,mathrsfs,amscd}
\usepackage[latin1]{inputenc}
\usepackage[all]{xy}
\usepackage{latexsym}
\usepackage{longtable}
\usepackage{url}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{examplewr}[theorem]{Example}
\newtheorem{ass}[theorem]{Assumption}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{remark}[theorem]{Remark}

%\theoremstyle{remark}
%\newtheorem{obswr}[theorem]{Observation}
%\newtheorem{remarkwr}[theorem]{Remark}



%\newenvironment{obs}{\begin{obswr}\begin{upshape}}{\end{upshape}\end{obswr}}
%\newenvironment{remark}{\begin{remarkwr}\begin{upshape}}{\end{upshape}\end{remarkwr}}
%\newenvironment{example}{\begin{examplewr}\begin{upshape}}{\end{upshape}\end{examplewr}}

\DeclareMathOperator{\fil}{fil}
\DeclareMathOperator{\fr}{fr}

 \DeclareMathOperator{\an}{an}
 \DeclareMathOperator{\nh}{nh}
 \DeclareMathOperator{\cusps}{cusps}
 \DeclareMathOperator{\ffm}{ffm}
 \DeclareMathOperator{\Ext}{Ext}
 \DeclareMathOperator{\comp}{comp}
  \DeclareMathOperator{\Eis}{Eis}

 \DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\dR}{dR}
\DeclareMathOperator{\fp}{fp}
 \DeclareMathOperator{\cris}{cris}
 \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\et}{et}
 \DeclareMathOperator{\oc}{oc}
 \DeclareMathOperator{\ur}{ur}
 \DeclareMathOperator{\ah}{ah}
 \DeclareMathOperator{\para}{par}

\DeclareMathOperator{\cts}{cts} \DeclareMathOperator{\bal}{bal}
\DeclareMathOperator{\KS}{KS}
\DeclareMathOperator{\sym}{\mathrm{sym}}
\DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\spec}{spec} \DeclareMathOperator{\la}{la}
\DeclareMathOperator{\dlog}{dlog}

 \newcommand{\reg}{{\mathbf {reg}}}
%\DeclareMathOperator{\reg}{reg}

\newcommand{\cS}{{\mathcal S}}
\newcommand{\cZ}{{\mathcal Z}}
\newcommand{\hf}{{\mathbf{f}}}
\newcommand{\hE}{{\mathbf{E}}}
\newcommand{\hg}{{\mathbf{g}}}
\newcommand{\hh}{{\mathbf {h}}}
% For below, try \breve, \check, \dot, \hat, \mathring, \ddot
\newcommand{\testf}{\breve{f}}
\newcommand{\testg}{\breve{g}}
\newcommand{\testh}{\breve{h}}
\newcommand{\testG}{\breve{G}}
\newcommand{\testH}{\breve{H}}
\newcommand{\htf}{\breve{\mathbf{f}}}
\newcommand{\htg}{\breve{\mathbf{g}}}
\newcommand{\hth}{\breve{\mathbf {h}}}
\newcommand{\uS}{{\mathbf{S}}}
\newcommand{\hphi}{\boldsymbol{\phi}}
\newcommand{\uhp}{{\mathfrak{H}}}
\newcommand{\PPhi}{{\underline{\phi}}}
\newcommand{\ff}{{\mathbf{f}}}
\newcommand{\aaa}{{\mathbf{a}}}
\newcommand{\LL}{{\mathscr{L}}}
\newcommand{\Lp}{{\mathscr{L}_p}}
\newcommand{\rhop}{\rho(P,\omega_1,\omega_2)}

 \newcommand{\hB}{{\mathbf B}}
  \newcommand{\bV}{{\mathbb V}}

\newcommand{\St}{{\mathrm{St}}}
\newcommand{\cl}{{\mathrm{cl}}}
\DeclareMathOperator{\red}{red}

\newcommand{\varrhoP}{{\varrho_{_P}}}
\newcommand{\cA}{\mathcal A}
\newcommand{\cN}{\mathcal N}
\newcommand{\cW}{\mathcal W}
\newcommand{\cV}{\mathcal V}
\newcommand{\cF}{\mathcal F}
\newcommand{\cX}{\mathcal X}
\newcommand{\cH}{\mathcal H}
\newcommand{\cC}{\mathcal C}
\newcommand{\V}{\mathcal V}
\newcommand{\E}{\mathcal E}
\newcommand{\cE}{\mathcal E}
\newcommand{\g}{\gamma}
% \newcommand{\G}{\Gamma}
\newcommand{\Gap}{\Gamma_0^D(p M)}
\newcommand{\Ga}{\Gamma_0^D(M)}
\newcommand{\hGa}{\hat{\Gamma }_0(M)}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\G}{\mathbb{G}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\Ql}{\Q_{\ell}}
\newcommand{\GQ}{G_{\Q}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\Sel}{\mathrm{Sel}}
\newcommand{\Sha}{\mathrm{Sha}}
\newcommand{\Gal}{\mathrm{Gal\,}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\Cl}{\mathrm{Cl}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\Div}{\mathrm{Div}}
\newcommand{\Fil}{\mathrm{Fil}}
\newcommand{\cond}{\mathrm{cond}}
\newcommand{\Frob}{\mathrm{Frob}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Jac}{\mathrm{Jac}}
\newcommand{\disc}{\mathrm{disc}}
\newcommand{\Id}{\mathrm{Id}}
\newcommand{\MS}{\mathrm{MS}}
\newcommand{\Aut}{\mathrm{Aut}}
\newcommand{\Fr}{\mathrm{Fr}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\Up}{\mathrm{U}_p}
\newcommand{\ord}{{\mathrm{ord}}}
\newfont{\gotip}{eufb10 at 12pt}
\newcommand{\gn}{\mbox{\gotip n}}
\newcommand{\gm}{\mbox{\gotip m}}
\newcommand{\ga}{\mbox{\gotip a}}
\newcommand{\gb}{\mbox{\gotip b}}
\newcommand{\gf}{\mbox{\gotip f}}
\newcommand{\go}{{\mathcal O}_{A_f}}
\newcommand{\OAf}{{\mathcal O}_{A_f}}
\newcommand{\gp}{\mbox{\gotip p}}
\newcommand{\gL}{\mbox{\gotip L}}
\newcommand{\gd}{\mbox{\gotip d}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\GG}{{\mathcal G}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\om}{{\omega}}
\newcommand{\ra}{\mathrm{ra}}
\newcommand{\lra}{\longrightarrow}
\newcommand{\hra}{\hookrightarrow}
\newcommand{\qbar}{\bar\Q}
\newcommand{\Norm}{\mathrm{N}}
\newcommand{\NS}{\mathrm{NS}}
\newcommand{\SL}{{\mathrm {SL}}}
\newcommand{\SO}{{\mathrm {SO}}}
\newcommand{\n}{{\mathrm{n}}}
\newcommand{\Pic}{{\mathrm{Pic}}}
\newcommand{\CH}{{\mathrm{CH}}}
\newcommand{\AJ}{{\mathrm{AJ}}}
\newcommand{\tr}{{\mathrm{tr}}}
\newcommand{\si}{\sigma}
\newcommand{\eps}{\epsilon}
\newcommand{\ta}{{\tau}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\M}{{\mathrm{M}}}
\newcommand{\HC}{\mathcal F_{\rm har}}
\newcommand{\PGL}{{\mathrm{PGL}}}
\newcommand{\Supp}{\mathrm{Supp}}
\newcommand{\ep}{\varepsilon}
\newcommand{\chione}{{\chi}_{_{\!1}}}
\newcommand{\chitwo}{{\chi}_{_{2}}}
\newcommand{\sk}{\medskip}
\def \mint {\times \hskip -1.1em \int}

%\newcommand{\mint}{\times\!\!\!\!\!\hskip-0.3em\int}

\newcommand{\SP}{\vspace{0.5cm}}
\newcommand{\T}{\mathbb T}
\newcommand{\new}{\mathrm{new}}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Meas}{Meas}
\newcommand{\X}{\mathbb X}
\newcommand{\Y}{\mathbb Y}
\newcommand{\U}{\mathbb U}
\newcommand{\Rat}{{\rm Rat}\bigl(\PP^1(\Q_p),\C_p\bigr)}
\newcommand{\Ker}{{\mathrm Ker}}
\newcommand{\res}{\mathrm{res}}
\DeclareMathOperator{\rig}{rig}
\DeclareMathOperator{\cmp}{comp}
\DeclareMathOperator{\triv}{triv}
\DeclareMathOperator{\alg}{alg}


\DeclareMathOperator{\can}{can}
\DeclareMathOperator{\fin}{fin}
\DeclareMathOperator{\even}{even}
\DeclareMathOperator{\spl}{Spl}
\DeclareMathOperator{\hdg}{hdg}
 \DeclareMathOperator{\noc}{n-oc}
 \DeclareMathOperator{\hol}{hol}

\newcommand{\TT}{\mathbb{T}}
\newcommand{\uTT}{\underline{\mathbb{T}}}
\newcommand{\D}{\mathbb D}

\newcommand{\sumprime}{\sum_{(m,n) \in N\Z\times \Z}^{\phantom{spac} _{_{_{ \prime}}}}}
\newcommand{\sumprimeprime}{\sum_{(m,n) \in N\Z\times \Z} ^{\phantom{spac} _{_{_{\prime \prime}}}}}

\newcommand{\rom}{\mathrm}


\newcommand{\longmono}{\mbox{$\lhook\joinrel\longrightarrow$}}
\newcommand{\longleftmono}{\mbox{$\longleftarrow\joinrel\rhook$}}
\newcommand{\longepi}{\mbox{$\relbar\joinrel\twoheadrightarrow$}}
\newcommand{\mat}[4]{\left(\begin{array}{cc}#1&#2\\#3&#4\end{array}\right)}
\newcommand{\smallmat}[4]{\bigl(\begin{smallmatrix}#1&#2\\#3&#4\end{smallmatrix}\bigr)}
\newcommand{\dirlim}{\mathop{\varinjlim}\limits}
\newcommand{\invlim}{\mathop{\varprojlim}\limits}
 \newcommand{\eis}{{\mathrm{Eis}}}


\newcommand{\Corr}{\mathrm{Corr}}

\include{thebibliography}

\begin{document}

\title[Syntomic regulators and $p$-adic Rankin L-series]{Beilinson-Flach elements and Euler systems I:\\ Syntomic regulators and $p$-adic Rankin L-series}




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



\begin{abstract} This article is the first in a series devoted to  the Euler
 system  arising from  $p$-adic families of
 {\em Beilinson-Flach elements} in  the  first $K$-group of
  the product   of two modular curves.
 It  relates the image of these elements under  the $p$-adic
 syntomic
 regulator (as described by Besser \cite{besser-K1S}) to the
 special values at the near-central point
  of Hida's   $p$-adic Rankin $L$-function
 attached to two Hida families of cusp forms.
 \end{abstract}


\address{M.B.: Universit\"at Duisburg-Essen, Institut f\"ur Experimentelle Mathematik, Ellernstr. 29 
45326 Essen, Germany}
\email{massimo.bertolini@uni-due.de}
\address{H. D.: McGill University, Burnside Hall, Room 1111, Montr\'eal, Canada}
\email{darmon@math.mcgill.ca}
\address{V.R.: Universitat Polit\`{e}cnica de Catalunya, MA II, Despatx 413, C. Jordi Girona 1-3, 08034 Barcelona, Spain}
\email{victor.rotger@upc.edu}





\maketitle


 \tableofcontents

\section{Introduction}
This article
is the first in a series devoted to  the Euler
system of {\em Beilinson-Flach elements}
   in the motivic cohomology of a product
 of two modular curves.  Its main result (see Theorem \ref{thm:main}
 and Corollary \ref{cor:main}
 of \S \ref{sec:main-padic})  is a $p$-adic analogue of the
 formula of Beilinson \cite[Ch.\,2, \S 6]{beilinson-K1S} expressing
  special values of Rankin $L$-series in terms of complex regulators.
  Beilinson's theorem (cf.\,\S \ref{sec:main-C} for an explicit version) relates:
 \begin{enumerate}
 \item the Rankin $L$-series  $L(f\otimes g, s)$
 attached to the convolution of  weight $2$ newforms $f$ and $g$
 on $\Gamma_1(N)$, evaluated  at the {\em near-central point} $s=2$;
      \item  the image under the complex regulator of
     certain explicit elements in the  motivic cohomology group
   $H^3_{\cM}(X_1(N)^2,\Q(2))$, or, equivalently, in the higher
   Chow group $\CH^2(X_1(N)^2,1)\otimes \Q$.
   These elements, whose definition is recalled in Section \ref{sec:global-elements},
     are constructed from
    modular units and  are referred to in the sequel as
   {\em Beilinson-Flach elements}.
   \end{enumerate}
In the $p$-adic setting, the complex $L$-series $L(f\otimes g,s)$ is replaced
by Hida's $p$-adic Rankin $L$-series   attached to two ordinary
families of modular forms interpolating $f$ and $g$, whose definition
 is briefly recalled in Section
\ref{sec:padic-L}. The  role of the  complex regulator is played
by the $p$-adic syntomic regulator on  $K_1$ of a surface.
Besser's   description of it in terms of Coleman integration
\cite{besser-K1S}, which is summarised in \S \ref{sec:besser}, is
a key ingredient in the proof of Theorem \ref{thm:main}.

