% Paper written in AMS-LaTeX.
%\documentclass[12pt]{article}
%\usepackage{amstex}
\documentstyle[12pt,amstex,amssymb]{article}
% New definitions
\newcommand{\bp}{\ast}
\newcommand{\emb}{{\rm {Emb}}}
\newcommand{\Emb}{{\rm {Emb}}}
\newcommand{\res}{{\rm {res}}}
\newcommand{\fo}{{\frak{o}}}
\newcommand{\FP}{{\rm{FP}}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cX}{{\cal X}}
\newcommand{\cY}{{\cal Y}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cG}{{\cal G}}
\newcommand{\uR}{{\underline{R}}}
\newcommand{\Lie}{{\rm {Lie}}}
\newcommand{\Ima}{{\rm {Im}}}
\newcommand{\Div}{{\rm {Div}}}
\newcommand{\Supp}{{\rm {Supp}}}
\newcommand{\mx}{{\rm {max}}}
\newcommand{\Hom}{{\rm {Hom}}}
\newcommand{\Aut}{{\rm {Aut}}}
\newcommand{\aut}{{\rm {Aut}}}
\newcommand{\sing}{{\rm {sing}}}
\newcommand{\ord}{{\rm {ord}}}
\newcommand{\fin}{{\rm {fin}}}
\newcommand{\G}{{\Bbb G}}
\newcommand{\T}{{\Bbb T}}
\newcommand{\PP}{{\Bbb P}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cC}{{\cal C}}
\newcommand{\cT}{{\cal T}}
\newcommand{\cE}{{\cal E}}
\newcommand{\oedge}{{\stackrel{\rightarrow}{\cal E}}}
\newcommand{\Char}{{\rm {Char}}}
\newcommand{\Fitt}{{\rm {Fitt}}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} {#1} & {#2} \\ {#3} & {#4}
\end{array} \right)}
% End of new definitions
%\newcommand{\Bbb}{{\bf}}
%\newcommand{\bold}{{\bf}}
%\font\cyr=mcyr10.300
%\input diagram
\newtheorem{mtheorem}{Theorem}
\newtheorem{mlemma}[mtheorem]{Lemma}
\newtheorem{mcorollary}[mtheorem]{Corollary}
\newtheorem{mproposition}[mtheorem]{Proposition}
\newtheorem{assumptions}[mtheorem]{Assumption}
\newtheorem{assumption}[mtheorem]{Assumption}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newtheorem{lemma}[theorem]{Lemma}
\newcommand{\TT}{{\Bbb T}}
\newcommand{\That}{{\hat \TT}}
\newcommand{\Tate}{{ {\rm Ta}}}
\newcommand{\Sel}{{\rm Sel}}
\newcommand{\Gal}{{\rm Gal}}
\newcommand{\ds}{\displaystyle}
\newcommand{\disc}{{\rm Disc}}
\newcommand{\sk}{\vspace{0.1in}}
\newcommand{\lra}{\longrightarrow}
\newcommand{\gal}{\mbox{Gal}}
\newcommand{\Pic}{\mbox{Pic}}
\newcommand{\norm}{\mbox{norm}}
\newcommand{\Norm}{\mbox{N}}
\newcommand{\Ann}{\mbox{Ann}}
\newcommand{\PGL}{{\rm \bold{PGL}}}
\newcommand{\GL}{{\rm \bold{GL}}}
\newcommand{\End}{{\rm{End}}}
\newcommand{\PSL}{{\rm {\bold PSL}}}
\newcommand{\SL}{{\rm {\bold SL}}}
\newcommand{\gm}{{\rm {\bold G}_m}}
\newcommand{\ga}{{\rm {\bold G}_a}}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\F}{{\Bbb F}}
\newcommand{\N}{{\Bbb N}}
\newcommand{\qbar}{{\bar\Q}}
\newcommand{\A}{{\underline{A}}}
\newcommand{\B}{{\underline{B}}}
\newcommand{\X}{{\underline{X}}}
\newcommand{\sha}{{\underline{III}}}
%\newcommand{\sha}{\mbox{{\cyr{X}}}}
\newcommand{\osum}{{\oplus}}
\newcommand{\sel}{{\rm Sel}}
\newcommand{\DD}{{\mbox{D}}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cI}{{\cal I}}
\newcommand{\cJ}{{\cal J}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cB}{{\cal B}}
%\newcommand{\PP}{{\bf P}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cO}{{\cal O}}
\newcommand{\frob}{{\mbox{Frob}}}
\newcommand{\barrho}{{\bar\rho}}
\newcommand{\bL}{{\Bbb L}}
\newcommand{\barL}{{\overline{\Bbb L}}}
\newcommand{\notdiv}{{\not\!|}}
\newcommand{\Trace}{{\mbox{Tr}}}
\newcommand{\uV}{{\underline{V}}}
\newcommand{\rank}{{\mbox rank}}
%\newcommand{\nmid}{{\not|}}
\newcommand{\liminv}{\lim\limits_{\buildrel\leftarrow\over n}}
\newcommand{\limdir}{\lim\limits_{\buildrel\rightarrow\over n}}
\newcommand{\fm}{{\frak{m}}}
\newcommand{\fa}{{\frak{a}}}
\newcommand{\fp}{{\frak{p}}}
\title{Iwasawa's Main Conjecture
for elliptic curves \\
over anticyclotomic $\Z_p$-extensions}
\author{M.\ Bertolini\footnote{Partially supported by
GNSAGA (INdAM), M.U.R.S.T.,\ and
the EC.}
\\ H.\ Darmon\footnote{Partly supported by CICMA and by an NSERC research
grant.}}
\begin{document}
\maketitle
\tableofcontents
\section*{Introduction}
\sk
\noindent
Let $E$ be an elliptic curve over $\Q$,
let $p$ be an ordinary
prime for $E$, and let $K$ be an imaginary quadratic field.
Write $K_\infty/K$ for the anticyclotomic $\Z_p$-extension of $K$ and
set $G_\infty=\Gal(K_\infty/K)$.
Following a
construction of sec.~2 of
\cite{BD1} which is recalled in section \ref{sec:p-adic_L-function},
one attaches to
the data $(E,K,p)$ an anticyclotomic $p$-adic $L$-function $L_p(E,K)$
which interpolates special values of the complex
$L$-function of $E/K$ twisted by characters of $G_\infty$.
Here $L_p(E,K)$ is viewed as an element
of the Iwasawa algebra $\Lambda:=\Z_p[\![G_\infty]\!]$.
Let $\Sel(K_\infty,E_{p^\infty})$ be the $p$-primary
Selmer group attached
to $E$ over $K_\infty$, equipped with its natural $\Lambda$-module
structure, as defined in section \ref{sec:selmer}.
Write $\cC$ for the characteristic power series of the Pontrjagin
dual of
$\Sel(K_\infty,E_{p^\infty})$ (which is well-defined up to units in
$\Lambda$).
The main goal of the present work is to prove theorem
\ref{thm:main_conjecture}
below, a weak form of the Main Conjecture of Iwasawa Theory for elliptic
curves in the anticyclotomic setting.
Assume from now on
that $(E,K,p)$ satisfy the mild technical hypotheses
(assumption \ref{assumptions})
stated at the end of this introduction.
\begin{mtheorem}
\label{thm:main_conjecture}
The characteristic power series $\cC$ divides the $p$-adic $L$-function
$L_p(E,K)$.
\end{mtheorem}
Denote by $L_p(E,K,s)$ the $p$-adic Mellin transform of
the measure defined by the element $L_p(E,K)$ of $\Lambda$.
Let $r$ be the rank of the Mordell-Weil group $E(K)$.
The next result follows by
combining theorem \ref{thm:main_conjecture} with standard techniques
of Iwasawa theory.
\begin{mcorollary}
\label{cor:main1}
${\rm{ord}}_{s=1}L_p(E,K,s)\ge r$.
\end{mcorollary}
A program of study of $L_p(E,K,s)$ in
the spirit of the work of Mazur, Tate and Teitelbaum \cite{mtt} is outlined
in
\cite{BD1}, and partially carried out in \cite{BD2}--\cite{BD5}. In
particular, section 4 of
\cite{BD1} formulates a conjecture predicting the exact order of vanishing
of
$L_p(E,K,s)$ at $s=1$.
More precisely, set $E^+=E$ and let $E^-$
be the elliptic curve over $\Q$ obtained by twisting $E$ by $K$.
Write $r^\pm$ for
the rank of $E^\pm(\Q)$, so that
$r=r^+ + r^-$.
Set
$\tilde r^\pm=r^\pm+\delta^\pm$, where
$$\delta^\pm= \left\{
\begin{array}{ll}
1 & \mbox{ if $E^\pm$ has split
multiplicative reduction at $p$}, \\
0 & \mbox{ otherwise}.
\end{array} \right. $$
Finally set $\rho:=\max(\tilde r^+,\tilde r^-)$ and $\tilde r:=\tilde r^+
+\tilde r^-$. Conjecture
4.2 of \cite{BD1} predicts that
\begin{equation}
\label{eqn:conjeq}
\ord_{s=1}L_p(E,K,s)=2\rho=\tilde r+|\tilde r^+
-\tilde r^-|.
\end{equation}
This conjecture indicates that
$L_p(E,K,s)$ vanishes to order strictly greater than $r$,
if either $\tilde
r> r$ or if
$\tilde r^+\not=\tilde r^-$. The first source of extra vanishing is
accounted
for by the phenomenon of exceptional
zeroes arising when $p$ is a prime of
split multiplicative reduction for $E$ over $K$, which was
discovered by Mazur, Tate and
Teitelbaum in the cyclotomic setting
\cite{mtt}. The second source of extra vanishing is specific to the
anticyclotomic setting, and may be accounted for by certain predictable
degeneracies in the anticyclotomic
$p$-adic height. (Cf. for example \cite{bd_derived_heights}.)
A more careful study of the $\Lambda$-module structure of
$\Sel(K_\infty, E_{p^\infty})$, which, in the good
ordinary reduction case is carried out in
\cite{bd_derived_regulators} and \cite{bd_derived_heights}
yields the following refinement of corollary
\ref{cor:main1} which is consistent with the conjectured equality
(\ref{eqn:conjeq}).
\begin{mcorollary}
\label{cor:main2}
If $p$ is a prime of good ordinary reduction for $E$, then
$${\rm{ord}}_{s=1}L_p(E,K,s)\ge 2\rho.$$
\end{mcorollary}
\sk\noindent
Let $\cO$ be a finite extension of $\Z_p$, and let
$\chi:G_\infty\lra \cO^\times$ be a finite order character,
extended by $\Z_p$-linearity to a homomorphism of $\Lambda$ to
$\cO$.
If $M$ is any $\Lambda$-module, write
$$ M^\chi = M\otimes_{\chi} \cO,$$
where the tensor product is taken over
$\Lambda$ via the map $\chi$.
Let $\sha(E/K_\infty)$ denote the $p$-primary part of the
Shafarevich-Tate group of $E$ over $K_\infty$.
A result of Zhang (\cite{zhang}, \S 1.4) generalising a formula
of Gross established in \cite{gross_montreal} for prime $N$ and
unramified $\chi$, relates $\chi(L_p(E,K))$ to a non-zero multiple of
the classical $L$-value $L(E/K,\chi,1)$
(where one views
$\chi$ as a complex-valued character by
choosing an embedding of $\cO$ into $\C$).