    Our approach also
    relies crucially on techniques developed in  \cite{DR}
    for relating $p$-adic Abel-Jacobi
 images of diagonal cycles to values of the Garrett-Rankin
 triple product $p$-adic $L$-function attached to a triple
 $(\hf,\hg,\hh)$ of Hida families of cusp forms.
 Corollary \ref{cor:main} deals with the setting where  the cuspidal family
$\hh$ in the triple $(\hf,\hg,\hh)$ is replaced by a Hida family
of Eisenstein series. The reader will
   also note the close parallel between Theorem \ref{thm:main} and the main
result of
  \cite{BD1}, in which the $p$-adic regulators of certain elements
in $K_2(X_1(N))$ are related to the value at $s=2$ of the Mazur-Swinnerton-Dyer
$p$-adic $L$-functions attached to weight two cusp forms. The  results of the present article
are in fact  intermediate between those of
\cite{DR} and \cite{BD1},  the latter treating the case where {\em both}  $\hg$ and $\hh$ are
replaced by Hida
families of Eisenstein series---a setting in which the resulting $p$-adic Rankin $L$-function
factors as a product of two
Mazur-Kitagawa  $L$-functions attached to $\hf$.


We also remark that a  function field analogue  of Beilinson's Theorem
   involving Drinfeld modular curves is described in
  \cite{sreekantan-K1}, based on a description of non-archimedean
  regulators  given in \cite{sreekantan-regulators}. See also the related work of
  Ambrus P\`al in the setting of the $K_2$ of Mumford curves \cite{pal}.



Let us conclude this introduction by briefly discussing some
eventual  arithmetical applications of the main result of this paper.

\medskip\noindent
{\bf I.  The Euler system of Beilinson-Flach elements}.
The image of   Beilinson-Flach elements under the $p$-adic \'etale
regulator map gives rise to   classes in the global cohomology group
$H^1(\Q,V_{f}\otimes V_{g}(2))$, where $V_{f}$ and $V_{g}$
are the $p$-adic Galois representations attached to $f$ and $g$,
respectively. The work in preparation \cite{BDR2} explores the
theme of the $p$-adic variation of the Beilinson-Flach classes
attached to Hida families of cusp forms $\hf$ and $\hg$. In
particular, when  $\hg$ specialises in weight one to a classical cusp form attached
to an odd irreducible Artin representation $\rho$,  and $\hf$ specialises  in
weight two to the cusp form associated with an elliptic curve $E$
over $\Q$, we expect the associated cohomology class to
yield    new cases of the Birch and Swinnerton-Dyer
conjecture for the complex $L$-series $L(E,\rho,s)$, proving in particular
that $\rho$ does not
occur in the representation $E(\bar\Q)\otimes\C$ when $L(E,\rho,1)\ne 0$.

The idea of using Beilinson elements in Euler system arguments
occurs much earlier in the work of   Flach \cite{Fl},
who used them  to construct classes in $H^1(\Q,{\rm
  Sym}^2(E)(2))$ which are cristalline at $p$ but  ramified at a single prime $\ell\not=p$.
Applying  Kolyvagin's method to these classes leads to the finiteness
of the  Shafarevich-Tate group of ${\rm
Sym}^2(E)(2)$ and an explicit annihilator of this group related to the
special value $L({\rm Sym}^2(E),2)$, which is critical in the sense of Deligne, unlike
the special values $L(f\otimes g,2)$ when $f$ and $g$ are distinct normalised newforms.


\medskip\noindent
{\bf II.  Hida's $L$-function for   the symmetric square of a modular
form}.  Theorem \ref{thm:main},
specialised to   the case $f=g$, is exploited
by S.\ Dasgupta   \cite{dasgupta} to study the
Hida $L$-function $L(f\otimes f,s)$ and express it as the product
of the  Coates-Schmidt
$p$-adic $L$-function attached to ${\rm Sym}^2(f)$ and a Kubota-Leopoldt
$p$-adic $L$-function. This factorisation, which can be viewed as
another  manifestation of the Artin formalism for $p$-adic $L$-series, is analogous to a formula
of Gross \cite{gross} expressing the restriction to the cyclotomic line  of the Katz two-variable
$p$-adic $L$-function attached to an imaginary quadratic field as a product of
two Kubota-Leopoldt  $L$-functions. The Beilinson-Flach elements play the same role in Dasgupta's proof as elliptic units  in the work of Gross.



\medskip\noindent
{\bf III.  $p$-adic L-functions and Euler systems over $\Z_p^2$-extensions}.
The paper in preparation \cite{llz} of A.\ Lei,
 D.\ Loeffler and S.L.\ Zerbes builds on the
methods of this paper, in the setting  where $g$ varies over a collection
of theta series attached to
  Hecke
characters of an imaginary quadratic field $K$,
to construct an Euler system for  $V_f$ over the various layers
of the {\em two variable } $\Z_p$-extension $K_\infty$ of $K$,
thus supplying the global input for their extension
\cite{loeffler-zerbes}
of Perrin-Riou's machinery  in which the cyclotomic
$\Z_p$-extension of $\Q$ is replaced by $K_\infty$.


  \medskip\noindent
 {\em Acknowledgements}: The authors are grateful to Amnon Besser, Fran\c cois
 Brunault, Samit Dasgupta, Antonio Lei, David Loeffler, Nicola Mazzari
  and Sarah Zerbes for their useful comments and stimulating  conversations
   about their research in connection to this work. The third author was financially supported by  MTM20121-34611 during the ellaboration of this work.



\section{Rankin $L$-series}
Let $f$ and $g$ be normalised newforms of
 weights $k,\ell$, levels $N_f, N_g$,  and nebentypus characters $\chi_f, \chi_g$ respectively.
 The $p$-adic representations  $V_f$ and $V_g$ are part of a  compatible system  of
  representations which we continue to denote by the same symbol.
  Let
  $$ L(V_f\otimes V_g,s) = \prod_p \det( (1-\sigma_p p^{-s}){|(V_f\otimes V_g)^{I_p}})^{-1} $$
be   the  motivic $L$-function attached to the tensor product
  $V_f\otimes V_g$,
  where $I_p$ denotes the inertia subgroup of a decomposition
   group   at $p$, and $\sigma_p$  a
  corresponding geometric Frobenius element.


 The goal of this first chapter is to briefly recall the basic analytic properties of this
 $L$-series, describe Hida's construction of a $p$-adic avatar, and---
  in the special case where
  $f$ and $g$ are both of weight two---present
   parallel formulae for their special
 values at the near-central point $s=2$, which
is  {\em not} critical in the sense of Deligne.



\subsection{Complex $L$-series}\label{sec:Lseries}

Set $N := \lcm(N_f,N_g)$ and replace $\chi_f$ and $\chi_g$ by
their counterparts of modulus $N$ sending any prime $r|N$ to $0$.
It is also convenient to replace  $f$, as well as $g$, by a
normalised eigenform  of level $N$ which
\begin{enumerate}
\item
has the same eigenvalues for   the  good
  Hecke operators   $T_r$ with $\gcd(r,N)=1$;
  \item
  is  also an eigenvector  for the Hecke operators   $U_r$
  attached to the primes $r$ dividing $N$.
  \end{enumerate}
This substitution having been made, let
\begin{equation}\label{fourier}
 f(z) = \sum_{n=1}^\infty a_n(f) e^{2\pi i n z}, \qquad
  g(z) = \sum_{n=1}^\infty a_n(g) e^{2\pi i nz}
\end{equation}
  be the Fourier expansions of $f$ and $g$,
  let
  $K_f$ and $K_g\subset \bar\Q$ denote the subfields generated by the coefficients $a_n(f)$
  and $a_n(g)$ respectively, and let $K_{fg}$ denote the compositum of the two fields.
   The
  Hecke polynomials attached to $f$  can be factored as
  $$x^2-a_p(f) x+ \chi_f(p) p^{k-1} = (x-\alpha_p(f))(x-\beta_p(f)),$$
   where
    $(\alpha_p(f),\beta_p(f))=(a_p(f),0)$ when $p|N$. Similar notations are adopted
     for $g$.
  The Rankin $L$-function attached to the pair $(f,g)$ is defined by the formula
  \begin{eqnarray*}
   L(f\otimes g,s) &:=&  \prod_p L_{(p)} (f\otimes g,s), \qquad \mbox{ where } \\
  L_{(p)}(f\otimes g,s) &:= &   (1-\alpha_p(f) \alpha_p(g) p^{-s})^{-1}
   (1-\alpha_p(f) \beta_p(g) p^{-s})^{-1}\\
   && \times (1-\beta_p(f) \alpha_p(g) p^{-s})^{-1}
     (1-\beta_p(f) \beta_p(g) p^{-s})^{-1}.
     \end{eqnarray*}
 The Euler factors at $p$  defining  $L(V_f\otimes V_g,s)$ and $L(f\otimes g,s)$ agree for all $p\nmid N$,
 and hence the special values of $L(V_f\otimes V_g,s)$ and $L(f\otimes g,s)$ at integer points
 differ by elementary quantities in  $K_{fg}^\times$.
 It will be more convenient, for the sequel, to focus attention on $L(f\otimes g,s)$.
 Assume without loss of generality that the forms $f$ and $g$ have been ordered in such a way that
 $k\ge \ell$.



 \subsubsection{Rankin's method}
 We begin by recalling the general formula for $L(f\otimes g,s)$  coming out
 of Rankin's method, involving the  {\em non-holomorphic Eisenstein series}
 \begin{equation}
\label{eqn:def-Eisenstein}
\widetilde E_{k-\ell,\chi}(z,s) = \sumprime
\frac{\chi^{-1}(n)}{ (mz+n)^{k-\ell}} \cdot \frac{y^s}{|mz+n|^{2s}}
\end{equation}
 of weight $k-\ell$, level $N$
 and character
$$\chi := \chi_f^{-1} \chi_g^{-1},$$
where the superscript $'$  in \eqref{eqn:def-Eisenstein}
indicates that the sum is taken over the
non-zero lattice vectors $(m,n)\in N\Z\times\Z$.
For fixed complex $s$ with  $\mbox{Re}(s)>\!\!>0$, the product
$\widetilde E_{k-\ell,\chi}(z,s)\times g(z)$ is a real-analytic
$\C$-valued function on the Poincar\'e upper half-plane $\cH$ which transforms like a
 modular form of weight $k$, level $N$ and
character $\chi_f^{-1}$ and is of rapid decay at infinity. The space of such functions,
denoted  $S^{\ra}_k(N,\chi_f^{-1})$,   is
equipped with the  Petersson scalar product
   $$ \langle \ , \ \rangle_{k,N}: S_k^{\ra}(N,\chi_f^{-1})
 \times S_k^{\ra}(N,\chi_f^{-1})\lra \C$$
    given
  by the formula
 \begin{equation}
 \label{eqn:petersson}
 \left\langle f_1,f_2\right\rangle_{k,N} :=
 \int_{\Gamma_0(N)\backslash \cH} y^k \overline{f_1(z)} f_2(z) \frac{dxdy}{y^2},
 \end{equation}
 which is hermitian linear in the first argument and $\C$-linear in the second.
 Let $f^*\in S_k(N,\chi_f^{-1})$ denote the modular form obtained from $f$ by applying complex conjugation
 to its Fourier coefficients.
\begin{proposition}[Shimura]
\label{prop:rankin-primo}
 For all $s\in \C$ with $\Re(s)>\!\!>0$,
  \begin{equation}
  \label{eqn:rankin}
  L(f\otimes g,s)=\frac{1}{2}
  \frac{(4\pi)^s}{\Gamma(s)}   \left\langle  f^*(z), \widetilde E_{k-\ell,\chi}(z,s-k+1)\cdot g(z) \right\rangle_{k,N}.
 \end{equation}
\end{proposition}
This well-known formula for the Rankin $L$-series is taken from equation (14) of \cite{BD1}.