Theorem
\ref{thm:main_conjecture} combined with Zhang's
formula leads to the following corollary,
a result
which lends some new evidence for
the classical
Birch and Swinnerton-Dyer conjecture.
\begin{mcorollary}
\label{cor:mainzhang}
If $L(E/K,\chi,1)\ne 0$, then
$E(K_\infty)^\chi$ and $\sha(E/K_\infty)^\chi$
are finite.
\end{mcorollary}
{\em Remark}: The techniques used in the proof of theorem
\ref{thm:main_conjecture} could be adapted with
little extra effort to establish corollary
\ref{cor:mainzhang} for the $\ell$-primary part of the Shafarevich-Tate
group of $E$ for all primes $\ell$, and for arbitrary anticyclotomic
characters. Such general
results (in particular, concerning the finiteness of
the Shafarevich-Tate group) could not be obtained by the methods
of \cite{BD2}.
Another immediate consequence of theorem
\ref{thm:main_conjecture} is that $\Sel(K_\infty,E_{p^\infty})$ is
a cotorsion $\Lambda$-module whenever $L_p(E,K)$ is not
identically $0$, so that in particular one has
\begin{mcorollary}
\label{cor:mainvatsal}
If $L_p(E,K)$ is non-zero, then the
Mordell-Weil group $E(K_\infty)$ is finitely generated.
\end{mcorollary}
{\em Remark}: The non-vanishing of
$L_p(E,K)$ has been established by Vatsal
\cite{vatsal1},
\cite{vatsal2} under assumption
\ref{assumptions} combined with the assumption that $K$
has prime discriminant.
The assumption that $K$ has prime discriminant does not appear to be
essential and will be removed in forthcoming work
of Vatsal.
\sk\noindent
{\em Assumptions}:
Let $N_0$ denote the conductor of $E$, set $N=pN_0$ if $E$ has good
ordinary reduction at $p$, and set $N=N_0$ if $E$ has multiplicative
reduction at $p$ so that $p$ divides $N_0$ exactly.
Let $E_p$ be the
mod $p$ representation of $G_\Q$ attached to $E$.
For simplicity, it is assumed throughout the paper that
$(E,K,p)$ satisfies the following conditions.
\begin{assumptions}
\label{assumptions}
\begin{enumerate}
\item The prime $p$ is $\ge 5$.
\item
The Galois representation attached to
$E_p$ has image isomorphic to $\GL_2(\F_p)$.
\item
The prime $p$ does not divide the minimal degree of a modular
parametrisation $X_0(N_0)\lra E$.
\item For all primes $\ell$ such that $\ell^2$ divides $N$, and
$p$ divides $\ell+1$, the module $E_p$ is an irreducible
$I_\ell$-module.
\item
The discriminant of $K$ is prime to $N$.
\item
Write $N=p N^+N^-$,
where $N^+$ (resp.~$N^-$) is divisible only by primes
different from $p$
which are split (resp.~inert) in $K$.
Assume that $N^-$ is the square-free
product of an odd number of primes.
\end{enumerate}
\end{assumptions}
{\em Remarks}:
\noindent 1. Note that conditions 1--4 are satisfied by all but
finitely many primes once $E$ is fixed, provided that
$E$ has no complex multiplications.
Conditions 1--5 are imposed to simplify the argument and
could in principle be
relaxed. This is unlike condition
6, which -- although it may appear less natural to the
uninitiated -- is an essential feature of the situation
being studied. Indeed,
for square-free $N^-$, condition 6 on the parity of the
number of primes appearing in its factorisation is equivalent
to requiring that the sign in the functional equation of $L(E,K,\chi,s)$,
for $\chi$ a ramified character of $G_\infty$, is equal to $1$.
Without this condition, the $p$-adic $L$-function $L_p(E,K,s)$ would vanish
identically.
See \cite{BD1} for a discussion of this case where it becomes
necessary to interpolate the first derivatives $L'(E,K,\chi,1)$.
\sk
\noindent 2.
The analogue of theorem
\ref{thm:main_conjecture} for the cyclotomic
$\Z_p$-extension has been proved by Kato.
Both the proof of theorem \ref{thm:main_conjecture}
and Kato's proof of the
cyclotomic counterpart are based on
Kolyvagin's theory of Euler systems.
\sk\noindent
3. The original ``Euler system" argument of Kolyvagin
relies on the presence of a systematic supply of algebraic
points on $E$ - the so-called {\em Heegner points} defined over $K$ and
over abelian extensions of $K$.
As can be seen from corollaries
\ref{cor:mainzhang} and \ref{cor:mainvatsal},
the situation in which we have placed ourselves through
assumption \ref{assumptions} (particularly, the sixth of these)
precludes the existence
of a
norm-compatible system of points in $E(K_\infty)$.
One circumvents this difficulty by resorting to the
theory of congruences between modular forms and the Cerednik-Drinfeld
interchange of
invariants,
which, for each $n\ge 1$,
realises the Galois representation $E_{p^n}$
in the $p^n$-torsion of the Jacobian of certain
Shimura curves for which the Heegner point construction
becomes available.
By varying the Shimura curves, a compatible
collection of cohomology classes in $H^1(K_\infty,E_{p^n})$ is produced,
a collection which can be related to special values of $L$-functions and is
sufficient to control the Selmer group $\Sel(K_\infty,E_{p^\infty})$.
It should be noted that this geometric approach to the theory of Euler
Systems
does not require working
with Heegner points defined over auxiliary ring class field
extensions of $K_\infty$; in particular, Kolyvagin's derivative operators do
not appear in this argument.
\sk\noindent
\sk\noindent
{\em Acknowledgements:} It is a pleasure to thank Professor Ihara
for some useful information on his work.
\section{$p$-adic $L$-functions}
\label{sec:p-adic_L-function}
\subsection{Modular forms on quaternion algebras}
\label{subsec:modularquaternion}
Let $N^{-}$ be an arbitrary square-free integer which is
the product of an odd number of primes, and let $N^{+}$ be
an integer prime to $N^{-}$.
Let $p$ be a prime which does not divide $N^+N^-$ and write
$N=pN^+N^-$.
Let $B$ be the definite quaternion algebra
ramified at all the primes dividing $N^-$,
and let $R$ be an Eichler $\Z[1/p]$-order of level $N^+$ in $B$.
The algebra $B$ is unique up to isomorphism, and the Eichler
order $R$ is unique up
to conjugation by $B^\times$, by strong approximation.
Denote by $\cT$ the Bruhat-Tits tree of $B_p^\times/\Q_p^\times$,
where
$$B_p:= B\otimes\Q_p \simeq M_2(\Q_p).$$
The set
$\cV(\cT)$ of vertices of $\cT$ is indexed by the maximal
$\Z_p$-orders in $B_p$, two vertices being
adjacent if their intersection is an Eichler order of level $p$.
Let $\oedge(\cT)$ denote the set of ordered edges of $\cT$, i.e.,
the set of ordered pairs $(s,t)$ of adjacent vertices of $\cT$.
If $e=(s,t)$, the vertex $s$ is called the {\em source} of $e$ and the
vertex $t$ is called its {\em target}; they are denoted by $s(e)$ and
$t(e)$ respectively.
The tree $\cT$ is endowed with a natural left action of
$B_p^\times/\Q_p^\times$ by
isometries corresponding to
conjugation of maximal orders by elements of $B_p^\times$.
This action is transitive on both $\cV(\cT)$ and $\oedge(\cT)$.
Let $R^\times$ denote the group of invertible elements of $R$.
The group
$\Gamma:= R^\times/\Z[1/p]^\times$ -- a discrete
subgroup of $B_p^\times/\Q_p^\times$ in the $p$-adic topology --
acts naturally on $\cT$ and the quotient $\cT/\Gamma$ is a
finite graph.
\begin{definition}
A modular form (of weight two) on $\cT/\Gamma$ is a function
$ f:\oedge(\cT)\lra \Z_p$ satisfying
$$ f(\gamma e) = f(e), \quad \mbox{for all }\gamma\in
\Gamma.$$
\end{definition}
Denote by $S_2(\cT/\Gamma)$ the space of such modular forms. It is a
free $\Z_p$-module of finite rank.
More generally, if $Z$ is any ring,
denote by $S_2(\cT/\Gamma, Z)$ the space of
$\Gamma$-invariant functions on $\oedge(\cT)$ with values in $Z$.
\sk\noindent
{\bf Duality}. Let $e_1,\ldots,e_s$ be a set of representatives for the
orbits of $\Gamma$ acting on $\oedge(\cT)$, and let $w_j$ be the
cardinality of the finite group Stab$_\Gamma(e_j)$.
The space $S_2(\cT/\Gamma)$ is endowed with a $\Z_p$-bilinear
pairing defined
by
\begin{equation}
\label{eqn:formpairing}
\langle f_1,f_2 \rangle = \sum_{i=1}^s w_i f_1(e_i) f_2(e_i).
\end{equation}
This pairing is non-degenerate so that it identifies
$S_2(\cT/\Gamma)\otimes \Q_p$
with its $\Q_p$-dual.
\sk\noindent
{\bf Hecke operators}.
Let $\ell\ne p$ be a prime which does not divide $p$.
Choose an element $M_\ell$ of reduced norm $\ell$
in the $\Z[1/p]$-order $R$ that was used to define $\Gamma$.
The double coset $\Gamma M_\ell \Gamma$ decomposes as a disjoint
union of left cosets:
\begin{equation}
\Gamma M_\ell \Gamma = \gamma_1 \Gamma \cup \cdots \cup
\gamma_{t}\Gamma.
\end{equation}
Here $t=\ell+1$ (resp. $\ell$, $1$) if $\ell$ does not divide $N^+N^-$
(resp. divides $N^+$, $N^-$).
The function $f_{|\ell}$ defined on $\oedge(\cT)$ by the rule
\begin{equation}
f_{|\ell}(e) = \sum_{i=1}^{t} f(\gamma^{-1} e)
\end{equation}
is independent of the choice of $M_\ell$ or of the representatives
$\gamma_1,\ldots,\gamma_{t}$, and the assignment
$f\mapsto f_{|\ell}$ is a linear endomorphism of
$S_2(\cT/\Gamma)$, called
the
$\ell$-th Hecke operator at $\ell$ and denoted
$T_\ell$ if $\ell$ does not divide $N$, and $U_\ell$ if
$\ell$ divides $N^+N^-$.
Associated to the prime $p$ there is a Hecke operator denoted $U_p$
and
defined by the rule
\begin{equation}
(U_p f)(e) =
\sum_{s(e')=t(e)} f(e').
\end{equation}
The Hecke operators $T_\ell$ (with $\ell\notdiv N$) are called the
{\em good} Hecke operators. They are self-adjoint for the pairing
on $S_2(\cT/\Gamma)$ defined in (\ref{eqn:formpairing}):
\begin{equation}
\langle T_\ell f_1,f_2\rangle = \langle f_1,T_\ell f_2\rangle.
\end{equation}
\sk\noindent
{\bf Oldforms and Newforms}.
A form $f\in S_2(\cT/\Gamma)$ is said to be $p$-old
if there exist $\Gamma$-invariant functions $f_1$ and $f_2$ on
$\cV(\cT)$ such that
\begin{equation}
f(e) = f_1(s(e)) + f_2(t(e)).
\end{equation}
A form which is orthogonal to all the old forms is said to be $p$-new.
The form $f$ is $p$-new if and only if $f$ is
{\em harmonic} in the sense that it satisfies
\begin{equation}
\sum_{s(e)=v} f(e) = 0, \quad
\sum_{t(e)=v} f(e) = 0, \quad \forall v\in \cV(\cT).
\end{equation}
\sk\noindent
{\bf $p$-isolated forms}.
Let $\TT$ be the Hecke algebra
acting on the space $S_2(\cT/\Gamma)$. An
eigenform $f$ in this space
determines a maximal ideal $\fm_f$ of $\cT$.
The eigenform $f$
is said to be $p$-isolated if the completion of $S_2(\cT/\Gamma)$ at
$\fm_f$ is a free $\Z_p$-module of rank one. In other words,
$f$ is $p$-isolated if there are
no non-trivial congruences between $f$ and other modular forms in
$S_2(\cT/\Gamma)$.
\sk\noindent
{\bf The Jacquet-Langlands correspondence}.
The space $S_2(\cH/\Gamma_0(N))$ of classical modular forms of
weight $2$ on $\cH/\Gamma_0(N))$ is similarly endowed with an action
of Hecke operators, which will also be denoted by the
symbols $T_\ell$, $U_\ell$ and $U_p$ by abuse of notation.
Let $\phi$ be an eigenform on $\Gamma_0(N)$ which arises
from a newform $\phi_0$ of level $N_0$. It is
a simultaneous
eigenfunction for all the good Hecke operators $T_\ell$.
Assume that it is also an eigenfunction
for the Hecke operator $U_p$.
Write $a_\ell$ for the eigenvalue of
$T_\ell$ acting on $\phi$, and $\alpha_p$ for the
eigenvalue of $U_p$ acting on $\phi$.
\sk\noindent
{\em Remark}:
If $p$ does not divide $N_0$, so that $\phi$ is not new at $p$,
then the eigenvalue $\alpha_p$ is a root of the polynomial
$x^2-a_p x+p$, where $a_p$ is the eigenvalue of
$T_p$ acting on $\phi_0$.
If $p$ divides $N_0$, then $\phi=\phi_0$ and the eigenvalue $\alpha_p$
is equal to $1$ (resp $-1$) if the abelian variety attached to $\phi$
by the Eichler-Shimura construction has
split (resp. non-split) multiplicative reduction at $p$.
\begin{proposition}
\label{prop:jacquet_langlands}
There exists a modular form $f$ in $S_2(\cT/\Gamma,\C)$
satisfying
$$ T_\ell f = a_\ell f \mbox{ for all } \ell\not| N, \qquad
U_p f = \alpha_p f.$$
The form
$f$ with these properties is unique up to
multiplication by a non-zero complex number.
\end{proposition}
{\em Proof}: Suppose first that $p$ divides $N_0$, so that $\phi$ is a
newform on $\Gamma_0(N)$. Let $R_0$ be an Eichler $\Z$-order of level
$pN^+$ in the definite quaternion algebra
of discriminant $N^{-}$.
Write ${\hat R_0}= R_0\otimes {\hat \Z} =
\prod_\ell R_0\otimes \Z_\ell$,
and ${\hat B}:= {\hat R_0} \otimes\Q$.
The Jacquet-Langlands correspondence (which, in this case, can be
established using the Eichler trace formula as in \cite{eichler})
implies the existence of a unique function
\begin{equation}
\label{eqn:dblcst}
f: B^\times\backslash {\hat B^\times} /{\hat R_0}^\times \lra
\C
\end{equation}
satisfying $T_\ell f = a_\ell f$ for all $\ell\notdiv N$, and
$U_p f = \alpha_p f$.
(Where the operators $T_\ell$ and $U_p$ are the general Hecke operators
defined in terms of double cosets as in \cite{shimura_book}.)
Strong approximation identifies the double coset space appearing
in (\ref{eqn:dblcst}) with the space $R^\times \backslash B_p^\times
/(R_0)_p^\times.$
The transitive action of
$B_p^\times$ on the set of maximal orders in $B_p$ by conjugation
yields an action of $B_p^\times$ on $\cT$ by isometries, for which
the subgroup $(R_0)_p^\times$ is equal to the stabiliser of a certain
oriented edge. In this way
$B_p^\times/(R_0)_p^\times$ is identified with $\oedge(\cT)$,
and $f$ can thus be viewed as an element of $S_2(\cT/\Gamma)$.
If $p$ does not divide $N_0$, let $a_p$ denote the eigenvalue of
$T_p$ acting on $\phi_0$, and let $R_0$ denote now the
Eichler order of level $N^+$ in the quaternion
algebra $B$. As before, to the form $\phi_0$
is associated a unique function
\begin{equation}
\label{eqn:dblcstn}
f_0: B^\times\backslash {\hat B^\times} /{\hat R_0}^\times \lra
\C
\end{equation}
satisfying $T_\ell f = a_\ell f$ for all $\ell\notdiv N_0$.
As before, strong approximation makes it possible to identify
$f_0$ with a $\Gamma$-invariant function on $\cV(\cT)$.
In this description, the action of $T_p$ on $f_0$ is given by the
formula
$$T_p(f_0(v)) = \sum_{w} f_0(w),$$
where the sum is taken over the $p+1$ vertices $w$ of $\cT$ which are
adjacent to $v$.
Define functions $f_s, f_t:\oedge(\cT)\lra \C$ by the rules:
$$ f_s(e) = f_0(s(e)), \qquad f_t(e) = f_0(t(e)).$$
The forms $f_s$ and $f_t$ both satisfy
$T_\ell(g) = a_\ell g$ for all $\ell\notdiv N$, and span the
two-dimensional eigenspace of forms with this property.
A direct calculation reveals that
$$ U_p f_s = p f_t, \quad U_p f_t = -f_s + a_p f_t.$$
The function $f = f_s - \alpha_p f_t$ satisfies
$U_p f = \alpha_p f,$ and is the unique eigenform
in $S_2(\cT/\Gamma)$ with this property.
\sk\sk\noindent
{\bf The Shimura-Taniyama conjecture}.
Let $E$ be an elliptic curve as in the introduction.
For each prime $\ell$ which does not divide $N$, set
$$ a_\ell = \ell+1 - \#E(\F_\ell).$$
If $E$ has good ordinary reduction at $p$, let
$\alpha_p\in \Z_p$ be the unique root of the polynomial $x^2- a_p x+p$
which is a $p$-adic unit.
Set $\alpha_p=1$ (resp.~$-1$) if $E$ has
split (resp. non-split)
multiplicative reduction at $p$.
The following theorem is a consequence of the Shimura-Taniyama
conjecture in view of proposition \ref{prop:jacquet_langlands}.
\begin{proposition}
\label{prop:stw}
There exists an eigenform $f$ in $S_2(\cT/\Gamma)$ satisfying
$$ T_\ell f = a_\ell f, \mbox{ for all } \ell\not| N \quad
U_p f = \alpha_p f,$$
$$ f\notin p S_2(\cT/\Gamma).$$
The form
$f$ with these properties
is unique up to multiplication by a scalar in
$\Z_p^\times$.
\end{proposition}
{\em Proof}: Proposition \ref{prop:jacquet_langlands}
shows that there exists a form $f\in S_2(\cT/\Gamma,\C)$
satisfying the conclusion of proposition \ref{prop:stw}.
The eigenvalues $a_\ell$ belong to $\Z$, and, since $E$
is ordinary at $p$, the eigenvalue
$\alpha_p$ belongs to the ring of integers $\cO$
of a quadratic
extension of $\Q$ in which $p$ splits completely.
Hence, the form $f$ can be chosen to lie in $S_2(\cT/\Gamma,\cO)$.
After applying the unique embedding of $\cO$ into $\Z_p$ which sends
$\alpha_p$ to a $p$-adic unit, and rescaling $f$ appropriately,
one obtains a form in $S_2(\cT/\Gamma)$ satisfying the conclusion of
proposition \ref{prop:stw}.
\subsection{$p$-adic Rankin $L$-functions}
\label{subsec:p-adic_L-functions}
An eigenform $f$ in $S_2(\cT/\Gamma)$ is said to
be {\em ordinary} if the eigenvalue
$\alpha_p$ of $U_p$ acting on $f$
is a $p$-adic unit.
This section recalls the definition of the
$p$-adic Rankin $L$-function attached to an ordinary
form on $\cT/\Gamma$
and a quadratic algebra
$K\subset B$.
\sk\noindent
If $A$ is any $\Z$-algebra, let
\begin{equation}
\label{eqn:completion}
A_\ell = A\otimes\Z_\ell,
\quad {\hat A}=A\otimes {\hat \Z} =
\prod_\ell A_\ell.
\end{equation}
Let $K$ be a quadratic algebra which embeds in $B$.
Since $B$ is definite
of discriminant $N^-$, the algebra
$K$ is an imaginary quadratic field
in which all prime divisors of $N^-$ are inert.
Let $\cO_K$ denote the ring of integers of $K$ and let $\cO=\cO_K[1/p]$
be the maximal $\Z[1/p]$-order in $K$.
Let ${\tilde G_\infty}$ denote the group
\begin{equation}
\label{eqn:defGinftytilde}
{\tilde G_\infty}=
{\hat K}^\times/({\hat \Q^\times}
\prod_{\ell\ne p}\cO_\ell^\times K^\times)
\end{equation}
Fix an embedding
\begin{equation}
\Psi: K\lra B \quad\mbox{satisfying}\quad \Psi(K)\cap R = \Psi(\cO).
\end{equation}
Such a $\Psi$ exists if and only if all the primes dividing
$N^+$ are split in $K$.
By passing to the adelisation the embedding $\Psi$ induces a map
\begin{equation}
{\hat\Psi}:{\tilde G_\infty}\lra
B^\times\backslash {\hat B}^\times
/ \left({\hat\Q}^\times \prod_{\ell\ne p} R_\ell^\times\right).
\end{equation}
By strong approximation, the double coset space appearing
on the right has a fundamental region contained in
$B_p^\times\subset{\hat B}^\times$. In fact,
strong approximation yields a canonical identification
\begin{equation}
\eta:
B^\times\backslash {\hat B}^\times
/ \left({\hat\Q}^\times \prod_{\ell\ne p} R_\ell^\times\right)
\lra \Gamma\backslash B_{p}^\times / \Q_p^\times.
\end{equation}
The modular form $f\in S_2(\cT/\Gamma)$ determines
a pairing between ${\tilde G_\infty}$ and $\oedge(\cT)$ by the rule
\begin{equation}
\label{eqn:palphae}
[\sigma,e]_f := f(\eta{\hat\Psi}(\sigma)e) \in \Z_p.
\end{equation}
The embedding $\Psi$ induces an embedding of $K_p^\times$ into
$B_p^\times$ and hence yields an action of $K_p^\times/\Q_p^\times$
on
$\cT$. This action fixes a single vertex if $p$ is inert in $K$,
and no vertex if $p$ is split in $K$. Let
\begin{equation}
U_n := (1+p^n\cO_K\otimes\Z_p)^\times /(1+p^n\Z_p)^\times
\end{equation}
denote the standard compact subgroup of $K_p^\times/\Q_p^\times$
of level $n$.