 \subsubsection{Critical values}\label{subsec:critical}

Assume here and in \S \ref{subsec:Deligne} that $\ell <k$.
 The functional equation for $L(f\otimes g,s)$ arising from
 Proposition \ref{prop:rankin-primo} reveals that the integer
 $j$ is critical for
 $L(f\otimes g, s)$ if any only if it lies in the closed interval $[\ell,k-1]$.
 We now describe a further closed formula for the value at an integer $j$
 belonging to the ``right half critical segment" $[\frac{\ell+k-1}{2},k-1]$, which will be
 useful
 in deriving the algebraicity (up to periods) of $L(f\otimes g,j)$ predicted by the
 Deligne conjectures, and ultimately in constructing Hida's $p$-adic Rankin
 $L$-function  by  interpolating these quantities $p$-adically.

Having fixed an integer $j\in [\frac{\ell+k-1}{2},k-1]$, let
  $t \ge 0$ and  $m \ge 1$ be  given by
$$ t := k-1-j, \qquad m:= k-\ell-2t.$$

If $m\leq 2$, let us assume also that $\chi$ is nontrivial. Then the series
\begin{equation}
 \label{eqn:qexpek}
 E_{m,\chi}(z)=
 2^{-1}L(\chi,1-m)+
 \sum_{n=1}^\infty\sigma_{m-1,\chi}(n)q^n,\qquad
 \sigma_{m-1,\chi}(n)=\sum_{d|n}\chi(d)d^{m-1}
 \end{equation}
is the $q$-expansion of a {\em holomorphic} Eisenstein series of weight $m$ and character $\chi$.


The Shimura-Maass derivative operator
$$ \delta_m := \frac{1}{2\pi i}
\left( \frac{d}{dz}  + \frac{i m}{2y}\right) $$ transforms modular
forms of weight $m$ into (real analytic) modular forms of weight
$m+2$ which are {\em nearly holomorphic}
    in the sense of \cite{shimura}, and its $t$-fold iterate
  $\delta_m^t := \delta_{m+2t-2} \cdots \delta_{m+2} \delta_m$
  maps the space $M_m(N,\chi)$ to the space
  $M_{m+2t}^{\nh}(N,\chi)$ of nearly holomorphic modular forms of weight
  $m+2t$.
   Let
  $$ C(k,\ell,j) := \frac{ (-1)^t 2^{k-1}(2\pi)^{k+m-1}
  \iota_\chi (iN)^{-m}\tau(\chi^{-1}) }{(m+t-1)!(j-1)!}$$
  be the elementary constant (in which $\iota_\chi=1$ when $\chi$ is primitive)
  appearing in equation (18) of \cite{BD1}.
The following formula for $L(f\otimes g, j)$,
 is obtained by
setting $c=j$ in equation (18) of loc.\ cit.  (See also
Theorem 2 of \cite{shimura76}.)
  \begin{proposition}
  \label{prop:rankin-bis}
The special
   value $L(f\otimes g, j)$ is given by the formula
   \begin{equation}
  L(f\otimes g,j)= C(k,\ell,j)
  \left \langle f^*(z), \delta_m^t E_{m,\chi}(z)\times g(z) \right\rangle_{k,N}.
 \end{equation}
 \end{proposition}

\subsubsection{Algebraicity  and Deligne's conjecture}\label{subsec:Deligne}
 Let
 $S_k^{\rm nh}(N,\chi_f^{-1}; K_{fg}) \subset S_k^{\rm nh}(N,\chi_f^{-1})$
 denote the  space of
  nearly-holomorphic cusp forms which are {\em
 defined over $K_{fg}$} in the sense of Shimura (cf.\
 Section 2.4 of \cite{DR}).
 The cusp form
 \begin{equation}
 \label{eqn:xikell}
 \Xi(f,g,j):=\delta_m^t
 E_{m,\chi}\times g \in S_k^{\nh}(N,\chi_f^{-1})
 \end{equation}
 which appears in Proposition \ref{prop:rankin-bis}
 belongs to the $K_{fg}$-rational structure
 $S_k^{\nh}(N,\chi_f^{-1};K_{fg})$.   Hence, its image
 \begin{equation}
 \label{eqn:xikellhol}
 \Xi(f,g,j)^{\rm hol}:=\Pi_{N}^{\rm
 hol}(\Xi(f,g,j))
 \end{equation}
 under the holomorphic
 projection $\Pi_{N}^{\rm hol}$ of loc.\ cit.\ belongs to the space
 $S_k(N,\chi_f^{-1};K_{fg})$ of holomorphic cusp forms with
 Fourier coefficients in $K_{fg}$.  In particular,  the  ratio
  \begin{equation}
 \label{eqn:projection}
L^{\alg}(f\otimes g,j) :=
C(f,g,j)^{-1} \frac{L(f\otimes g, j)}{\langle f^*,f^*\rangle_{k,N}}
 =  \frac{\left\langle f^*, \Xi(f,g,j)^{\phantom{x}}  \right\rangle_{k,N}}
 {\langle f^*,f^*\rangle_{k,N}}=
 \frac{\left\langle f^*,  \Xi(f,g,j)^{\rm hol}
 \right\rangle_{k,N}}
 {\langle f^*,f^*\rangle_{k,N}},
 \end{equation}
   belongs to $K_{fg}$.
 This algebraicity result is consistent with Deligne's conjecture
 which predicts that the period $C(f,g,j) \langle f^*,f^*\rangle_{k,N}$ is the
 `transcendental part" of the special value $L(f\otimes g,j)$.
  The associated ``algebraic part"
  appearing in \eqref{eqn:projection}  will later be interpolated
   $p$-adically to obtain Hida's $p$-adic Rankin $L$-function attached to $f$ and $g$.

In order to do this, it will be convenient to  give a more geometric description of the
 quantity $L^{\alg}(f\otimes g,j) $ appearing in
 \eqref{eqn:projection},
  in terms of the Poincar\'e duality on the
 de Rham cohomology of the   modular curve $X_1(N)$ with values in
 appropriate sheaves with connection, as described in \cite[\S\,2.3]{DR}.
 To lighten the notations, denote by  $Y$ and by $X$   the open modular curve $Y_1(N)$
 and the complete modular curve $X_1(N)$ respectively, classifying (generalised) elliptic curves
equipped with an embedding of the finite flat group scheme $\mu_N$ of $N$-th roots of unity.
 
 
 Let $K$ be any
 field containing $K_{fg}$. Denote by $\cE\lra Y$ the universal elliptic curve
 over $Y$, and by $\omega$ the
 sheaf of relative differentials on $\cE$ over $Y$,
 extended to $X$ as in \cite[\S\,1.1]{BDP1}.
 Recall the Kodaira-Spencer isomorphism
 $\omega^2\simeq\Omega^1_X(\log{\rm cusps})$, where
 $\Omega^1_X(\log{\rm cusps})$ is
 the sheaf of regular differentials on $Y$ with log poles at
 the cusps.

A modular form $\phi$  on $\Gamma_1(N)$ of weight $k=r+2$  with Fourier
 coefficients in $K$  corresponds to a global section of the sheaf
 $\omega^{r+2}=\omega^r\otimes\Omega_X^1(\log{\rm cusps}) $ over the
  base-change $X_K$ of $X$ to $K$.  The sheaf $\omega^r$ can be viewed as a subsheaf of
 $\cL_r:= \Sym^r \cL$,
 where
 $$ \cL:= R^1 \pi_\ast (\cE\lra Y)$$
 is the relative de Rham cohomology sheaf on $Y$,  extended
 to $X$ as in loc.\ cit., equipped
 with the filtration
 \begin{equation}
 \label{eqn:hodge-fil-fibers} 0 \lra \omega \lra \cL \lra
 \omega^{-1} \lra 0
 \end{equation}
 arising from the Hodge filtration on the  fibers. The sheaf $\cL_r$
 is a coherent sheaf over $X$ of rank $r\!+\!1$,
 endowed with the Gauss-Manin connection
 $$ \nabla: \cL_r \lra \cL_r\otimes \Omega^1_X(\log{\rm cusps}). $$
 Let $H^1_{\dR}(X_K,\cL_r,\nabla)$  be the de Rham cohomology of
  $\cL_r$. It is equipped with the  perfect Poincar\'e pairing
 \begin{equation}
 \label{eqn:pd}
 \left\langle \ , \  \right\rangle_{k,X}:
 H^1_{\dR}(X_K,\cL_r,\nabla) \times H^1_{\dR}(X_K,\cL_r,\nabla)
 \lra K,
 \end{equation}
 which is compatible with the  exact sequence
 \begin{equation}
\label{eqn:hodge-filtration-k}
 0 \lra H^0(X_K,\omega^r\otimes \Omega^1_X)
 \lra  H ^1_{\dR}(X_K,\cL_r,\nabla) \lra
 H^1(X_K,\omega^{-r} ) \lra 0,
 \end{equation}
 in the sense that $H^0(X_K,\omega^r\otimes \Omega^1_X)$ is an
 isotropic subspace.
   (Cf.\ Sections 2 and 3
 of \cite{coleman-bu}, for a more detailed account.)
 In particular, Poincar\'e duality  induces a perfect pairing
\begin{equation}
\label{eqn:pp-descended}
\left \langle \ ,\ \right \rangle_{k,X}: H^1(X_K,\omega^{-r} )
 \times H^0(X_K,\omega^r\otimes \Omega^1_X) \lra K,
 \end{equation}
 which is denoted by the same symbol by a slight abuse of notation.

Set $\omega_f = f(z)dz$ and $\overline{\omega}_f = \bar f^*(z)d\bar z$.
 The antiholomorphic differential $\eta_f^{\ah}$
  defined by
 \begin{equation}
 \label{eqn:etafah}
 \eta_f^{\ah}:=\frac{\overline{\omega}_f}{\langle \overline{\omega}_f,\omega_f\rangle_{k,X}}\ .
 \end{equation}
  gives
 rise to a class
 in $H ^1_{\dR }(X_\C,\cL_r,\nabla)$, whose image $\eta_f$
 in $H^1(X_\C,\omega^{-r} )$ belongs to
 $H^1(X_K,\omega^{-r} )$ (cf.\ Corollary 2.13 of
 \cite{DR}).  The
 following expression for the algebraic part $L^{\alg}(f\otimes g, j)$
 in terms of the class $\eta_f$
 follows directly from  \eqref{eqn:projection}
 in light of the discussion above:
  \begin{proposition}
 \label{prop:rankin-ter}
 The  algebraic part $L^{\alg}(f\otimes g, j)$
is equal to
 \begin{equation}
 \label{eqn:projection2}
 L^{\alg}(f\otimes g, j) =
\left \langle \eta_f,  \Xi(f,g,j)^{\rm
 hol}  \right\rangle_{k,X}.
 \end{equation}
 \end{proposition}


 \subsubsection{The value at the near central point}
  \label{subsec:near}
Consider now  the case where
  $k=\ell=2$ and $\chi_f\ne \chi_g^{-1}$, so that the character
   $\chi = \chi_f^{-1} \chi_g^{-1}$ is not the trivial one.
    The functional equation for $L(f\otimes g,s)$ relates $L(f\otimes g,s)$ to
  $L(f^*\otimes g^*,3-s)$ and this $L$-series has no critical points
   in the sense of Deligne.
Proposition
\ref{prop:rankin-quater} below
describes its value at the near-central point $s=2$ in terms of logarithms of
{\em modular units}.