Choose a sequence $e_1, e_2, \ldots, e_n,\ldots$
of consecutive edges on $\cT$
satisfying
\begin{equation}
\mbox{Stab}_{K_p^\times/\Q_p^\times}(e_j) = U_j, \quad
j=1,\ldots,n,\ldots.
\end{equation}
Since $\alpha_p$ is a $p$-adic unit,
the pairing defined by equation (\ref{eqn:palphae}) can be
used to
define a $\Z_p$-valued distribution $\tilde \nu_f$
on ${\tilde G_\infty}$
by the rule
\begin{equation}
\label{eqn:tildenu}
{\tilde \nu}_f(\sigma U_j) := \alpha_p^{-j}[\sigma,e_j]_f,
\end{equation}
for all compact open subsets of $\tilde G_\infty$ of the form
$\sigma U_j$ with $\sigma\in \tilde G_\infty$.
The distribution relation for $\tilde\nu_f$ is ensured by the
fact that $f$ is an
eigenform for the $U_p$ operator with eigenvalue $\alpha_p$.
The distribution
${\tilde \nu}_f$ gives rise to an element
$\tilde \cL_f$ in the completed integral group ring
$Z_p[\![{\tilde G_\infty}]\!]$
by the rule
$$ (\tilde \cL_f)_n := \sum_{g\in {\tilde G_n}} \tilde\nu_f(g U_n) \cdot
g,$$
where ${\tilde G_n}:= {\tilde G_\infty}/U_n$ so that
${\tilde G_\infty}$ is the inverse limit of the finite groups
${\tilde G_n}$.
\sk\noindent
Let $\Delta$ denote the torsion subgroup of ${\tilde G_\infty}$, and
let
\begin{equation}
\label{eqn:defginfty}
G_\infty = {\tilde G}_\infty/\Delta\simeq \Z_p.
\end{equation}
Write $\cL_f$ for the natural image of
$\tilde\cL_f$ in
the Iwasawa algebra
$$\Lambda = \Z_p[\![G_\infty]\!]
\simeq \Z_p[\![T]\!],$$ and denote by $\nu_f$ the associated
measure on $G_\infty$.
\sk\noindent
{\bf Functional equations}.
The Iwasawa algebra is equipped with the involution
$\theta\mapsto \theta^\ast$ sending
any $\sigma\in G_\infty$ to $\sigma^{-1}$.
Let $\epsilon=\pm 1$ be the sign in the functional equation
of the classical
$L$-function $L(E/\Q,s)$ attached to
$E/\Q$. Conjecturally, the value of $\epsilon$ determines
the parity of the rank of $E/\Q$. More precisely, this rank should
be even if $\epsilon=1$, and odd if $\epsilon=-1$.
Set $\epsilon_p=\epsilon$ if
$E$ does not have split multiplicative reduction over $\Q_p$, and
set $\epsilon_p=-\epsilon$ otherwise.
The sign $\epsilon_p$ is interpretted in \cite{mtt} as the sign
in the functional equation for the Mazur-Swinnerton-Dyer
$p$-adic $L$-function attached to $E/\Q$.
While this $L$-function differs markedly from the
$p$-adic Rankin $L$-function considered in this article,
one still has
\begin{lemma}
\label{lemma:sign_fe}
The equality
$$ \cL_f^\ast = \epsilon_p \cL_f$$
holds in $\Lambda$, up to multiplication by an element of
$G_\infty$.
\end{lemma}
{\em Proof}: See \cite{BD1}, proposition 2.13 and equation (11).
\sk\noindent
{\em Remark}: The ambiguity in lemma
\ref{lemma:sign_fe}
is
unavoidable, since to begin with
$\cL_f$ is only well-defined up to multiplication
by elements of $G_\infty$.
\begin{definition}
\label{def:rankin}
The anti-cyclotomic Rankin $L$-function attached to $f$ and $K$ is the
element $L_p(f,K)$ of $\Lambda$ defined by
$$ L_p(f,K) = \cL_f \cL_f^*.$$
\end{definition}
\sk\noindent
{\em Remark}: 1. Note that $L_p(f,K)$ is a well-defined element of
$\Lambda$, since multiplying $\cL_f$ by $\sigma\in G_\infty$ has the
effect of multiplying $\cL_f^\ast$ by $\sigma^{-1}$, so that
the ambiguity in the definition of $\cL_f$
arising from the choice of end in $\cT$ is cancelled out.
\sk\noindent
2. Definition
\ref{def:rankin} extends naturally, {\em mutatis mutandis},
to any eigenform $g$ in $S_2(\cT/\Gamma,Z)$, where $Z$ is
any ring in which the eigenvalue of $U_p$ acting on $g$ is invertible.
In this case the anti-cyclotomic Rankin $L$-function
$L_p(g,K)$ is simply an element
of the completed group ring $Z[\![G_\infty]\!]$.
\sk
Let $\mu_{f,K}$ be the $\Z_p$-valued
measure on $G_\infty$ associated to
$L_p(f,K)$. The
function $L_p(f,K,s)$ is defined to be the $p$-adic Mellin
transform of $\mu_{f,K}$:
$$ L_p(f,K,s) := \int_{G_\infty} g^{s-1} d\mu_{f,K}(g).$$
where
$g^{s-1}:=\exp((s-1)\log(g))$,
and $\log: G_\infty\to \Q_p$ is a choice of $p$-adic logarithm.
\sk\noindent
{\bf Interpolation properties.}
Let $\varphi$ be the normalised eigenform on $\Gamma_0(N)$
attached to $f$ via the Jacquet-Langlands correspondence, and
let $\Omega_f=\langle \varphi,\varphi\rangle$
denote the Peterson scalar product of $\varphi$ with itself.
It is known (cf.~\cite{zhang}, sec.~1.4) that the
measure $\mu_{f,K}$ on $G_\infty$ satisfies the following
$p$-adic interpolation property:
$$| \int_{\tilde G_\infty}\chi(g)d\mu_{f,K}(g) |^2 \stackrel{.}{=}
L(f,K,\chi,1)/ (\sqrt{\mbox{Disc}(K)}\Omega_f),$$
for all ramified finite order characters of $\tilde G_\infty$.
Here, the symbol $\stackrel{.}{=}$ indicates
an equality up
to a simple algebraic fudge factor expressed as a product of terms
comparatively less
important than the quantities explicitly described in the formulas.
The values of $\chi$ and $\mu_{f,K}$ are viewed as
complex
numbers by fixing an embedding of $\bar\Q_p$ in $\C$.
Note in particular that dividing
$L(f,K,\chi,1)$ by the complex period $\Omega_f$
yields an algebraic number.
\sk\noindent
{\bf Elliptic curves}. If $E$ is an elliptic curve as in the
introduction, let $f_E$ be the modular form in
$S_2(\cT/\Gamma)$ attached to it by proposition
\ref{prop:stw}. The $p$-adic $L$-function attached to $E$ and $K$ is
defined by:
\begin{equation}
L_p(E,K) := L_p(f_E,K), \quad L_p(E,K,s) := L_p(f_E,K,s).
\end{equation}
{\em Remark}: Note that $L_p(E,K)$
is only well-defined up to multiplication by
a unit in $\Z_p^\times$, since the same is true of the form
$f_E$ attached to it by proposition \ref{prop:stw}.
\section{Selmer Groups}
\label{sec:selmer}
\subsection{Galois representations and cohomology}
\label{subsec:cohomology}
Let $f$ be an ordinary eigenform in $S_2(\cT/\Gamma)$ with
coefficients in $\Z_p$, and let $K$ be a quadratic imaginary field
in which all primes dividing $N^{-}$ (resp. $N^{+}$) are
inert (resp. split). To these two objects a
$p$-adic $L$-function $L_p(f,K)$ was attached in section
\ref{sec:p-adic_L-function}. This section introduces an invariant of
a
more arithmetic nature -- the so called {\em
Selmer group} attached to
$f$ and $K$.
\sk\noindent
{\bf Galois representations}.
To $f$ is attached a continous representation
of the Galois group $G_\Q$:
$$V_f \simeq \Q_p^2,$$
satisfying
\begin{equation}
\mbox{trace}((\frob_\ell)_{|V_f}) = a_\ell(f), \quad\mbox{for all }
\ell\not|N.
\end{equation}
The action of the compact group $G_\Q$ is continous for the $p$-adic
topology on $V_f$ and hence preserves a $\Z_p$-lattice $T_f$.
Let
\begin{equation}
A_f = V_f/T_f \simeq (\Q_p/\Z_p)^2
\end{equation}
be the divisible $G_\Q$-module attached to $f$,
and let $A_{f,n}:= A_f[p^n]$ denote the $p^n$-torsion submodule of
$A_f$.
It will be occasionally convenient to denote $A_{f}$ by
$A_{f,\infty}$. Likewise
write $T_{f,n}=T_f/p^n T_f$
and set $T_{f,\infty}:= T_f$.
Note that for $n<\infty$, the modules
$A_{f,n}$ and $T_{f,n}$ are isomorphic as
$G_\Q$-modules, but the
$A_{f,n}$ fit naturally into an inductive system while the
$T_{f,n}$ are part of a projective system. It is therefore useful to
maintain the notational distinction between the two.
\sk
The fact that $f$ is ordinary at $p$ implies that $A_{f,n}$ is ordinary,
in the sense that it has a quotient $A_{f,n}^{(1)}$ which is free of rank
one over $\Z/p^n\Z$ and on which the inertia group $I_p$ at $p$
acts trivially. The kernel of the natural projection $A_{f,n}\lra
A_{f,n}^{(1)}$ is a free module of rank one over $\Z/p^n\Z$, denoted
$A_{f,n}^{(p)}$, on which $I_p$ acts via
the the $p$-adic cyclotomic character
$$ \epsilon: G_\Q\lra \Z_p^\times$$
describing the action of $G_\Q$ on the $p$-power roots of unity.
\sk
In our treatment of the Selmer group attached to $f$ and
$K$, it is convenient to make the following
technical assumption on $f$:
\begin{assumption}
\label{ass:selmer}
The Galois representation $A_{f,1}$ is surjective.
Furthermore, for all $\ell$ dividing $N_0$ exactly, the
Galois representation
$A_{f,1}$ has a unique one-dimensional subspace
$A_{f,1}^{(\ell)}$ on which
$\Gal(\bar\Q_\ell/\Q_\ell)$ acts via $\epsilon$ or
$-\epsilon$.
\end{assumption}
{\em Remark}:
1. Note that assumption \ref{ass:selmer} is automatically satisfied
for $\ell$ if $A_{f,1}$ is ramified at $\ell$,
because $A_{f,n}$ arises from
the Tate module of an abelian variety which acquires purely
toric reduction over the quadratic unramified extension of $\Q_\ell$.
If $A_{f,1}$ is
unramified at $\ell$, then the Frobenius element at $\ell$ acts on
$A_{f,1}$ with eigenvalues $\pm 1$ and $\pm\ell$, and the condition
in assumption \ref{ass:selmer}
stipulates that $p$ should not divide $\ell^2-1$.
\sk\noindent
2. For the same reason as explained in remark 1,
the maximal submodule $A_{f,n}^{(\ell)}$ on which $G_{\Q_\ell}$
acts via $\pm \epsilon$ is free of rank one over $\Z/p^n\Z$.
\begin{lemma}
\label{lemma:ribet_appl}
Assumption \ref{ass:selmer} is satisfied by the
modular form $f$ attached to the elliptic curve $E$ as in the
introduction.