Enlarge $K$ so that it contains the field which is cut out by all the
 Dirichlet characters of modulus $N$,
 and let $F$ be the field generated over $K$ by the values of these characters.
 Let $\Eis_\ell(\Gamma_1(N);F)$ denote the
 subspace of $M_\ell(\Gamma_1(N);F)$ spanned by the weight $\ell$
 Eisenstein series with coefficients in $F$.
  The logarithmic
 derivative gives a surjective homomorphism
 \begin{equation}
 \label{eqn:dlog}
 \xymatrix{
 \cO(Y_K)^\times\otimes F\ \ar[r]^{\dlog} &
 \Eis_2(\Gamma_1(N); F),}
 \end{equation}
whose kernel is the subspace $K^\times \otimes F$ spanned by the nonzero constant functions.

\begin{definition}\label{def-uchi}
Let $u_{\chi}$ be the modular
 unit satisfying
 \begin{equation}
 \label{eqn:uchi}
 \dlog(u_\chi)=E_{2,\chi}
 \end{equation}
whose value at $\infty$ is $1$ in the sense of \cite[\S 5]{brunault}.
\end{definition}


\begin{proposition}
 \label{prop:rankin-quater}
Given weight two eigenforms $f$ and $g$  as above,
 \begin{equation}
  \label{eqn:rankin2quater}
  L(f\otimes g,2)=
  16 \pi^3 N^{-2} \tau(\chi^{-1})  \left\langle f^*(z),
 \log|u_{\chi}(z)|
  \cdot g(z) \right\rangle_{2,N}.
 \end{equation}
 \end{proposition}
 \begin{proof}
 By Proposition \ref{prop:rankin-primo},
 \begin{equation}
  \label{eqn:rankin2bis}
  L(f\otimes g,2)=\frac{1}{2}
  (4\pi)^2  \left\langle f^*(z),
  \widetilde E_{0,\chi}(z,1)\cdot g (z) \right\rangle_{2,N}.
 \end{equation}
 A direct calculation (cf. equation (26) of \cite{BD1}) shows that
 \begin{equation}
 \label{eqn:shimura-maassbis}
\frac{1}{2\pi i}\frac{d}{dz}\widetilde  E_{0,\chi}(z,1) =
 -\frac{1}{4\pi}
 \widetilde E_{2,\chi}(z)=
 2\pi N^{-2}\tau(\chi^{-1}) E_{2,\chi}(z).
 \end{equation}
Having normalized $u_\chi$ as in Definition \ref{def-uchi}, one obtains the equality
 \begin{equation}
 \label{eqn:shimura-maasster}
 \widetilde  E_{0,\chi}(z,1) =
 2\pi N^{-2}\tau(\chi^{-1}) \log|u_{\chi}(z)|,
 \end{equation}
 which is compatible with \eqref{eqn:uchi}.
Combining \eqref{eqn:rankin2bis} with
 \eqref{eqn:shimura-maasster}
 completes the proof of the proposition.
 \end{proof}


\subsection{$p$-adic $L$-series}
\label{sec:padic-L}
Let $p\geq 3$ be a prime, and fix an embedding of $K$ into $\C_p$.
 This section recalls the definition of  the Rankin $p$-adic $L$-function associated
by Hida  \cite{hida} to the convolution of two Hida families of cusp forms.
 For the sake of brevity, we proceed here---just as in \cite{BD1}--- by specialising
  the approach and notations of \cite{DR},
 which constructs the $p$-adic $L$-function associated to
 a triple product of  three Hida families $(\hf,\hg,\hh)$ of cusp forms.
  The setting considered here
 consists, essentially, in  letting $\hh$ be a Hida family of Eisenstein series.

 \subsubsection{Ordinary projections}
 \label{subsec:ordinary}
 Let $f$, $g$ be eigenforms of level $N$,
weights $k>\ell$ and nebentypus
 $\chi_f$, $\chi_g$ as in \eqref{fourier}.
  Let also $j\in [\frac{\ell+k-1}{2},k-1]$ be an integer and set
    $t = k-1-j\geq 0$ and $m=k-\ell-2t\geq 1$ as in \S \ref{subsec:critical}.
     The following ordinariness assumption is important for the constructions
described in this section.
 \begin{assumption}
 \label{ass:ordinariness}
 The cuspidal eigenforms $f$ and $g$ are ordinary at $p$, and $p\nmid N$.
 \end{assumption}
 Under this assumption, the $f$-isotypic part of the exact sequence
 \eqref{eqn:hodge-filtration-k} with $K=\C_p$
 admits a canonical {\em unit root} splitting, arising
 from the action of Frobenius on   de Rham cohomology.
 Let $\eta_f^{\rm ur}$ be the lift of $\eta_f$ to the unit
 root subspace $H ^1_{\dR}(X_{\C_p},\cL_r,\nabla)^{f,{\rm ur}}$.
 The right-hand side of \eqref{eqn:projection2}
 is then equal to
 \begin{equation}
 \label{eqn:projection3}
 \left\langle \eta_f,  \Xi(f,g,j)^{\rm
 hol} \right \rangle_{k,X}=
\left \langle \eta_f^{\ur},  \Xi(f,g,j)^{\rm
 hol} \right \rangle_{k,X}.
 \end{equation}
Now let
   $e_{\ord}$ be Hida's ordinary projector to
 $H ^1_{\dR}(Y_K,\cL_r,\nabla)^{\ord}$.
 By Proposition 2.11 of \cite{DR},
 the right-hand side of \eqref{eqn:projection3}
 can be re-written, after viewing $\Xi(f,g,j)^{\rm
 hol} $ as an overconvergent $p$-adic
 modular form and setting $\Xi(f,g,j)^{\ord} :=e_{\ord}\Xi(f,g,j)^{\rm
 hol}$, as
 \begin{equation}
 \label{eqn:projection4}
 \left\langle \eta_f^{\ur},  \Xi(f,g,j)^{\rm
 hol} \right\rangle_{k,X}=
\left  \langle \eta_f^{\ur}, \Xi(f,g,j)^{\ord} \right\rangle_{k,X}.
 \end{equation}
 By Proposition 2.8 of \cite{DR},
 \begin{equation}
 \label{eqn:projection5}
 \Xi(f,g,j)^{\ord}  =
 e_{\ord}(d^{t}
 E_{m,\chi}\cdot g),
 \end{equation}
 where $d=q\frac{d}{dq}$ is Serre's derivative
 operator on $p$-adic modular forms.



 Given a $p$-adic modular form $\phi=\sum c_n q^n$,
 let $\phi^{[p]}:=\sum_{p\nmid n} c_n q^n$
 denote its ``$p$-depletion", and set
  \begin{equation}
 \label{eqn:projectionp5}
 \Xi(f,g,j)^{\ord,p}  :=
 e_{\ord}(d^{t}
 E_{m,\chi}^{[p]}\cdot g).
 \end{equation}



 \begin{proposition}
 \label{prop:pdepletion}
 Let
  $e_{f^\ast}$ be the projector to the $f^\ast$-isotypic subspace
  of $H ^1_{\dR}(Y_K,\cL_{k-2},\nabla)$.
 Then
 $$e_{f^\ast}\Xi(f,g,j)^{\ord,p} =
 \frac{\cE(f,g,j)}{\cE(f)}\cdot
 e_{f^\ast}\Xi(f,g,j)^{\ord},
 $$
 where
 \begin{eqnarray*}
 \cE(f,g,j)&=&
 (1-\beta_p(f)\alpha_p(g)  p^{t-k+1})
 (1-\beta_p(f) \beta_p(g)  p^{t-k + 1})\\
 && \times (1-\beta_p(f) \alpha_p(g) \chi (p)p^{t-k+m})
 (1-\beta_p(f)\beta_p(g) \chi(p)p^{t-k+m}),\\
 \cE(f)&=&1-\beta_p(f)^2\chi_f^{-1}(p) p^{-k}.
 \end{eqnarray*}
 \end{proposition}
 \begin{proof}
 This follows from Corollary 4.17 of \cite{DR}, in
 light of Proposition 2.8 of loc.\ cit.
 \end{proof}

\subsubsection{Hida's $p$-adic L-series}\label{subsec:Hidap}


 Let $\hf$ and $\hg$  be Hida families of ordinary $p$-adic modular forms of tame level $N$,
 indexed by   weight variables $k$  and $\ell$ in suitable neighborhoods
 $U_{\hf}$ and $U_{\hg}$  of $\Z/(p-1)\Z\times\Z_p$,  contained
 in a single residue class modulo $p-1$. (These families may be obtained,
  as shall be the case considered in \S \ref{sec:main-padic},
   by deforming two given ordinary classical eigenforms $f$ and $g$  of possibly equal weights.)
   Assume likewise that the parameter $j=k-1-t$ belongs to a
   single residue class modulo $p-1$, so that the same holds true for
   the weight $m=k-\ell-2t$ of the Eisenstein series $E_{m,\chi}$.

 For $k\in U_{\hf}\cap
 \Z^{\geq 2}$ and $\ell\in U_{\hg}\cap \Z^{\geq 2}$, let
 $$ f_k\in  S_k(N,\chi_f), \qquad g_\ell \in S_\ell(N,\chi_g)$$
   be the classical cusp forms whose
 $p$-stabilisations  are the weight $k$ and $\ell$ specialisations
  of $\hf$ and $\hg$ respectively. (We denote by $\chi_f$, resp.\
  $\chi_g$ the common character of the modular forms $f_k$, resp.\ $g_k$.)

 The collection of $p$-adic modular forms $\Xi(f_k,g_\ell,j)^{\ord,p}$
 (defined as in equation \eqref{eqn:projectionp5}) indexed by
 \begin{equation}
 \label{eqn:good-triples}
  \{  (k,\ell, j), \qquad k\in U_{\hf}\cap \Z^{\ge 2}, \quad
 \ell \in U_{\hg}\cap \Z^{\ge 2}, \quad  \frac{\ell+ k-1}{2} \le j \le k-1 \}
 \end{equation}
  has Fourier coefficients which
 extend analytically to $U_{\hf}\times U_{\hg} \times  \Z_p$,
 as functions in
 $k$, $\ell$ and $j$. Hence, they can be viewed as a
 (three-variable) $\Lambda$-adic family of modular forms of level $N$ in the sense of
 \cite[\S 2.7]{DR}.

 Set
 $$\cE^\ast(f_k):=1-\beta_p(f_k)^2\chi_f^{-1}(p) p^{1-k}.$$
 Proposition 4.10 of loc.\ cit.\ shows that
 the expression
 \begin{equation}
 \label{eqn:lpfE}
 L_p(\hf,\hg)(k,\ell,j):=\frac{1}{\cE^\ast(f_k)}
\left \langle \eta_{f_k}^{\ur},
 \Xi(f_k,g_\ell,j)^{\ord,p}
 \right\rangle_{k,X},
 \end{equation}
 defined on the triples $(k,\ell,j)$  in the set in
 \eqref{eqn:good-triples}
 extends to an analytic function $L_p(\hf,\hg)$ on
 $U_{\hf}\times U_{\hg}  \times\Z_p$, which we refer to as the Hida
 {\em $p$-adic Rankin $L$-function} attached to $\hf$ and
 $\hg$.
 This appelation is justified by noting that, for all triples $(k,\ell,j)$ in 
 the range of ``classical interpolation", i.e, belonging to 
 \eqref{eqn:good-triples},
 the function $L_p(\hf,\hg)(k,\ell,j)$ satisfies the interpolation 
 property
 $$ L_p(\hf,\hg)(k,\ell,j) = \frac{\cE(f_k,g_\ell,j)}{\cE^\ast(f_k) \cE(f_k)}  L^{\alg}(f_k\otimes g_\ell,j).$$
 This follows from a direct calculation combining  \eqref{eqn:lpfE}, Proposition \ref{prop:pdepletion},
 \eqref{eqn:projection4},
 \eqref{eqn:projection3}
and \eqref{eqn:projection2}.

 
 


  Note that the point
  $(2,2,2)$ lies outside the region of classical interpolation for this
  function. (In fact, there are no critical values for the pair of weights
  $(2,2)$.)
 Corollary \ref{cor:main} of Section \ref{sec:main-padic}
 relates the value of $L_p(\hf,\hg)$ at $(2,2,2)$ to the $p$-adic
 regulator attached in Section \ref{sec:besser} to the triple of
 modular forms $(f=f_2,g=g_2,E_{2,\chi})$.

 Generalising our setting somewhat, we do not assume now that
  $g\in S_2(N,\chi_g)$ is ordinary, so that $g$ may not necessarily
  be viewed as the weight $2$ specialisation of a Hida family.
 In this case, the above
 construction still allows us to define a two-variable $p$-adic $L$-function
 $L_p(\hf,g)(k,j)$ on $U_{\hf}\times\Z_p$, by the equation
 \begin{equation}
 \label{eqn:lpfEbis}
 L_p(\hf,g)(k,j):=\frac{1}{\cE^\ast(f_k)}
 \left \langle \eta_{f_k}^{\ur},
 \Xi(f_k,g,j)^{\ord,p}
 \right\rangle_{k,X},
 \end{equation}
 for $k\in U_{\hf}\cap \Z^{\ge 2}$ and $(k+1)/2 \le j \le k-1$.
 Theorem \ref{thm:main} relates
 $L_p(\hf,g)(2,2)$ to the $p$-adic regulator attached to
 $(f,g,E_{2,\chi})$.





\section{Beilinson-Flach elements}


\subsection{Definition and basic properties}
\label{sec:global-elements}
Let $S$ be a quasi-projective variety over a field $K$,
 and $K_j(S)$ denote Quillen's algebraic $K$-groups of $S$.
The motivic cohomology groups $H^i_{\cM}(S,\Q(n)) =
K_{2n-i}^{(n)}(S)$ of $S$ were defined
 by Beilinson \cite[\S 2]{beilinson-K1S} as the $n$-th graded piece
  of the Adams filtration on $K_{2n-i}(S)\otimes \Q$.
   In parallel with Beilinson's motivic cohomology groups,
    Bloch \cite{bloch86} introduced the higher Chow groups $\CH^i(S,n)$ of $S$.