\end{lemma}
{\em Proof}: The assumption that $p$ does not divide the
degree of the modular parametrisation of $E$ implies that
the newform on $\Gamma_0(N_0)$ attached to $E$ is $p$-isolated.
By Ribet's level-lowering theorem, it follows that the Galois
representation attached to $A_{f,1}$ is ramified at all primes
dividing $N_0$, and hence lemma \ref{lemma:ribet_appl}
follows from remark 1 after the statement of
assumption \ref{ass:selmer}.
\sk\noindent
{\bf $\Z_p$-extensions}.
Class field theory identifies the group
${\tilde G}_\infty$ of (\ref{eqn:defGinftytilde})
with the
Galois group of the maximal abelian extension
$\tilde K_\infty$ of $K$
which is unramified outside
of $p$ and which is of ``dihedral type" over $\Q$.
The subfield
$K_\infty :=
\tilde K_\infty^\Delta$
is called the
{\em anticyclotomic $\Z_p$-extension }
of $K$. Its Galois group over $K$
is identified with the group
$G_\infty\simeq \Z_p$ of equation (\ref{eqn:defginfty}).
Let $K_m$ be the $m$-th layer of $K_\infty/K$, so that
$\gal(K_m/K)\simeq \Z/p^m\Z$.
\sk\noindent
{\bf Galois Cohomology}.
For each $m\in \N$ and $n\in \N\cup \{\infty\}$, let
$H^1(K_m,A_{f,n})$ and
$H^1(K_m, T_{f,n})$ denote the usual continuous
Galois cohomology groups
of Gal$({\bar K_m}/K_m)$ with values in these modules.
(Note that
$$ H^1(K_m, A_f) :=
\lim_{\stackrel{\longrightarrow}{n}} H^1(K_m,A_{f,n}),
\qquad
H^1(K_m,T_f) := \lim_{\stackrel{\longleftarrow}{n}}
H^1(K_m, T_{f,n}).)$$
To study the behaviour of these groups
as $K_m$ varies over the finite
layers of the anticyclotomic $\Z_p$-extension, it is
convenient to introduce the groups
$$H^1(K_\infty,A_{f,n}):=
\lim_{\stackrel{\longrightarrow}{m}} H^1(K_m,A_{f,n}),\qquad
\hat H^1(K_\infty,T_{f,n}) = \lim_{\stackrel{\longleftarrow}{m}}
H^1(K_m, T_{f,n}),$$
where the direct limit is taken with respect to the
natural restriction maps, and the inverse limit is
taken with respect to the norm (corestriction)
maps.
The compatible actions of the group rings $\Z_p[G_m]$ on the groups
$H^1(K_m,A_{f,n})$ and
$H^1(K_m,T_{f,n})$
yield
an action of the Iwasawa algebra $\Lambda=\Z_p[\![G_\infty]\!]$
on both of
the groups $H^1(K_\infty, A_{f,n})$ and
$\hat H^1(K_\infty, T_{f,n})$.
\sk\noindent
{\bf Local cohomology groups}.
For each rational prime $\ell$, set
$$K_{m,\ell} :=
K_m\otimes \Q_\ell = \osum_{\lambda|\ell} K_{m,\lambda},$$
where the direct
sum is taken over all primes $\lambda$ of $K_m$ dividing $\ell$,
and write for any $G_{K_m}$-module X:
$$H^1(K_{m,\ell},X) := \osum_{\lambda|\ell}
H^1(K_{m,\lambda},X).$$
Set
$$H^1(K_{\infty,\ell},A_{f,n}) =
\lim_{\stackrel{\longrightarrow}{m}} H^1(K_{m,\ell},A_{f,n}),
\quad
{\hat H}^1(K_{\infty,\ell},T_{f,n}) =
\lim_{\stackrel{\longleftarrow}{m}} H^1(K_{m,\ell}, T_{f,n}) $$
for the local counterparts of $H^1(K_\infty,A_{f,n})$ and
$\hat H^1(K_\infty,T_{f,n})$.
The Iwasawa algebra $\Lambda$ acts naturally on these
modules
in a manner which is compatible with the restriction maps.
For each rational prime $\ell$, write
$$ H^1(I_{m,\ell},A_{f,n}):=
\osum_{\lambda|\ell}H^1(I_{m,\lambda},A_{f,n}),$$
where
$I_{m,\lambda}$ denotes the inertia group at $\lambda$.
\sk\noindent
{\bf Tate duality}.
Let $\ell$ be a rational prime, and let $n\in \N\cup\{\infty\}$.
The finite Galois modules $T_{f,n}=A_{f,n}$ are isomorphic to their
own Kummer duals: the Weil pairing gives rise to a canonical
$G_\Q$-equivariant pairing
$$ T_{f,n}\times A_{f,n} \lra \Z/p^n\Z(1) = \mu_{p^n}.$$
Combining this with the cup product pairing in cohomology
gives rise to the
collection of local Tate pairings at the primes
above $\ell$
over the finite layers $K_m$ in
$K_\infty$:
\begin{equation}
\langle\ ,\ \rangle_{m,\ell}: H^1(K_{m,\ell}, T_{f,n})
\times H^1(K_{m,\ell},A_{f,n}) \lra \Q_p/\Z_p
\end{equation}
gives rise, after passing to the limit with $m$, to a perfect
pairing
$$ \langle\ ,\ \rangle_\ell: \hat H^1(K_{\infty,\ell}, T_{f,n})
\times H^1(K_{\infty,\ell},A_{f,n}) \lra \Q_p/\Z_p.$$
These pairings satisfy the rule
$$ \langle \lambda\kappa,s\rangle_\ell
= \langle \kappa, \lambda^* s\rangle_\ell,$$
for all $\lambda\in \Lambda$,
and hence give an isomorphism of $\Lambda$-modules
$$ \hat H^1(K_{\infty,\ell},T_{f,n}) \lra
H^1(K_{\infty,\ell},A_{f,n})^\vee,$$
where the Pontrjagin dual $X^\vee$ of a $\Lambda$-module $X$ is
itself endowed
with a $\Lambda$-module structure by the rule
$$ \lambda f(x) := f(\lambda^* x), \quad\mbox{ for all }
\lambda\in\Lambda,
f\in X^\vee, x\in X.$$
\subsection{Definition of the Selmer group}
\noindent
{\bf Finite/singular structures}. Let $\ell\not|N$ be a rational prime.
The {\em singular part} of $H^1(K_{m,\ell},A_{f,n})$ is the group
$$ H^1_\sing(K_{m,\ell},A_{f,n}) :=
H^1(I_\ell,A_{f,n}).$$
There is a natural map
arising from restriction -- the so-called {\em residue map} --
$$\partial_\ell:H^1(K_{m,\ell},A_{f,n})
\lra H^1_\sing(K_{m,\ell},A_{f,n}). $$
Let $H^1_\fin(K_{m,\ell},A_{f,n})$ denote the kernel of
$\partial_\ell$. The classes in $H^1_\fin(K_{m,\ell},A_{f,n})$
are sometimes called the {\em finite} or {\em unramified classes}.
Of course, identical definitions can be made in
which $A_{f,n}$ is replaced by
$T_{f,n}$.
By passing to the limit as $m\lra \infty$ (taking either
a direct or an inverse limit) the definition of the
residue map $\partial_\ell$
extends both to $H^1(K_{\infty,\ell},A_{f,n})$ and
to $\hat H^1(K_{\infty,\ell},T_{f,n})$ and
the groups
$$H^1_{\fin}(K_{\infty,\ell},A_{f,n}), \qquad
\hat H^1_{\fin}(K_{\infty,\ell},T_{f,n}),$$
$$H^1_{\sing}(K_{\infty,\ell},A_{f,n}),\qquad
\hat H^1_{\sing}(K_{\infty,\ell},T_{f,n})$$
are defined in the natural way.
\sk
Let $\ell$ be a prime dividing $N_0$ exactly.
Recall in this case
(in view of assumption \ref{ass:selmer})
the distinguished line
$A_{f,n}^{(\ell)}$ consisting of elements on which $G_{\Q_\ell}$
acts via $\pm \epsilon$.
The {\em ordinary part} of
$H^1(K_{\infty,\ell},A_{f,n})$ is defined to be
the group
$$ H^1_\ord(K_{\infty,\ell},A_{f,n}) :=
H^1(K_{\infty,\ell},A_{f,n}^{(\ell)}). $$
Finally, at the prime $p$, set
$$ H^1_\ord(K_{\infty,p},A_{f,n}) := \mbox{res}_p^{-1}\left(
H^1(I_{\infty,p},A_{f,n}^{(p)}) \right), $$
where $\res_p:
H^1(K_{\infty,p},A_{f,n})\lra
H^1(I_{\infty,p},A_{f,n}) $
is induced from the restriction maps at the (finitely many)
primes of $K_\infty/K$ above $p$.
\begin{proposition}
\label{prop:isotropy}
If $\ell$ is a prime not dividing $N$, the
groups $H^1_\fin(K_{\infty,\ell}, A_{f,n})$ and
$\hat H^1_\fin(K_{\infty,\ell},T_{f,n})$ are annihilators of
each other under the
local Tate pairing $\langle\ ,\ \rangle_\ell$.
The same is true of $H^1_\ord(K_{\infty,\ell}, A_{f,n})$
and $\hat H^1_\ord(K_{\infty,\ell}, T_{f,n})$
for $\ell |\!| N$.
In particular,
$H^1_\fin(K_{\infty,\ell},A_{f,n})$ and
$\hat H^1_\sing(K_{\infty,\ell},T_{f,n})$ are the Pontryagin duals
of each other.
\end{proposition}
{\em Proof}: The result over the finite layers $K_{m}$ follows
from standard
properties of the local Tate
pairing, and is then deduced over $K_\infty$ by
passage to the limit.
\sk\sk\noindent
Proposition \ref{prop:isotropy}
yields a perfect pairing of $\Lambda$-modules (denoted
by the same symbols $\langle\ ,\ \rangle_\ell$ by abuse of notation)
$$ \langle\ ,\ \rangle_\ell:
\hat H^1_{\sing}(K_{\infty,\ell}, T_{f,n})
\times H^1_{\fin}(K_{\infty,\ell},A_{f,n}) \lra \Q_p/\Z_p.$$
\sk\noindent
The following lemma makes explicit the structure of the local
cohomology
groups $\hat H^1(K_{\infty,\ell},T_{f,n})$
and $H^1(K_{\infty,\ell},A_{f,n})$.
Suppose that $\ell$ is a rational prime which does not divide $N$.
\begin{lemma}
\label{lemma:struct_l_split}
If $\ell$ is split in $K/\Q$, then
$$
\hat H^1_{\sing}(K_{\infty,\ell},T_{f,n})= 0,
\quad
H^1_{\fin}(K_{\infty,\ell},A_{f,n})=0.
$$
\end{lemma}
{\em Proof}: Because $(\ell)=
\lambda_1\lambda_2$ is split in $K/\Q$,
the frobenius element attached to $\lambda_1$ topologically
generates a subgroup of finite index in $G_\infty$. Hence
$K_{\infty,\ell}$ is isomorphic to a direct sum of a
finite number of copies of the unramified
$\Z_p$-extension of $\Q_\ell$. Since $A_{f,n}$ is of exponent
$p^n$, any unramified cohomology class
in $H^1(K_{m,\ell},A_{f,n})$ becomes trivial after restriction to
$H^1(K_{m',\ell},A_{f,n})$ for $m'$ sufficiently large.
This implies the second assertion; the first follows from the
non-degeneracy of the local Tate pairing.
\sk
The primes $\ell$ which are inert in $K/\Q$ exhibit a markedly
different
behaviour, because they split completely in the anticyclotomic tower.
It is the presence of such primes which accounts for some of the
essential differences between the anti-cyclotomic theory and the more
familiar Iwasawa theory of the cyclotomic $\Z_p$-extension.
\begin{lemma}
\label{lemma:struct_l_inert}
If $\ell$ is inert in $K/\Q$, then
$$
\hat H^1_{\sing}(K_{\infty,\ell},T_{f,n}) \simeq
H^1_{\sing}(K_\ell,T_{f,n})\otimes \Lambda,
$$
and
$$H^1_\fin(K_{\infty,\ell},A_{f,n}) \simeq
\Hom(H^1_{\sing}(K_\ell,T_{f,n})\otimes \Lambda, \Q_p/\Z_p).$$
\end{lemma}
{\em Proof}: Since $\ell$ is inert in $K$ and $K_\infty/\Q$ is an
extension of dihedral type, the frobenius element at $\ell$ in
$\gal(K_\infty/\Q)$ is of order two and hence $\ell$ splits
completely in $K_\infty/K$.
The choice of a prime $\lambda_m$ of $K_m$ above $\ell$ thus
determines an isomorphism
$H^1(K_{m,\ell},T_{f,n}) \lra H^1(K_\ell,T_{f,n})\otimes
\Z_p[G_m]$.
Choosing a compatible sequence of primes $\lambda_m$ of $K_m$
which lie above each other, one obtains
an isomorphism
$$
\hat H^1(K_{\infty,\ell},T_{f,n}) \simeq
H^1(K_\ell,T_{f,n})\otimes \Lambda,
$$
from the definition of the completed group ring
$\Lambda$.
The first isomorphism of the lemma now follows by passing to
the singular parts of the cohomology, while the second is a
consequence of proposition \ref{prop:isotropy}.
\sk\noindent
{\bf Selmer groups}.
Let $\ell$ be a prime not dividing $N$.
Composing restriction from $K_\infty$ to $K_{\infty,\ell}$
with $\partial_\ell$ yields residue maps on the global cohomology
groups,
still
denoted $\partial_\ell$ by an abuse of notation,
\begin{eqnarray}
\partial_\ell&:&H^1(K_\infty,A_{f,n})\to
H^1_{\sing}(K_{\infty,\ell},A_{f,n}), \\
\label{eqn:resproj}
\partial_\ell &:& \hat H^1(K_\infty,T_{f,n})\to
\hat H^1_{\sing}(K_{\infty,\ell},T_{f,n}).
\end{eqnarray}
Note that if $\ell$ is split in $K/\Q$, the residue map of
(\ref{eqn:resproj}) is $0$ by lemma \ref{lemma:struct_l_split}.
If $\partial_\ell(\kappa)=0$ for $\kappa\in H^1(K_\infty,A_{f,n})$
(resp.~$\hat H^1(K_{\infty},T_{f,n})$),
let
$$v_\ell(\kappa)\in H^1_{\fin}(K_{\infty,\ell},A_{f,n}) \quad (
\mbox{resp. }
\hat H^1_{\fin}(K_{\infty,\ell},T_{f,n}))$$
denote
the natural image of $\kappa$ under the restriction map at $\ell$.
\begin{definition}
The {\em Selmer group} $\Sel_{f,n}$
attached to $f$, $n$ and $K_\infty$ is
the group of elements $s$ in $ H^1(K_\infty,A_{f,n})$
satisfying
\begin{enumerate}
\item $\partial_\ell(s)=0$ for all rational primes $\ell$ not
dividing $N$.
\item The class $s$ is ordinary at the primes
$\ell |\!| N$
with the property that $A_{f,1}$ is ramified at $\ell$.
\item The class $s$ is trivial at the primes
$\ell |\!| N$
with the property that $A_{f,1}$ is unramified at $\ell$.
\item The class $s$ is trivial at the
primes $\ell$ such that $\ell^2$ divides $N^+$.
\end{enumerate}
\end{definition}
{\em Caveat}: Note that the group
$\Sel_{f,n}$ depends on the value of $N$, hence
on the modular form $f$ itself,
and not just
on the Galois representation $A_{f,n}$ attached to it.
The same remark holds for the {\em compactified Selmer group}
${\hat H}^1_S(K_\infty,T_{f,n})$ defined below:
\begin{definition}
Let $S$ be a square-free integer which is relatively prime to $N$.
The {\em compactified Selmer group} $\hat H^1_S(K_\infty, T_{f,n})$
attached to $f$, $S$ and $K_\infty$ is
the group of elements $\kappa$ in
$\hat H^1(K_\infty,T_{f,n})$
satisfying
\begin{enumerate}
\item $\partial_\ell(\kappa)=0$ for all rational primes $\ell$ not
dividing $S N$.
\item The class $\kappa$ is ordinary at the primes
$\ell |\!| N$
with the property that $T_{f,1}$ is ramified at $\ell$.
\item The class $\kappa$ is arbitrary, at the primes
$\ell |\!| N$
with the property that $T_{f,1}$ is unramified at $\ell$.
\item The class $\kappa$ is arbitrary at the
primes $\ell$ such that $\ell^2$ divides $N^+$.
\item The class $\kappa$ is arbitrary at the
primes $\ell$ dividing $S$.
\end{enumerate}
\end{definition}
\sk\noindent
{\bf Global reciprocity}.
Let $\kappa\in \hat H^1(K_\infty,T_{f,n})$ and let
$s\in H^1(K_\infty, A_{f,n})$ be
global cohomology classes.
For each rational prime $q$, let $\kappa_q$ and $s_q$ denote the
restrictions of these cohomology classes to the (semi-)local
cohomology group attached to the prime $q$.
The
global reciprocity law of class field theory
implies that
\begin{equation}
\label{eqn:globalrec}
\sum_{q} \langle\kappa_q,s_q\rangle_q = 0,
\end{equation}
where the sum is taken over all the rational primes.
In particular, if $\kappa$ belongs to $\hat H^1_S(K_\infty,T_{f,n})$
and
$s$ belongs to $\Sel_{f,n}$, then
since the local conditions defining these two groups are
orthogonal at the primes not dividing $S$,
and since $s$ has trivial residue at the primes dividing $S$,
formula (\ref{eqn:globalrec}) becomes:
$$ \sum_{q|S} \langle \partial_q(\kappa), v_q(s)\rangle_q=0.$$
Of particular interest is the following special case:
\begin{proposition}
\label{prop:global_duality}
Suppose that $\kappa$ belongs to
$\hat H^1_\ell(K_\infty,A_{f,n})$.
Then
$$ \langle \partial_\ell(\kappa),v_\ell(s)\rangle_\ell = 0,$$
for all $s\in \Sel_{f,n}$.
\end{proposition}
The strategy of the proof of theorem
\ref{thm:main_conjecture}
is to produce,
for sufficiently many primes $\ell$ that are inert in $K$,
cohomology classes
$\kappa(\ell)\in \hat H^1_\ell(K_\infty,A_{f,n})$ whose
residue
$\partial_\ell(\kappa(\ell))$
can
be related to the $p$-adic
$L$-function $L_p(f,K)$ constructed in
section \ref{sec:p-adic_L-function}.
Thanks to proposition
\ref{prop:global_duality},
the elements $\partial_\ell(\kappa(\ell))$
yield relations in a presentation for
$\sel_{f,n}^\vee$.
\section{Some Preliminaries}
\subsection{$\Lambda$-modules}
If $X$ is any module over a ring $R$, let
$\Fitt_R(X)$ denote the Fitting ideal of $X$ over $R$.
If $R=\Lambda$ and $X$ is finitely generated, let
$\Char(X)$ denote the characteristic ideal attached to $X$.
\begin{proposition}
\label{prop:Lambda}
Let $X$ be a finitely generated
$\Lambda$-module and let $\cL$ be an
element of $\Lambda$. Suppose that
$\varphi(\cL)$ belongs to
$\Fitt_{\cO}(X\otimes_\varphi\cO)$,
for all surjective
homomorphisms
$\varphi:\Lambda\lra \cO$,
where $\cO$ is a discrete valuation ring.
Then $\cL$ belongs to $\Char(X)$.
\end{proposition}
{\em Proof:} If $X$ is not $\Lambda$-torsion, then $\Fitt_\Lambda(X)=0$.
Since $$\Fitt_{\cO}(X\otimes_\varphi \cO)
=\varphi(\Fitt_\Lambda(X)),$$ it follows that
$\varphi(\cL)=0$ for
all $\varphi$, so that $\cL=0$.
Hence one may assume without loss of generality
that $X$ is a $\Lambda$-torsion module.
In that case the
structure theory of $\Lambda$-modules ensures the existence of
an exact sequence of $\Lambda$-modules:
\begin{equation}
\label{eqn:pseudo}
0\lra C_1 \lra X \lra \oplus_i \Lambda/(g_i)
\lra C_2 \lra 0,
\end{equation}
where $C_1$ and $C_2$ are finite $\Lambda$-modules and
the $g_i$ are non-zero distinguished polynomials or powers of $p$.
By definition, $g:=\prod_i g_i$ is a generator of $\Char(X)$.
By tensoring the exact sequence (\ref{eqn:pseudo}) with $\cO$
one finds that
there exist $\iota_1$ and $\iota_2$ in
$\Lambda$, not depending on $\varphi$,
such that $\iota_1$ and $\iota_2$ do not have
common irreducible factors and
$$ \varphi(\iota_i) \Fitt_{\cO}(X\otimes_\varphi \cO) \subset (\varphi(g))$$
for $i=1,2$.
It follows therefore from the assumption that $\varphi(g)$ divides
$\varphi(\iota_i \cL)$ for all $\varphi$.
Hence $g$ divides
$\iota_i \cL$ for $i=1,2$, and therefore $g$ divides $\cL$.
\subsection{Auxiliary primes}
A rational prime $\ell$ is said
to be {\em $n$-admissible} relative to
$f$ if it
satisfies the following conditions:
\begin{enumerate}
\item $\ell$ does not divide $N=pN^+N^-$;
\item $\ell$ is inert in $K/\Q$;
\item $p$ does not divide $\ell^2-1$;
\item $p^n$ divides $\ell+1- a_\ell$ or $\ell+1+a_\ell$.
\end{enumerate}
Note that if $\ell$ is $n$-admissible, the module $T_{f,n}$
is unramified at $\ell$ and
the Frobenius element over $\Q$ at $\ell$ acts
semisimply on this module with eigenvalues
$\pm\ell$ and $\pm 1$ which are distinct modulo $p$.
>From this a direct calculation shows that:
\begin{lemma}
\label{lemma:local_calc} The local cohomology groups
$H^1_{\sing}(K_\ell,T_{f,n})$ and
$H^1_{\fin}(K_\ell,T_{f,n})$ are
both isomorphic to $\Z/p^n\Z$.
\end{lemma}
{\em Proof}: The group $H^1_\sing(K_\ell,T_{f,n})$ is
identified with $H^1(I_\ell,T_{f,n})^{G_{K_\ell}}$.