In this note we shall focus on the smooth projective surface
 $S:= X\times X$, where $X$ is the modular curve
  over the field $K$ of \S \ref{subsec:near}. For $i=3$ and
   $n=2$, $H^3_{\cM}(S,\Q(2))=K_1^{(2)}(S)$ is identified with
    $\CH^2(S,1)\otimes \Q$. The higher Chow group $\CH^2(S,1)$
     may be explicitly described (cf.\,also \cite{Scholl}) as the first homology of the
{\em Gersten complex}
\begin{equation}
\label{eqn:gersten}
\xymatrix{
K_2(K(S))  \ar[r]^{\partial} & \bigoplus_{Z\subset S} K(Z)^\times
 \ar[r]^{{\rm div}} & \bigoplus_{P\in S} \Z,  }
\end{equation}
where
\begin{enumerate}
\item $K_2(K(S))$ denotes the second Milnor $K$-group of the
 rational function field $K(S)$, and
 $\partial$ is the map whose ``component at $Z$" is the
   tame symbol attached to the valuation
$\ord_Z$;
\item the group
 $$\Theta:=\bigoplus_{Z\subset S} K(Z)^\times$$
 is the set of finite formal linear combinations
$ \sum_i (Z_i,u_i)$, where the $Z_i$ are irreducible curves
 in $S$ and $u_i$ is a rational function on $Z_i$;
\item  the map $\rm div$ is the divisor map and the direct
 sum defining its target is taken over all closed points
$P\in S_K$.
\end{enumerate}


Given a closed point $P\in X$ and a rational function $u$ on $X$,
 an element of $\Theta$ of the form $(\{P\}\times X,u)$
(resp. of the form $(X\times \{P\}, u)$) is said to be
 {\em vertical} (resp. {\em horizontal}). A
linear combination of vertical and horizontal terms is said to be {\em negligible}.
Similar definitions apply to the tensor product $\Theta\otimes F$ over any field $F$.

Let $\Delta\subset S$ be a copy of the curve $X$ diagonally embedded in $S$. Let $F$ denote the field introduced in \S \ref{subsec:near}, $u\in
\cO(Y_K)^\times\otimes F$
 be a modular unit with coefficients in $F$, and consider the element
$(\Delta,u)\in \Theta\otimes F$.
\begin{lemma}
\label{lemma:negligible}
There exists
a negligible element $\theta_u\in \Theta\otimes F$ satisfying
$$ {\rm div}(\theta_u) = {\rm div}(\Delta,u).$$
\end{lemma}
\begin{proof}
Let  $D_u={\rm div}(\Delta,u)\in \coprod_{P\in S} F$  be the image of
 the element $(\Delta,u)\in\Theta$ under the divisor map. Since $D_u$ is an $F$-linear
  combination of elements of the form
 $(c_1,c_1) - (c_2,c_2)$ where $c_1$ and $c_2$ are cusps
  of the modular curve $X_K$, it is enough to construct a
   negligible element $\theta\in \Theta\otimes \Q$
 satisfying
 \begin{equation}
 \label{eqn:div-negligible}
  \rm div(\theta) = (c_1,c_1) - (c_2,c_2).
  \end{equation}
 By the Manin-Drinfeld theorem, there is an element
  $\alpha\in \cO(Y_K)^\times\otimes \Q$ whose divisor is $c_1-c_2$,
and  the negligible element  given by
 $$ \theta = (\{c_1\}\times X,\alpha) + (X\times \{c_2\}, \alpha) $$
 satisfies \eqref{eqn:div-negligible}. The lemma follows.
 \end{proof}
Thanks to Lemma \ref{lemma:negligible}, we can associate to any
 element of the form $(\Delta,u)\in \Theta\otimes F$ the element
\begin{equation}\label{def-Du}
\Delta_u := \mbox{ class of } \quad (\Delta,u) - \theta_u  \quad \mbox{ in } H^3_{\cM}(S,F(2)).
\end{equation}

These elements were introduced by Beilinson in \cite[Ch.\,2, \S 6]{beilinson-K1S}.
 A variant (\cite[Prop\,2.1]{Fl}) of the above construction was later exploited
  by Flach in loc.\,cit.\,to prove the finiteness of the Tate-Shafarevic group
   of the symmetric square of an elliptic curve, using the method of Kolyvagin.
 We call $\Delta_u$ the {\em Beilinson-Flach element} attached to the modular
  unit $u\in\cO(Y_K)^\times\otimes F$.
Strictly speaking, $\Delta_u$ is not a well-defined element in
$H^3_{\cM}(S,F(2))$,
 as it is only  well-defined modulo the
$F$-vector space
generated by the classes of negligible elements. However, this inherent ambiguity will not
lead to  problems
because the image of $\Delta_u$ under the relevant piece of
the regulator maps will turn out to depend only on $u$ and not on the choice of
 $\theta_u$ made in defining
$\Delta_u$. See Proposition \ref{main-prop} below for more details.


\subsection{Complex regulators} Fix an  embedding of $K$ into the field of complex numbers.
Following the definitions in \cite[\S 2]{beilinson-K1S}, \cite[\S 2]{DS}, the
{\em complex regulator} on $H^3_{\cM}(S_\C,\Q(2))$
may be regarded as a map
\begin{equation}\label{regC}
\reg_\C: H^3_{\cM}(S_\C,\Q(2)) \lra (\Fil^1 H^2_{\dR}(S/\C))^\vee,
\end{equation}
where here the superscript $\vee$ denotes the complex linear dual.
It sends the class of $ \theta = \sum_{i} (Z_i,u_i)$ to the
element $\reg_\C(\theta) $ defined by
$$
\reg_\C(\theta)(\omega) = \frac{1}{2\pi i}
 \sum_i \int_{Z_i-Z_i^{\mathrm{sing}}} \omega \log|u_i|.
$$

Recall the modular unit $u_\chi$ associated to the
 Dirichlet character $\chi$, and the class $\eta_f^{\ah}\in H^1_{\dR}(X/\C)$
  attached to the cusp form $f$. Moreover, write as customary
 $\omega_g\in \Fil^1 H^1_{\dR}(X/\C)$ for the class associated to
 the regular differential $2\pi i g(z)dz$.

  The tensor product
$\omega_g\otimes\eta_f^{\ah}$ of these classes gives rise,
 via the K\"unneth decomposition of $H^2_{\dR}(S/\C)$,  to an element
of $\Fil^1 H^2_{\dR}(S/\C)$.

\begin{proposition}
\label{prop:regulator-C} With notations as above, we have
$$
\reg_\C(\Delta_{u_\chi})(\omega_g\otimes\eta_f^{\ah}) =
(-2i)[\Gamma_0(N):\Gamma_1(N)(\pm1)] \langle f^*,f^*\rangle_{2,N}
^{-1}  \left\langle  f^*(z), \log|u_\chi(z)| \cdot g(z)
\right\rangle_{2,N}.
$$
\end{proposition}

\begin{proof}
Since the differential $\omega_g\otimes\eta_f^{\ah}$ vanishes
 identically on the horizontal and vertical curves on $S$,
 the negligible element  $\theta_{u_\chi}$ arising in the
  definition of $\Delta_{u_\chi}$  does not contribute to the
 value of the regulator at that class. Hence
 \begin{eqnarray*}
  \reg_\C(\Delta_{u_\chi})(\omega_g\otimes\eta_f^{\ah}) &=&
    \int_{X(\C)} \frac{ \bar f^*(z)}{\langle f^*,f^*\rangle_{2,N}}
  g(z) \log|u_\chi(z)| dz d\bar z  \\
  &=&  (-2i)[\Gamma_0(N):\Gamma_1(N)(\pm1)]\langle f^*,f^*\rangle_{2,N} ^{-1}
  \left\langle  f^*(z), \log|u_\chi(z)| \cdot g(z) \right\rangle_{2,N},
  \end{eqnarray*}
where the last equality follows from the explicit formula
 for the Petersson scalar product on
$S_k^{\ra}(N,\chi_f^{-1})$.
\end{proof}



\subsection{$p$-adic regulators}
\label{sec:besser}

Let $K_p$ be a finite extension of $\Q_p$ containing $K$ and fix an embedding
 of $K_p$ in $\C_p$. Write $\cO_p$, resp.\ $k_p$ for the ring
  of integers, resp.\ the residue field of $K_p$.
   Let $\mathcal X$ denote the (Deligne-Rapoport) smooth model of $X$ over $\cO_p$,
    and $\tilde{\mathcal X}/k_p$ its special fiber. Define
     $\cS = \mathcal X \times \mathcal X$, which is a smooth
      projective model of $S_{K_p}$ over $\cO_p$.

In analogy with the complex regulator \eqref{regC}, there is
 a $p$-adic syntomic regulator map
\begin{equation}\label{def-regp}
\reg_p: H^3_{\cM}(S_{K_p},\Q(2)) \lra (\Fil^1 H^2_{\dR}(S/K_p))^\vee :=
 \Hom(\Fil^1 H^2_{\dR}(S/K_p),K_p)
\end{equation}
arising from the syntomic Chern character in $K$-theory
 (cf.\,\cite{Gr}, \cite{Ni}, \cite{besser-K1S}).

 After possibly enlarging the field $K_p$,
  let $\{P_1,...,P_t\}\subset \mathcal X(\cO_p)$ be
  a set of points consisting of the cusps and of a choice of a lift of every
 supersingular point in $\tilde{\mathcal X}(\bar{\F}_p)$.
  Set
 $$ \mathcal X'=\mathcal X\setminus \{P_1,...,P_t\}, \qquad
 X'=\mathcal X'\times_{\spec \cO_p} \spec K_p.$$
 Let $\mathrm{red}: \mathcal X(\cO_p)\lra \tilde{\mathcal X}(k_p)$
  denote the reduction map and let  $\cA\subset X(K_p)$ be the affinoid subspace
  of the rigid
   analytic variety underlying $X$ defined by
   $$ \cA :=   X(K_p) - \mathrm{red}^{-1}(\{\tilde{P}_1, \ldots, \tilde{P}_t\}), \qquad \tilde{P}_j :=
   \mathrm{red}(P_j).$$
 Fix a system $\{ \cW_\epsilon\}_{\epsilon>0}$ of wide open neighborhoods
 of $\cA$ as in \cite[\S 2.1]{DR} and denote $\Phi$ the canonical lift
  of Frobenius on $X$ as in \cite[\S 2.2]{DR}. As explained in loc.\,cit.,\,restriction
   from $X'$ to $\cW_\epsilon$ gives rise to an isomorphism
 \begin{equation}
 \xymatrix{
 H^1_{\dR}(X') \ar[r]^-{\mathrm{comp}_\epsilon} &  H^1_{\rig}(\cW_\epsilon)}
 \end{equation}
 between the de Rham cohomology of the open curve $X'$ and the rigid
 cohomology $H^1_{\rig}(\cW_\epsilon)$ of $\cW_\epsilon$.
% If we let $Y=Y_1(N)/K_p$ denote the affine modular curve obtained by removing the cusps from $X$,
The inclusion $X'\subset X$ yields by restriction a monomorphism
$$
H^1_{\dR}(X)\hookrightarrow H^1_{\dR}(X'),
$$
and the image of $H^1_{\dR}(X)$
under $\mathrm{comp}_\epsilon$ consists of those classes in
  $H^1_{\rig}(\cW_\epsilon)$  whose annular residues about all
  the points $\{P_i\}$ vanish.
  The lift $\Phi$ of Frobenius induces a linear endomorphism of
  $H^1_{\rig}(\cW_\epsilon)$ which preserves the subspace
  $H^1_{\dR}(X)$.


 Label now two copies of $X$ as $X_1$ and $X_2$, denote by $\Phi_1$
 and $\Phi_2$ the corresponding canonical  lifts of Frobenius on the system of wide open neighborhoods $\cW_\epsilon$, and write
 $\Phi_{12}:=(\Phi_1,\Phi_2)$
  for the associated lift of Frobenius on the product $X_1\times X_2$.

Choose a polynomial $P(x)\in \C_p[x]$ such that
\begin{enumerate}
\item[(i)] $P(\Phi_{12})$ annihilates the class of
 $\omega_g\otimes \frac{du_\chi}{u_\chi}$ in $H^2_{\rig}(\cW_\epsilon^2)$;

%\item[(ii)] $P(\Phi)$ is the zero endomorphism on $K_p(-1)$.

\item[(ii)] $P(\Phi)$ is an invertible endomorphism on $H^1_{\dR}(X')$.

\end{enumerate}

Such a polynomial exists, since the eigenvalues of $\Phi_{12}$
acting on the space spanned by the Frobenius translates
of $\omega_g\otimes \frac{du_\chi}{u_\chi}$ have complex absolute value
$p^{3/2}$, while $\Phi$ acts on $H^1_{\dR}(X')$ with eigenvalues of
complex absolute value $p^{1/2}$ and $p$.

Thanks to (i), there exists a rigid analytic one-form
\begin{equation}\label{condition-rho}
\varrhoP\in \Omega^1(\cW_\epsilon^2)\, \mbox{ such that } \, d(\varrhoP)=P(\Phi_{12})\left(\omega_g\otimes \frac{du_\chi}{u_\chi}\right).
\end{equation}
This form, which depends on the choice of $P$,   is  only  determined
 up to {\em closed} forms in $\Omega^1(\cW_\epsilon^2)$
 by \eqref{condition-rho}.