Since $T_{f,n}$ is unramified at $\ell$, this first cohomology group is
identified with a group of homomorphisms, which necessarily
factor through the tame inertia group at $\ell$. The
Frobenius element over $K$
at $(\ell)$ acts on this tame inertia group as multiplication
by
$\ell^2$, while it acts on $T_{f,n}$ with eigenvalues
$\ell^2$ and $1$. The result follows
from this, in light of the fact that
$p$ does not divide $\ell^2-1$.
Similarly, the group
$H^1_{\fin}(K_\ell,A_{f,n})$ is identified with the
$G_{K_\ell}$-coinvariants of $A_{f,n}$ which are also
isomorphic to $\Z/p^n\Z$.
\begin{lemma}
\label{lemma:local_free}
The local groups $\hat H^1_{\sing}(K_{\infty,\ell},T_{f,n})$
and ${\hat H}^1_{\fin}(K_{\infty,\ell},T_{f,n})$
are each free of rank one over $\Lambda/p^n\Lambda$.
\end{lemma}
{\em Proof}: Since $\ell$ is inert in $K/\Q$, lemma
\ref{lemma:struct_l_inert} implies that
$\hat H^1(K_{\ell,\infty},T_{f,n})$ is isomorphic to
$H^1(K_\ell,T_{f,n})\otimes \Lambda$.
The result now follows from lemma \ref{lemma:local_calc}.
\sk\noindent
{\em Remark}:
\noindent 1.
Note that the $n$-admissible primes
are not the primes appearing in
Kolyvagin's study of the Selmer groups of
elliptic curves, where the condition
that $p^n$ divides $\ell+1$ was imposed.
\sk\noindent 2.
The notion of admissible prime introduced here is similar
to the
one introduced in
\cite{bd_derived_regulators}, def.~2.20, the main difference arising from
the
fact that the local cohomology groups
$H^1_{\fin}(K_{\ell},A_{f,n})$ and
$H^1_{\sing}(K_{\ell},T_{f,n})$
are both free of rank one (and not two) over
$\Z/p^n\Z$.
\sk\noindent
Suppose now that $A_{f,1}$ satisfies the irreducibility condition
2 of assumption \ref{assumptions}.
\begin{theorem}
\label{thm:chebotarev1}
Let $s$ be a non-zero element of $H^1(K,A_{f,1})$.
There exist infinitely many $n$-admissible primes $\ell$
relative to $f$
such that $\partial_\ell(s)= 0$
and $v_\ell(s)\ne 0$.
\end{theorem}
{\em Proof}: Let $\Q(A_{f,n})$ be
the extension of $\Q$
fixed by the kernel of the Galois
representation $A_{f,n}$.
It is unramified at the primes not dividing $N$.
Since the discriminant of $K$ is assumed to be prime to $N$,
the fields $\Q(A_{f,n})$ and $K$ are linearly disjoint.
Letting $M$ denote the compositum of these fields, there is
therefore a natural
inclusion
$$ \gal(M/\Q) = \gal(K/\Q)\times \gal(\Q(A_{f,n})/\Q) \subset
\{1,\tau\} \times \aut_{\Z/p^n\Z}(A_{f,n}),$$
so that elements of $\gal(M/\Q)$ can be labelled by certain
pairs $(\tau^j,T)$ with $j\in \{0,1\}$
and $T \in \Aut_{\Z/p^n\Z}(A_{f,n})$.
Let $M_s$ be the extension of $M$ cut out by the image
$\bar s$
of $s$ under restriction to
$H^1(M, A_{f,1}) = \Hom(\gal(\bar M/M),A_{f,1})$.
Assume without loss of generality that $s$ belongs to a specific
eigenspace for the action of $\tau$, so that
$$ \tau s = \delta s, \mbox{ for some } \delta \in \{1,-1\}.$$
Under this assumption, the extension $M_s$ is Galois over $\Q$,
not merely over $K$. In fact, by the assumption that
$A_{f,1}$ is an irreducible $G_\Q$-module,
$\gal(M_s/\Q)$ is identified with the semi-direct product
\begin{equation}
\gal(M_s/\Q) = A_{f,1} \rtimes \gal(M/\Q),
\end{equation}
where the quotient $\gal(M/\Q)$ acts on the abelian normal subgroup
$A_{f,1}$ of $\gal(M_s/\Q)$ by the rule
\begin{equation}
\label{eqn:rule_action}
(\tau^j,T)(v) = \delta^j \bar Tv.
\end{equation}
Here $\bar T$ denotes the natural image
of $T$ in $\Aut_{\F_p}(A_{f,1})$.
By part 2 of assumption \ref{assumptions} on the Galois representation
$A_{f,1}$, the
group $\gal(M_s/\Q)$ contains an element of the form
$(v,\tau,T)$, where
\sk
\noindent
1. The automorphism $T$ has eigenvalues $\delta$ and $\lambda$, where
$\lambda\in (\Z/p^n\Z)^\times$
is not equal to $\pm 1$ mod $p$ and has order prime to $p$.
\sk
\noindent
2. The vector $v\in A_{f,1}$ is non-zero and
belongs to the $\delta$-eigenspace for $\bar T$.
\sk
\noindent
Let $\ell\not|N$ be a rational prime which is unramified in
$M_s/\Q$ and satisfies
\begin{equation}
\label{eqn:chebotarev_condition1}
\frob_\ell(M_s/\Q) = (v,\tau,T).
\end{equation}
By the Chebotarev density theorem, there exist infinitely many
such primes. In fact, the set of such primes has
positive density.
The fact, immediate from
(\ref{eqn:chebotarev_condition1}),
that $\frob_\ell(M/\Q) = (\tau,T)$ implies that
$\ell$ is an admissible prime. To see that
$v_\ell(s)\ne 0$,
choose a prime $\lambda$ of $M$ above $\ell$,
and let $d$ be the (necessarily even)
degree of the corresponding residue field extension.
Then
$$ \frob_\lambda(M_s/M) = (v,\tau,T)^d
= v+ \delta \bar Tv + \bar T^2 v + \cdots + \delta \bar T^{d-1} v
= d v.$$
Let $\bar s$ denote the image of $s$ in
$$H^1(M,A_{f,1}) = \hom(\gal(\bar M/M),A_{f,1})$$
under restriction.
Since $d$ is prime to $p$ by property 1 of $T$,
it follows that $\bar s(\frob_\lambda(M_s/M)) = dv\ne 0$,
so that the restriction at
$\lambda$ of $\bar s$ is non-zero. Hence, so is
$v_\ell(s)$, a fortiori.
\sk\noindent
{\bf Global cohomology groups}.
Following \cite{bd_derived_regulators},
definition 2.22, a finite set $S$ of primes
is said to be {\em $n$-admissible}
relative to $f$ if
\sk\noindent
1. All $\ell\in S$ are $n$-admissible relative to $f$.
\sk\noindent
2. The map $\sel(K,A_{f,n})\lra \osum_{\ell\in S} H^1_{\fin}(K_\ell,
A_{f,n})$ is injective.
\sk\noindent
A
direct argument based on theorem \ref{thm:chebotarev1}
shows that $n$-admissible sets exist.
(See also the proof
of lemma 2.23
of \cite{bd_derived_regulators}.)
In fact, any finite
collection of $n$-admissible primes can be enlarged
to an
$n$-admissible set.
\begin{proposition}
\label{prop:h1free}
If $S$ is an $n$-admissible set, then the group
$\hat H^1_S(K_\infty,T_{f,n})$ is free
of rank $\#S$ over
$\Lambda/p^n\Lambda.$
\end{proposition}
{\em Proof}: The fact that $H^1_S(K_m,T_{f,n})$ is free over
$\Z/p^n\Z[G_m]$ is essentially theorem
3.2 of \cite{bd_derived_regulators}, whose proof
carries over, mutatis mutandis, to the
present context with its slightly modified notion of admissible prime.
Proposition \ref{prop:h1free}
follows by passing to the limit as
$m\lra \infty$.
\begin{theorem}
\label{thm:control}
Let $\fm_\Lambda$ denote the maximal ideal of $\Lambda$.
Then
\begin{enumerate}
\item The natural map from
$H^1(K,A_{f,1})\to H^1(K_\infty,A_{f,n})[\fm_\Lambda]$
induced by restriction is an isomorphism.
\item If $S$ is an $n$-admissible set, the natural map from
$\hat H^1_S(K_\infty,T_{f,n})/\fm_\Lambda$ to $H^1(K,T_{f,1})$
induced by corestriction is injective.
\end{enumerate}
\end{theorem}
{\em Proof}: Let $I_\Lambda$ denote the augmentation ideal of
$\Lambda$.
The inflation-restriction sequence from $K$ to $K_m$
gives the exact sequence
\begin{eqnarray*}
\label{eqn:esGinv}
H^1(K_m/K,A_{f,n}^{G_{K_m}})& \lra &
H^1(K,A_{f,n})\stackrel{j}{\lra} H^1(K_m,A_{f,n})[I_\Lambda] \lra
\\ & \lra &
H^2(K_m/K, A_{f,n}^{G_{K_m}}).
\end{eqnarray*}
By part 2 of assumption \ref{assumptions} in the introduction,
the module $A_{f,n}^{G_{K_m}}$ is trivial. (Otherwise,
the Galois representation attached to
$A_{f,n}$ would have solvable image, contradicting the hypotheses that
were made in the introduction.) Hence the map $j$ is an isomorphism.
Taking the $G_K$-cohomology of the exact sequence
$$ 0\lra A_{f,1} \lra A_{f,n} \stackrel{p}{\lra} A_{f,n-1} \lra 0$$
and using the fact that $A_{f,1}^{G_K}=0$ once again, shows
that the natural map
\begin{equation}
H^1(K,A_{f,1}) \lra H^1(K,A_{f,n})[p]
\end{equation}
is an isomorphism. It follows that the natural map
$$H^1(K,A_{f,1}) \lra H^1(K_m,A_{f,n})[\fm_\Lambda]$$ is
an isomorphism as well.
Part 1 of theorem \ref{thm:control} follows
by taking the
direct
limit as $m\lra \infty$.
Part 2 of theorem \ref{thm:control}
follows directly from proposition
\ref{prop:h1free}.
\subsection{Admissible pairs}
\label{sec:admissible_pairs}
Let $W_f:=\mbox{ad}_0(A_{f,1})= \Hom_0(A_{f,1},A_{f,1})$ be the
adjoint representation attached to $A_{f,1}$, i.e., the
vector space of trace zero endomorphisms of $A_{f,1}$. It is a
three-dimensional $\F_p$-vector space endowed with a natural
action of $G_\Q$ arising from the conjugation of endomorphisms.
Write $W_f^*:= \Hom(W_f,\mu_p)$ for the Kummer dual of
$W_f$.
Recall that $A_{f,1}$ is ordinary at $p$, so that there is an
exact sequence of $I_p$-modules
$$ 0\lra A_{f,1}^{(p)} \lra A_{f,1} \lra A_{f,1}^{(1)} \lra 0
$$
where $A_{f,1}^{(p)}$ represents the subspace on which
$I_p$ acts via the cyclotomic character $\epsilon$, and $A_{f,1}^{(1)}$
represents the $I_p$-coinvariants of $A_{f,1}$.
Let $$W_f^{(p)}:= \Hom(A_{f,1}^{(1)}, A_{f,1}^{(p)}).$$
It is an $I_p$-submodule of $W_f$; let $W_f^{(1)}:= W_f/W_f^{(p)}$.