In order to adapt our calculations to Besser's in \cite{Bes-syn} and \cite{besser-K1S}, it will be convenient to fix a particular choice of polynomial $P$ and form $\varrhoP$. (In the next section we shall exploit the fact that the computations performed there hold independently of the choice of $P$, and will work with a different polynomial so that we can take advantadge of the results obtained in \cite{DR}.)
% can in fact be described explicitly, and the latter can then be
%normalized so that we work with a {\em canonical} choice of $\varrhoP$ up to exact one-forms,
%among its translates by closed forms in $\Omega^1(\cW_\epsilon^2)$.

Let $P_g(t)\in \C_p[t]$ be a polynomial such that $P_g(\Phi)$
 annihilates the class of $\omega_g$ in
 $H^1_{\rig}(\cW_\epsilon)$. Specifically, we may set
  $P_g(t):=t^2 -a_p(g) t+\chi_g(p) p$,
 and let $F_g \in \cO_{\rig}(\cW_\epsilon)$ be a Coleman integral
   of $\omega_g$, that is to say, a rigid analytic function such that
 \begin{equation}
 \label{eqn:Fg}
  p dF_g=p\omega_{g^{[p]}} = P_g(\Phi)\omega_g
  \end{equation}
  (cf.\ for example equation (127) of \cite{DR}).
  Likewise, let
 $P_{E_\chi}(t)\in \C_p[t]$ be a polynomial such that $P_{E_\chi}(\Phi)$
 annihilates the class of the Eisenstein series
 $E_\chi=\frac{du_\chi}{u_\chi}$ in $H^1_{\rig}(\cW_\epsilon)$.
 Here we make the specific choice $P_{E_\chi}(t):=t^h-p^h$, where $h$
 is the order of the root of unity $\chi(p)$ (in other words,
 $\Phi^h/p^h$ fixes the class of $E_\chi$). Although a more optimal choice for
 $P_{E_\chi}(t)$ would have been the linear polynomial $t-\chi(p) p$,
 we made here a choice corresponding to the one made in the definition of the
 modified syntomic regulator $\mathrm{reg}(u_\chi)$ of the function $u_\chi$ (cf.\,\cite[Prop. 10.3]{Bes-syn}). The rigid analytic function
\begin{equation}\label{FEchi}
F_{E_\chi} := p^{-h}P_{E_\chi}(\Phi)\log(u_{\chi}) \in \cO_{\rig}(\cW_\epsilon)
\end{equation}
 is a Coleman integral of
   $E_\chi$, satisfying
 $$ p^{h} d F_{E_\chi} =  P_{E_\chi}(\Phi) E_\chi.$$



Given two choices as above of polynomials $P_g(t)=\prod_i (t-\alpha_i)$
 and $P_{E_\chi}(t)=\prod_j (t-\beta_j)$, it is clear  that the polynomial
 \begin{equation}
 \label{eqn:P}
 P(t):=P_g(t)\star P_{E_\chi}(t) := \prod_{i,j} (t-\alpha_i \beta_j)
 \end{equation}
 satisfies (i) above.
 Moreover, as explained in \cite[Lemma 4.2, (4)]{Bes-inv},
 there exist polynomials $a(t_1,t_2)$, $b(t_1,t_2)$ such
  that $P(t_1\cdot t_2)=p^{-1} a(t_1,t_2)P_g(t_1)
   + p^{-h} b(t_1,t_2) P_{E_\chi}(t_2)\in \C_p[t_1,t_2]$
    and one checks that
\begin{equation}\label{description-rho}
\varrhoP \ \ =\ \   a(\Phi_1,\Phi_2)
 \left(F_g \otimes  \frac{du_\chi}{u_\chi}\right) \ \ + \ \
b(\Phi_1,\Phi_2) (\omega_g \otimes F_{E_\chi})
   \ \  \in \ \  \Omega^1(\cW_\epsilon^2)
\end{equation}
then satisfies \eqref{condition-rho}.

There is a certain degree of ambiguity in \eqref{description-rho}: neither the Coleman primitives $F_g$, $F_{E_\chi}$ nor the polynomials $a(t_1,t_2)$, $b(t_1,t_2)$ are unique. But all solutions of the differential equation  \eqref{condition-rho}
are of the form \eqref{description-rho}; moreover, given one such $\varrhoP$, all them can be written as $\varrhoP+\varrho_{_0}$
 with $\varrho_{_0}$ a closed $1$-form on $\cW_\epsilon^2$.


We can single out a canonical choice of $\varrhoP$
   (up to exact $1$-forms on $\cW_\epsilon^2$)
   by setting $F_g(\infty)=F_{E_\chi}(\infty)=0$ in \eqref{description-rho}; more precisely, in doing this, two different choices of pairs $(a(t_1,t_2), b(t_1,t_2))$, $(a'(t_1,t_2), b'(t_1,t_2))$ allowed by \cite[Lemma 4.2, (4)]{Bes-inv} give rise to forms $\varrho_{P,a,b}$, $\varrho_{P,a',b'}$ such that $\varrho_0=\varrho_{P,a,b}-\varrho_{P,a',b'}$ is exact on $\cW_\epsilon^2$ and therefore the class of $\varrho_0$ in $H^1_{\rig}(\cW_\epsilon^2)$ vanishes.

 Imposing $F_g(\infty)=0$ amounts to
    normalizing the
    $q$-expansion of $F_g$ to be
 \begin{equation}\label{q-exp}
 F_g(q)=\sum_{p\nmid n} \frac{a_n(g)}{n} q^n,
 \end{equation}
and the condition $F_{E_\chi}(\infty)=0$ is equivalent to normalizing the modular unit $u_\chi$ as was done in Definition \ref{def-uchi}. This way $F_{E_\chi}$ also equals the modified syntomic regulator $\mathrm{reg}(u_\chi)$ of $u_\chi$ defined in \cite[Prop. 10.3]{Bes-syn}.




Let $\Delta \subset \cW_\epsilon^2$ denote the diagonal and define
 \begin{equation}\label{defxiPx}
 \xi'_{P} := [{\varrhoP}_{|\Delta}] \in H^1_{\rig}(\cW_\epsilon) \simeq H^1_{\dR}(X').
 \end{equation}

The above discussion shows that the class $\xi'_{P}$
 in $H^1_{\rig}(\cW_\epsilon)
 =\frac{\Omega^1(\cW_\epsilon)}{d\cO(\cW_\epsilon)}$ is well-defined.
%Moreover, condition (ii) ensures that $\xi_{P}$ lands in
% the image of $H^1_{\dR}(X)$ in $H^1_{\rig}(\cW_\epsilon)$.
Moreover, in view of condition (ii), we can now set
\begin{equation}\label{defxix}
\xi' := P(\Phi)^{-1}\cdot \xi'_{P} \in H^1_{\dR}(X'),
\end{equation}
which is directly seen to be independent of the choice of $P$.


Finally, let $\mathrm{spl}_X: H^1_{\dR}(X')\lra H^1_{\dR}(X)$
denote the Frobenius equivariant splitting of the short exact sequence
\begin{equation}\label{ses}
0 \rightarrow H^1_{\dR}(X) \lra H^1_{\dR}(X') \lra K_p(-1)^{t-1} \rightarrow 0
\end{equation}
and set $\xi:=\mathrm{spl}_X(\xi')\in H^1_{\dR}(X)$.

%\footnote{Vic: I haven't thought carefully about that yet,
% but I'd say that the class of $\xi$ in $H^1_{\dR}(X)$ {\em does}
% depend on the choice of $o$. This is in fact coherent with my
% perception that the class of $\Delta_{u_\chi}$ also %depends on $o$.
% But I didn't think carefully about that, so you may easily
%convince me of the opposite...}



\begin{proposition}\label{main-prop}
With notations as above, we have
$$
\reg_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur})
 =  \langle \eta_f^{\ur}, \xi \rangle,
$$
where $\langle\ ,\ \rangle$
 is the pairing on $H^1_{\dR}(X)$ induced by Poincar\'e duality.
\end{proposition}

\begin{proof} Thanks to the work of Besser \cite{besser-K1S},
the $p$-adic syntomic regulator \eqref{def-regp} admits the
following description in terms of Coleman integration.
Let $\theta= \sum_{i} (Z_i,u_i)$ be an element in $K_1^{(2)}(S)$
 and write $\iota_i: Z_i\hookrightarrow S$ for the embedding of $Z_i$
  in $S$ given by inclusion.
Assume for simplicity that the curves $Z_i$ are all non-singular, and that
$\theta$ is {\em integral}, by what we mean that for each $i$:
\begin{itemize}

\item the curve $Z_i$ admits a smooth integral model $\mathcal Z_i$ over $\cO_p$, and

\item the divisor of $u_i$, when regarded as a function on $\mathcal Z_i$,
 does not contain the special fiber.
\end{itemize}
Note that these conditions are satisfied in our setting.

Under this assumption, $\theta$ lies in the image of the natural restriction
 map $K_1(\cS) \lra K_1(S)$.

Let $\Omega^{\mathrm{II}}(X_{K_p})$ denote the space of differential forms
 of the second kind on $X_{K_p}$, that is to say, the space of meromorphic
  $1$-forms whose residue at any point of the curve is zero. There is an exact sequence
$$
0 \rightarrow K_p(X)^\times \stackrel{d}{\lra }
 \Omega^{\mathrm{II}}(X_{K_p}) \lra H^1_{\dR}(X/K_p) \rightarrow 0
$$
and for any $\eta \in \Omega^{\mathrm{II}}(X_{K_p})$
 we write $[\eta]$ for its class in $H^1_{\dR}(X/K_p)$.

Instead of invoking the description of the $p$-adic syntomic
regulator in terms of Besser-de Jeu's global triple index as
stated in the main theorem of \cite{besser-K1S}, it will be more
convenient for us to exploit \cite[Prop.\,6.3]{besser-K1S},
which provides a formula for \eqref{def-regp} in the language
of Besser's finite polynomial cohomology \cite{Bes-inv}.
In order to state this formula, let $H^*_{\mathrm{ms}}$
and $H^*_{\fp}$ denote, respectively, Besser's modified
version of syntomic cohomology and finite polynomial cohomology:
cf.\,e.g.\,\cite[\S 2]{besser-K1S} for a quick review of both
and their interactions.

Let $\omega\in \Omega^1(X_{K_p})$ be a regular form on $X$
and $\eta\in \Omega^{\mathrm{II}}(X_{K_p})$ be a differential
of the second kind, regular on some affine curve $X^0\subset X$. Write
$$\omega_1=\pi_1^*(\omega)\in \Omega^1(S), \qquad \eta_2
=\pi_2^*(\eta)\in \Omega^{\mathrm{II}}(S) $$
for the pull-back of $\omega$ and $\eta$ under the projection
of $S$ into the first and second component, respectively.

Then the class $\omega_1\wedge [\eta_2]$ is an element of
 $\Fil^1 H^2_{\dR}(S)$ and, according to
  \cite[Theorem 1.1, Proposition \,6.3]{besser-K1S}:
\begin{equation}\label{Bes-regp}
\reg_p(\theta)(\omega_1 \otimes [\eta_2])
 = \sum_i \langle \iota_i^*\tilde{\eta}_2,
   \iota_i^*\tilde{\omega}_1\cup \mathrm{reg}(u_i)\rangle_{\mathcal Z^0_i,\fp},
\end{equation}
where
\begin{itemize}
\item $Z_i^0 =Z_i \cap (X\times X^0)$, $\mathcal Z^0_i$ is the
model for $Z_i^0$ deduced from $\mathcal Z_i$,
\item $\mathrm{reg}(u_i)\in H^1_{\mathrm{ms}}(\mathcal Z_i^0,1)
 \subseteq H^1_{\fp}(\mathcal Z_i^0,1,2)$ is the regulator of the
  function $u_i$ as defined in \cite[Prop. 10.3]{Bes-syn},

\item $\iota_i^* \tilde{\omega}_1\in H^1_{\fp}(\mathcal Z_i,1,1)$ is a Coleman
 primitive of $\iota_i^*\omega \in \Omega^1(Z_i)$,

\item  $\iota_i^* \tilde{\eta}_2\in H^1_{\fp,c}(\mathcal Z_i^0,0,1)$ is the single
 lift of $\iota_i^*([\eta_2])$ under the isomorphism
\begin{equation}\label{p-map}
\mathrm{p}: H^1_{\fp,c}
  (\mathcal Z_i^0,0,1)\overset{\sim}\rightarrow H^1_{\dR}(Z_i)
\end{equation}
of \cite[Lemma 6.2]{besser-K1S}, and
\end{itemize}
\begin{equation}\label{cup}
\langle \,,\rangle_{\mathcal Z^0_i,\fp}: H^1_{\fp,c}(\mathcal Z_i^0,0,1)
\times H^2_{\fp}(\mathcal Z_i^0,2,3)\lra H^3_{\fp,c}(\mathcal Z_i^0,2,4)
\simeq H^2_{\dR,c}(Z_i^0)\overset{\mathrm{tr}}{\simeq} K_p
\end{equation}
is the pairing induced by Poincar\'e duality in finite polynomial cohomology.
 Here $H^*_{\fp,c}$ stands for finite polynomial cohomology with
compact support, as introduced in \cite[\S 4]{besser-K1S}.
 The cup-product \eqref{cup} is constructed in loc.\,cit., where it is
also shown that it satisfies the projection formula.

At the time \cite{besser-K1S} was written, the results were subject
to the compatibility of pushforward maps in syntomic and motivic cohomology,
as specified in \cite[Conjecture 4.2]{besser-K1S}. At present this compatibility
has been checked  by D\'eglise and Mazzari \cite{DM}, and thus \eqref{Bes-regp}
holds unconditionally.

Let us now apply  \eqref{Bes-regp} to the Beilinson-Flach element
 $\Delta_{u_\chi}$ that was introduced in \eqref{def-Du}.
  Recall that the curves in $X\times X$ on which $\Delta_{u_\chi}$
   is supported are the images of $X$ under the diagonal embedding
    $\iota_{12}(x)=(x,x)$ and the various horizontal and vertical
     embeddings $\iota_{1,c}(x)=(x,c)$ and $\iota_{2,c}(x)=(c,x)$,
      where $c$ is a cusp on the modular curve $X$.

We firstly claim that the terms on the right-hand side of
  \eqref{Bes-regp} corresponding to $\iota_{1,c}$ and $\iota_{2,c}$
   vanish and the one corresponding to $\iota_{12}$ is independent
    of the choices of lifts to finite polynomial cohomology.

 To see that, put $\omega=\omega_g$ and $\eta=\eta_f^{\ur}$
 and recall $X'=\mathcal X'\times K_p$ is the curve obtained
  from $X$ by removing a finite set of points including all the cusps.
   Note first that $\iota_{1,c}^*([\eta_2])=0\in H^1_{\dR}(X)$,
    because the composition $\pi_2\circ \iota_{1,c}$ is the constant
     function $c$ on $X$. Hence, since the map $\mathrm{p}$ in
      \eqref{p-map} is an isomorphism, the class of the lift
       $\iota_{1,c}^*(\tilde{\eta}_2)$ is also trivial and
 $$\langle \iota_{1,c}^*\tilde{\eta}_2,
  \iota_{1,c}^*\tilde{\omega}_1\cup \mathrm{reg}(u)\rangle_{\mathcal X',\fp} =0,$$
 for any rational function $u$.

We similarly have $\iota_{2,c}^*(\omega_1)=0\in \Omega^1(X)$
 because $\pi_1\circ \iota_{2,c}=c$. Notice however that a lift
  of $0$ to $H^1_{\fp}(\cX,1,1)$ is not necessarily trivial,
   but represented by a pair in $\cO_{\rig}(\cW_\epsilon)\oplus
    \Omega^1(X)$ of the form $[(\lambda,0)]$, where $\lambda$
     is a constant. Then, if $u$ is a modular unit on $X$,
      the cup-product
$$\iota_{2,c}^*\tilde{\omega}_1\cup \mathrm{reg}(u)\in
 H^2_{\fp}(\cX',2,3)\simeq H^1_{\dR}(X')\simeq H^1_{\rig}(\cW_\epsilon)$$
may be represented by the pair $(\lambda \frac{du}{u}_{|\cW_\epsilon},0)$.
 But then
\begin{equation}\label{ind}
\langle \iota_{2,c}^*\tilde{\eta}_2,
 \iota_{2,c}^*\tilde{\omega}_1\cup \mathrm{reg}(u)\rangle_{\mathcal X',\fp}
  = \lambda \langle \eta_f^{\ur}, \frac{du}{u} \rangle_{\dR} = 0
\end{equation}
because the cusp form $f$ is orthogonal to the Eisenstein
 series $\frac{du}{u}$. This accounts for the vanishing
  of the horizontal and vertical terms, and explains why we call them negligible.

As for the diagonal term, let us show that
 $\langle \iota_{12}^*\tilde{\eta}_2,
   \iota_{12}^*\tilde{\omega}_1\cup
    \mathrm{reg}(u_\chi)\rangle_{\mathcal X',\fp}$
     is independent of the choices of lifts to
      finite polynomial cohomology.
       Since $\pi_1\circ \iota_{12}$
        and $\pi_2\circ \iota_{12}$
         are both the identity map on $X$, this is just
          $\langle \tilde{\eta}^{\ur}_f, \tilde{\omega}_g
          \cup \mathrm{reg}(u_\chi) \rangle_{\mathcal X',\fp}.$
           Again there is a single choice for $\tilde{\eta}^{\ur}_f$,
 but the Coleman integral $F_g$ of $\omega_g$ is only well-defined
  up to a constant. The difference between any two choices is then equal to
$$\langle \tilde{\eta}^{\ur}_f,
  [(\lambda,0)]\cup \mathrm{reg}(u_\chi)\rangle_{\mathcal X',\fp}
   = \lambda \left\langle \eta^{\ur}_f, \frac{du_\chi}{u_\chi}\right\rangle_{\dR}$$
 for some $\lambda\in K_p$, and the same orthogonality
  argument between cusp and Eisenstein forms
   again shows that this is $0$. The claim follows.

\vspace{0.2cm}

Summing up, we obtain from \eqref{Bes-regp} that
\begin{equation}\label{Bes-regDu}
\mathbf{reg}_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur})
 = \langle \tilde{\eta}^{\ur}_f,
\tilde{\omega}_g\cup \mathrm{reg}(u_\chi) \rangle_{\mathcal X',\fp}.
\end{equation}

Recall that $\tilde{\omega}_g$ may be represented
 by the pair $(F_g,\omega_g)$ where $F_g \in \cO_{\rig}(\cW_\epsilon)$
  is a Coleman integral  of $\omega_g$, which in light
   of the above claim we are entitled to normalize as it was done in \eqref{q-exp}.
 Besides, by \cite[Prop. 10.3]{Bes-syn} the class $\mathrm{reg}(u_\chi)$ is represented by the pair $(F_{E_\chi},\frac{du_\chi}{u_\chi}) \in \cO_{\rig}(\cW_\epsilon)
   \oplus \Omega^1(X')$ where $F_{E_\chi}$ is the Coleman integral of
    $\frac{du_\chi}{u_\chi}$ introduced in \eqref{FEchi} and normalized as we explained right after \eqref{q-exp}.


By definition, $\tilde{\omega}_g\cup \mathrm{reg}(u_\chi)$
 is the restriction to the diagonal of
  $\pi_1^*\tilde{\omega}_g\wedge \pi_2^*\mathrm{reg}(u_\chi)$.
   Note that the polynomial $P$ defined in equation \eqref{eqn:P} satisfies the properties (i) and (ii) above. The class
 $\pi_1^*\tilde{\omega}_g\wedge \pi_2^*\mathrm{reg}(u_\chi)$
  in $H^2_{\fp}(\mathcal {X'}^{2},2,3)$ may then be represented by the pair
\begin{equation}\label{repre}
(\varrhoP, \pi_1^*\omega_g\wedge \pi_2^*\frac{du_\chi}{u_\chi}) \in \Omega^1_{\rig}(\cW_\epsilon^2)\oplus \Omega^2({X'}^{2})
\end{equation}
where $\varrhoP$ is the form introduced in \eqref{description-rho}, which satisfies
\begin{equation}\label{diffeqn}
d\varrhoP=P(\Phi_{12})(\pi_1^*\omega_g\wedge \pi_2^*\frac{du_\chi}{u_\chi}).
\end{equation}

Let us again remark that this differential equation does
 not determine $\varrhoP$ uniquely, but that the above
  normalizations of $F_g$ and $F_{E_\chi}$
  completely determine it up to exact $1$-forms on $\cW_\epsilon^2$.
   Obviously, when we restrict \eqref{repre} to the diagonal,  this ambiguity does not affect
    the class we obtain  in $H^2_{\fp}(\mathcal X',2,3)$,
      because exact $1$-forms on $\cW_\epsilon$ vanish in $H^1_{\rig}(\cW_\epsilon)$.

In conclusion, the class $\tilde{\omega}_g\cup \mathrm{reg}(u_\chi)$ in $H^2_{\fp}(\mathcal X',2,3)$ may be represented by the pair
$$(\iota^*_{12}(\varrhoP),0)\in \Omega^1_{\rig}(\cW_\epsilon)\oplus \Omega^2(X'),$$
 where $\varrhoP$ is as above and
 $\iota^*_{12}(\varrhoP)$ is the form   denoted $\xi'_{P}$ in \eqref{defxiPx}.

As in  \cite[(14)]{Bes-inv} there is a commutative diagram
\begin{equation}\label{dia}
\xymatrix{
H^1_{\fp,c}(\cX',0,1) \times H^1_{\dR}(X') \ar[r]^{\mathrm{Id}\times \mathrm{i}}\ar[d]^{\mathrm{p}\times \mathrm{Id}}  & H^1_{\fp,c}(\cX',0,1) \times H^2_{\fp}(\cX',2,3)\ar[d]^{\langle\,,\,\rangle_{\fp}} \\
H^1_{\dR,c}(X')^{w=1} \times H^1_{\dR}(X') \ar[r]^{\langle\,,\,\rangle_{\dR}} & H^2_{\dR,c}(X')\simeq H^3_{\fp,c}(\cX',2,4),}
\end{equation}
where  $H^1_{\dR,c}(X')^{w=1}$ stands for the pure submodule of weight $1$ of $H^1_{\dR,c}(X')$. In fact both maps
$$
H^1_{\dR}(X')\overset{\mathrm{i}}{\longrightarrow} H^1_{\fp}(\cX',2,3) \,\mbox{ and }\, H^1_{\fp,c}(\cX',0,1) \overset{\mathrm{p}}{\longrightarrow} H^1_{\dR,c}(X')^{w=1}
$$
are isomorphisms, as it follows from \cite[(2.7) and the first assertion of Lemma 2.8]{besser-K1S}.

By definition of $\mathrm{i}$, the preimage of $\tilde{\omega}_g\cup \mathrm{reg}(u_\chi) = [( \xi'_{P},0)]$ under $\mathrm{i}$ is the class in $H^1_{\rig}(\cW_\epsilon)$ of the $1$-form $P(\Phi)^{-1}(\xi'_{Q}) = \xi'$. To conclude, we now deduce from the commutativity of \eqref{dia} that
$$\langle \eta_f^{\ur}, \xi'\rangle_{\dR} = \langle \tilde{\eta}_f^{\ur},
\mathrm{i}(\xi') \rangle_{\fp} = \langle \tilde{\eta}_f^{\ur},
\tilde{\omega}_g\cup \mathrm{reg}(u_\chi) \rangle_{\fp} .$$ Since the class $\eta_f^{\ur}$
is orthogonal to the complement of $H^1_{\dR}(X)$ in $H^1_{\dR}(X')$
under the Frobenius equivariant splitting of \eqref{ses},
we have $\langle \eta_f^{\ur}, \xi' \rangle_{\dR} = \langle \eta_f^{\ur}, \xi \rangle_{\dR}$
and the proposition follows.
\end{proof}

\section{The Beilinson formula}
\label{sec:main}

Let $f\in S_2(N, \chi_f)$, $g\in S_2(N, \chi_g)$
  be eigenforms of weight $2$ as in Section \ref{subsec:near}.
  Recall that $f$ and $g$ are not assumed to be newforms. Moreover, we
 insist on the condition
  $\chi_f\ne \chi_g^{-1}$, which implies that $\chi = \chi_f^{-1} \chi_g^{-1}$
  is non-trivial.


\subsection{The complex setting}
\label{sec:main-C}
 In \cite[Ch.\,2, \S\,6]{beilinson-K1S}, Beilinson relates the image
   of $\Delta_{u_\chi}$ under the complex regulator map to the value
   at $s=2$ of the Rankin $L$-series attached to $f\otimes g$.
  The following explicit version of Beilinson's theorem
  is a slight generalisation of the results of \cite{baba}.

\begin{proposition}
 For cusp forms $f$ and $g$ of weight two as in Section
\ref{subsec:near}, we have
$$ \frac{ L(f\otimes g,2)}{   \langle f^*,f^*\rangle_{2,N} }=
  (8i)\pi^3 [\Gamma_0(N):\Gamma_1(N)(\pm 1)]^{-1}N^{-2} \tau(\chi^{-1})
   \reg_{\C}(\Delta_{u_\chi})(\omega_g\otimes \eta_f^{\ah}).$$
\end{proposition}
\begin{proof} This follows by combining the explicit formula
 for $L(f\otimes g,2)$ obtained in
Proposition \ref{prop:rankin-quater} with the explicit expression
for $\reg_{\C}(\Delta_{u_\chi})$ given in Proposition
\ref{prop:regulator-C}.
\end{proof}


 \subsection{The $p$-adic setting}
 \label{sec:main-padic}

 Let $p\geq 3$ be a prime which does not divide $N$.
 Assume that $f$ is ordinary at $p$
 (with respect to a fixed embedding of the field $K_{f}$ in $\C_p$).
 Let $\hf$ be the Hida family whose specialisation
 in weight $2$ is the $p$-stabilisations of $f$,
  and let $L_p(\hf,g)(k,j)$ be the $p$-adic $L$-function
  defined in Section \ref{subsec:Hidap}.


 Let $\cE(f)$,
 $\cE^\ast(f)$ and $\cE(f,g,2)$ be the $p$-adic multipliers
 defined in Sections \ref{subsec:ordinary} and \ref{subsec:Hidap}.
 Recall that
 \begin{eqnarray*}
 \cE(f,g,2)&=&
 (1-\beta_p(f)\alpha_p(g)  p^{-2})
 (1-\beta_p(f) \beta_p(g)  p^{-2})\\
 && \times (1-\beta_p(f) \alpha_p(g) \chi (p)p^{-1})
 (1-\beta_p(f)\beta_p(g) \chi(p)p^{-1}).
 \end{eqnarray*}


 The following $p$-adic Beilinson formula is the main result of this paper.



\begin{theorem}
\label{thm:main}
 For cusp forms $f$ and $g$ of weight two as in Section
\ref{subsec:near}, we have
$$L_p(\hf,g)(2,2)=
\frac{\cE(f,g,2)}{\cE(f)\cdot \cE^{\ast}(f)}
 \times \reg_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur}).$$
\end{theorem}

\begin{proof} By the description of the $p$-adic $L$-function given
in equation \eqref{eqn:lpfEbis},
$$
L_p(\hf,g)(k,j)=\frac{1}{\cE^\ast(f_k)} \left \langle
\eta_{f_k}^{\ur},
 \Xi(f_k,g,j)^{\ord,p}
 \right\rangle_{k,X}
$$
for all triples $(k,\ell,j)$ belonging to the set
\eqref{eqn:good-triples}. Since the terms in the above
  expression vary analytically, taking the limit to $k=\ell=j=2$
  yields, in light of equation \eqref{eqn:projectionp5},
\begin{equation}\label{LHS}
L_p(\hf,g)(2,2)=\frac{1}{\cE^\ast(f)} \left \langle
\eta_{f}^{\ur},
 e_{\ord}(d^{-1}
 E_{2,\chi}^{[p]}\cdot g)
 \right\rangle_{2,X}.
\end{equation}
 On the other hand, by Proposition \ref{main-prop}
  $$\reg_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur}) =
  \langle \eta_f^{\ur}, \xi \rangle.$$ Since
  $$\Phi(\eta_f^{\ur})=
  \alpha_p(f) \eta_f^{\ur}, \qquad \langle \Phi(\eta_f^{\ur}), \Phi(\xi)\rangle
  = p\langle \eta_f^{\ur},\xi\rangle, \qquad \alpha_p(f) \beta_p(f)=\chi_f(p)
  p,$$
  we deduce by multi-linearity that
$$
\langle \eta_f^{\ur}, \xi \rangle = P\big(\chi^{-1}_f(p)\beta_p(f)\big)^{-1}
 \langle \eta_f^{\ur}, \xi'_P\rangle.
$$
 Since $f$ is an ordinary eigenform, the quantity
  $\langle \eta_f^{\ur}, \xi'_P\rangle$ only depends on the
  $f^\ast$-isotypical ordinary projection of $\xi'_P$, that is to say,
  $\langle \eta_f^{\ur}, \xi'_P\rangle = \langle \eta_f^{\ur}, e_{f^\ast} e_{\ord }\xi'_P\rangle$.

 Choose the polynomial $P(x)$ satisfying conditions (i) and (ii)
 to be
 $$P(x) := (x-\alpha_p(g)) \cdot  (x-\alpha_p(g) \chi(p) p)
  \cdot (x-\beta_p(g))\cdot (x-\beta_p(g) \chi(p) p).$$
 This choice of $P$ has the advantage of allowing us to directly invoke the
calculations already performed in
  \cite[Prop. 5.4]{DR}. They give
\begin{equation*}
 e_{f^\ast} e_{\ord }\xi'_P =  \chi_f(p)^{-2} p^4 \cE(f) \cdot e_{f^\ast} e_{\ord}
(d^{-1} E^{[p]}_{2,\chi} \cdot g).
\end{equation*}
 A direct calculation shows that
  $$\cE(f,g,2) = p^{-4}\chi_f(p)^{-2}
  P\big(\chi^{-1}_f(p)\beta_p(f)\big).$$
  By combining the above remarks, we find the following expression
  for the $p$-adic regulator:

  \begin{equation}
  \label{RHS}
  \reg_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur}) =
  \frac{\cE(f)}{\cE(f,g,2)}\times
  \left \langle \eta_{f}^{\ur},
  e_{f^\ast} e_{\ord}(d^{-1} E_{2,\chi}^{[p]}\cdot g)
  \right\rangle_{2,X}.
  \end{equation}
 The theorem follows by comparing equations \eqref{LHS} and \eqref{RHS}.
 \end{proof}
 \begin{remark}
Note that the modular form $g$ that arises in Theorem \ref{thm:main}
is fixed throughout the argument, and is
  thus not required to be ordinary at $p$.
\end{remark}
 Assume now that both $f$ and $g$
 are ordinary at $p$
 (with respect to a fixed embedding of the field $K_{fg}$ in $\C_p$).
 Let $\hf$ and $\hg$ be the Hida families whose specialisations
 in weight $2$ are the $p$-stabilisations of $f$ and $g$,
  respectively, and let $L_p(\hf,\hg)(k,\ell,j)$ be the $p$-adic $L$-function
  defined in Section \ref{subsec:Hidap}. The following corollary
  is an immediate consequence of Theorem \ref{thm:main}.

 \begin{corollary}
\label{cor:main}
 For cusp forms $f$ and $g$ of weight two as in Section
\ref{subsec:near}, we have
$$L_p(\hf,\hg)(2,2,2)=
\frac{\cE(f,g,2)}{\cE(f)\cdot \cE^{\ast}(f)}
 \times \reg_p(\Delta_{u_\chi})(\omega_g \otimes \eta_f^{\ur}).$$
\end{corollary}







\begin{thebibliography}{Gross}

\bibitem[BaSr]{baba} S.\,Baba, R.\,Sreekantan, {\em An analogue of
cyclotomic units for products of elliptic curves},
Proc. Edinb. Math. Soc. (2) {\bf 47} (2004), no. 1, 35--51.


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

\bibitem[BDP]{BDP1}
M.\ Bertolini, H.\ Darmon, K.\ Prasanna, {\em  Generalised Heegner
cycles and $p$-adic Rankin L-series}, Duke Math. J. {\bf 162} (2014), 1033--1148.

\bibitem[BDR]{BDR2} M.\ Bertolini, H.\ Darmon, V.\ Rotger, {\em
Beilinson-Flach elements and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series}, submitted.


\bibitem[Bei]{beilinson-K1S}
A.A.\,Beilinson, {\em Higher regulators and values of $L$-functions},
 in {\em Current problems in mathematics} {\bf 24}, 181--238, Akad. Nauk SSSR, Vsesoyuz,
Inst. Nauchn. Tekhn. Inform., Moscow, 1984.


\bibitem[Bes1]{Bes-inv} A. Besser, {\em A generalization of Coleman's $p$-adic integration theory}, Invent. Math. {\bf 142} (2000), 397--434.

\bibitem[Bes2]{Bes-syn} A. Besser, {\em Syntomic regulators and p-adic integration I: rigid syntomic regulators}, Israel J. Math. {\bf 120} (2000), 291--334.

\bibitem[Bes3]{besser-K1S}
A. Besser, {\em On the syntomic regulator for $K_1$ of a surface},
Israel J. Math. {\bf 190} (2012), 29--66.

 \bibitem[Bl]{bloch86}
S. Bloch, {\em Algebraic cycles and higher K-theory},
 Advances Math. {\bf 61} (1986) 267--304


\bibitem[Br]{brunault} Brunault, Fran{\c c}ois,
 {\em Valeur en $2$ de fonctions $L$ de formes modulaires de poids $2$:
  th\'eor\`eme de Beilinson explicite.} Bull.\ Soc.\ Math.\ France {\bf 135} (2007), no. 2, 215-246.

 \bibitem[Col]{coleman-bu} R.F.~Coleman, {\em A $p$-adic Shimura isomorphism
  and $p$-adic periods of modular forms}, in $p$-adic monodromy and the
   Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 21--51, Contemp. Math.
    {\bf 165}, Amer. Math. Soc., Providence, RI, 1994.


  \bibitem[DR]{DR}
 H. Darmon, V. Rotger, {\em Diagonal cycles and Euler systems I:  A $p$-adic Gross-Zagier formula},
  Ann. Sc. Ecole Norm. Sup. {\bf 47} (2014) 779--832.

\bibitem[Das]{dasgupta} S.\ Dasgupta, {\em Greenberg's conjecture on symmetric square $p$-adic $L$-functions}, preprint 2014.

\bibitem[DM]{DM} F.\,D\'eglise, N.\,Mazzari, {\em The rigid syntomic spectrum}, preprint 2012.

\bibitem[DS]{DS} C.\,Deninger, A.J.\,Scholl, {\em The Beilinson conjectures},
 in {\em L-functions and Arithmetic}, Proceedings
of the Durham Symposium, July, 1989 (J. Coates and M. J. Taylor, eds.), London
Math. Soc. Lecture Notes Series {\bf 153}, Cambridge University Press, 1991, 173--209.

\bibitem[Fl]{Fl} M.\,Flach, {\em A finiteness theorem for the symmetric
 square of an elliptic curve}, Invent. Math.\,{\bf 109} (1992), 307--328.


\bibitem[Gros]{Gr}
M.\,Gros, {\em Regulateurs syntomiques et valeurs de fonctions L
$p$-adiques II},
 Invent. Math.
{\bf 115} (1994), 61--79.

\bibitem[Gross]{gross}
B.H. Gross, {\em  On the factorization of $p$-adic $L$-series},
Invent. Math. {\bf 57}  (1980), 83--95.

\bibitem[Hi]{hida}
H. Hida, Elementary theory of $L$-functions and Eisenstein series.
London Mathematical Society Student Texts {\bf 26}, 1993.

\bibitem[LLZ]{llz}
A.\ Lei, D.\ Loeffler,  S.\  L.\ Zerbes, {\em Euler systems for Rankin-Selberg convolutions of modular forms}, Annals Math. {\bf 180} (2014), no. 2, 653--771.

 \bibitem[LZ]{loeffler-zerbes}
 D. Loeffler and S.L.\ Zerbes,
 {\em  Iwasawa theory and $p$-adic $L$-functions over $\Z_p^2$-extensions}, to appear in Int. J. Number Theory.


\bibitem[Ni]{Ni}
W. Nizio\l, {\em On the image of $p$-adic regulators}, Invent. Math. {\bf 127} (1997), 375--400.

\bibitem[Pa]{pal}
A.~P\`al. {\em The rigid analytical regulator for the $K_2$ of Mumford curves}.
Publications of the Research Institute for Mathematical Sciences {\bf 46} (2010), 219--253.


\bibitem[Sc]{Scholl} A. Scholl, {\em Integral elements in $K$-theory and products of modular curves}, preprint 2007.

\bibitem[Sh1]{shimura76}
G.\ Shimura. {\em The special values of the zeta functions associated with
cusp forms}, Communications on Pure and Applied Mathematics, vol. XXIX,
(1976) 783--795.


\bibitem[Sh2]{shimura}
G.\ Shimura, {\em On a class of nearly holomorphic automorphic
forms,} Annals Math.\ {\bf 123} (1986), 347-406.

\bibitem[Sre1]{sreekantan-regulators}
R. Sreekantan, {\em Non-Archimedean regulator maps and special values of L-functions},
 Proceedings of the International Colloquium on Motives and Shimura Varieties,
 TIFR Mumbai, January 2008.

\bibitem[Sre2]{sreekantan-K1}
 R. Sreekantan,  {\em  $K_1$ of products of Drinfeld modular curves and special values of $L$-functions},
 Compositio Math. {\bf 146:4} (2010), 886--918.


 \end{thebibliography}

\end{document}