The classes in $H^1(\Q_p,W_f)$ whose restriction at $p$ belong
to $H^1(I_p,W_f^{(p)})$ are called {\em ordinary} at $p$.
Likewise, if $\ell$ is a prime which divides $N$ exactly, recall the
submodules $A_{f,1}^{(\ell)}$ and $A_{f,1}^{(1)}$ on which
$G_{\Q_\ell}$ acts by $\pm \epsilon$ and $\pm 1$ respectively.
(These submodules are well-defined, by virtue of assumption
\ref{ass:selmer}.)
Set $$W_f^{(\ell)}:= \Hom(A_{f,1}^{(1)}, A_{f,1}^{(\ell)}).$$
The classes in $H^1(\Q, W_f)$ whose restriction at
$\ell$ belongs to $H^1(\Q_\ell, W_f^{(\ell})$ are called
{\em ordinary} at $\ell$.
\sk
If $\ell$ is an admissible prime for $f$, the eigenvalues of
$\frob_\ell$ acting on the Galois representation $A_{f,1}$
are $\pm 1$ and $\pm \ell$. Recall also that $\ell^2\ne 1$
belongs to
$\F_p^\times$.
Therefore, the eigenvalues of $\frob_\ell$ acting
on $W_f$ (resp. $W_f^*$) are the distinct elements
$1$, $\ell$, and $\ell^{-1}$ (resp. $\ell$, $1$ and $\ell^{2}$)
of $\F_p^\times$.
Let $W_f^{(\ell)}$ and $W_f^{*(\ell)}$
denote the one-dimensional $\F_p$-subspace on
which $\frob_\ell$ acts with eigenvalue $\ell$.
The classes in $H^1(\Q,W_f)$ whose
restriction at $\ell$ belong to
$H^1(\Q_\ell, W_f^{(\ell)})$
are called {\em ordinary} at $\ell$.
(In \cite{ramakrishna}, section 3,
these classes are referred to as {\em null cocycles}.)
Note that $H^1(\Q_\ell, W_f)$ decomposes as a direct sum of two
one-dimensional $\F_p$-subspaces,
$$ H^1(\Q_\ell,W_f) = H^1_\fin(\Q_\ell,W_f)
\osum H^1_\ord(\Q_\ell,W_f^{(\ell)}),$$
where
$$ H^1_\fin(\Q_\ell,W_f) :=
H^1(\Q_\ell^{nr}/\Q_\ell, W_f)$$
is the space of unramified cocycles. A similar remark holds for
$W_f^*$.
\sk\noindent
Let $S$ be a square-free product of $n$-admissible primes for $f$.
\begin{definition}
The $S$-{\em Selmer group} attached to $W_f$,
denoted $\Sel_S(\Q,W_f)$, is the subspace
of cohomology classes $\xi\in H^1(\Q,W_f)$ satisfying
\begin{enumerate}
\item
For all
$\ell$ which do not divide $NS$,
the image of $\xi$ in $H^1(\Q_\ell,W_f)$ belongs to
$H^1_\fin(\Q_\ell,W_f)$.
\item The class $\xi$ is ordinary at the primes
$\ell$ dividing
$NS$ exactly.
\item The class $\xi$ belongs to the kernel of the restriction
to $H^1(I_\ell,W_f)$ if $\ell$ is prime such that
$\ell^2$ divides $N^+$.
\end{enumerate}
\end{definition}
Similar definitions can be made for $\Sel_S(\Q,W_f^*)$. Note that
$H^1(\Q_\ell, W_f^{(\ell)})$ and $H^1(\Q_\ell,W_f^{*(\ell)})$ are
orthogonal to each other under the local Tate pairing.
\begin{proposition}
\label{prop:isolated_selmer}
The modular form $f$ is $p$-isolated if and only if
$\Sel_1(\Q,W_f)$ is trivial.
\end{proposition}
{\em Proof}: Let $R$ denote the universal ring attached to
deformations $\rho$ of the Galois representation $A_{f,1}$,
satisfying
\begin{enumerate}
\item The determinant of $\rho$ is the cyclotomic character
describing the action of $G_\Q$ on the $p$-power roots of unity.
\item $\rho$ is unramified outside $NS$.
\item $\rho$ is ordinary at $p$, i.e., the restriction of
$\rho$ to $I_p$ is of the form
$\mat{\epsilon}{*}{0}{1}$.
\item For all $\ell$ dividing $N^+N^-S$ exactly,
the restriction of $\rho$ to a decomposition group at $\ell$ is
ordinary, i.e., is
of the form $\mat{\epsilon}{*}{0}{1}$.
\end{enumerate}
The ring $R$ is a complete local
Noetherian $\Z_p$-algebra with residue field
$\F_p$.
Let $\fm$ denote the maximal ideal of $R$.
Standard results of deformation
theory (\cite{ddt})
identify $\fm/(p,\fm^2)$ with
the Pontryagin dual of
$\Sel_S(\Q,W_f)$.
It follows that $R=\Z_p$ if and only if
$\Sel_S(\Q,W_f)$ is trivial.
Taking $S=1$, a calculation as in \cite{wiles},
sec. 3 shows that
the ring $R$ surjects onto the ring $\TT_f$
of Hecke
operators acting on the space
$S_2(\cT/\Gamma)$, completed
at the maximal ideal attached to
$f$.
A deep result of Wiles (\cite{wiles}, \cite{ddt})
asserts that this surjection is
an isomorphism.
Hence the fact that $R=\Z_p$ is equivalent to the fact that
$\TT_f=\Z_p$, which in turn is
equivalent to the fact that the modular form
$f$ is $p$-isolated.
\sk
If $S$ is a square-free product of admissible primes for
$A_{f,1}$, let $\Sel_{(S)}(\Q,W_f)$ denote the Selmer group defined
in the same way as $\Sel_S(\Q,W_f)$ above, but with no condition imposed
at the primes of $S$.
Let $\Sel_{[S]}(\Q,W_f)$ denote the subgroup of $\Sel_S(\Q,W_f)$
consisting of classes that are trivial at the primes in $S$.
These notations can be combined: thus, if $S_1,S_2,S_3$
are pairwise coprime
square-free products of admissible primes, the group
$\Sel_{S_1(S_2)[S_3]}(\Q,W_f)$ is given the obvious meaning.
Similar definitions can be made with $W_f$ replaced by $W_f^*$.
Note that the Selmer groups
$\Sel_{(S)}(\Q,W_f)$ and $\Sel_{[S]}(\Q,W_f^{*})$ are dual
Selmer groups in the sense of \cite{ddt}, section 2.3,
and the same is true
of $\Sel_S(\Q,W_f)$ and $\Sel_S(\Q,W_f^*)$.
\begin{proposition}
\label{prop:selmeradjoint}
If $f$ is $p$-isolated, and $\ell$ is an $n$-admissible prime for $f$,
then $\Sel_{(\ell)}(\Q,W_f)$
and $\Sel_{(\ell)}(\Q,W_f^*)$
are one-dimensional $\F_p$-vector spaces.
\end{proposition}
{\em Proof}: It follows from a direct calculation of orders of
local cohomology groups, combined with theorem 2.18 of \cite{ddt},
that the groups $\Sel_1(\Q,W_f)$ and $\Sel_1(\Q,W_f^{*})$ have the same
cardinality. By proposition \ref{prop:isolated_selmer},
both these groups are trivial.
Applying theorem 2.18 of \cite{ddt} once more,
one finds that
$$\#\Sel_{(\ell)}(\Q,W_f)/\#\Sel_{[\ell]}(\Q,W_f^{*}) =p. $$
Hence $\Sel_{(\ell)}(\Q,W_f)$ is one-dimensional over $\F_p$.
The same argument, with $W_f$ and $W_f^*$ interchanged, shows
that $\Sel_{(\ell)}(\Q,W_f^*)$ is one-dimensional as well.
\sk
Suppose that $f$ is $p$-isolated so that the conclusion of
proposition \ref{prop:selmeradjoint} holds.
Given an $n$-admissible prime $\ell_1$, let $\xi_{1}$ be a
generator of the group $\Sel_{(\ell_1)}(\Q,W_f)$, and let
$\xi_{1}^*$ be a generator of
$\Sel_{(\ell_1)}(\Q,W_f^*)$.
Note that $\xi_{1}$ belongs to $\Sel_{\ell_1}(\Q,W_f)$
(ie., the restiction of $\xi_1$ at $\ell_1$ is ordinary)
if and only if $\Sel_{\ell_1}(\Q,W_f)$ is one-dimensional.
In this case, the restriction
of $\xi_1^{*}$ to $H^1(\Q_{\ell_1},W_f^*)$ is orthogonal to
$H^1(\Q_{\ell_1},W_f^{(\ell_1)})$, by
global reciprocity. Hence $\xi_1^*$
belongs to
$\Sel_{\ell_1}(\Q,W_f^*)$.
If $\ell_2\ne \ell_1$ is any $n$-admissible prime,
write $v_{\ell_2}(\xi_{1})$ for the natural image
of $\xi_{1}$ in the local cohomology group
$H^1_\fin(\Q_{\ell_2},W_f).$
\begin{proposition}
\label{prop:selmertrivadj}
\begin{enumerate}
\item If $\xi_1$ generates $\Sel_{\ell_1}(\Q,W_f)$, and
$v_{\ell_2}(\xi_1)$ and $v_{\ell_2}(\xi_1^*)$ are both non-zero, then
$\Sel_{\ell_1\ell_2}(\Q,W_f)=0$.
\item If $\xi_1$ does not belong to $\Sel_{\ell_1}(\Q,W_f)$,
and $v_{\ell_2}(\xi_1)$ is $0$,
then either $\Sel_{\ell_1\ell_2}(\Q,W_f)=0$ or
$\Sel_{\ell_2}(\Q,W_f)$ is one-dimensional.
\end{enumerate}
\end{proposition}
{\em Proof}:
1. By theorem 2.18 of \cite{ddt},
$$ \#\Sel_{\ell_1(\ell_2)}(\Q,W_f)/\#\Sel_{\ell_1[\ell_2]}(\Q,W_f^{*})=p.$$
The assumption that $v_{\ell_2}(\xi_1^*)\ne 0$ implies that
$\#\Sel_{\ell_1[\ell_2]}(\Q,W_f^*)=0$.
Hence $\Sel_{\ell_1(\ell_2)}(\Q,W_f) = \Sel_{\ell_1}(\Q,W_f)$
is one-dimensional and spanned by $\xi_1$. Now the assumption that
$v_{\ell_2}(\xi_1)\ne 0$ shows that
$\Sel_{\ell_1\ell_2}(\Q,W_f) = 0.$
\sk\noindent
2. Proposition \ref{prop:selmeradjoint} implies that
$\Sel_{(\ell_2)}(\Q,W_f)$ and $\Sel_{(\ell_2)}(\Q,W_f^{*})$ are
both one-dimensional over $\F_p$. A further application of theorem
2.18 of \cite{ddt} shows that
$\Sel_{(\ell_1\ell_2)}(\Q,W_f)$ and
$\Sel_{(\ell_1\ell_2)}(\Q,W_f^*)$ are two-dimensional.
Let $\xi_2$ and $\xi_2^{*}$ be generators for
$\Sel_{(\ell_2)}(\Q,W_f)$ and $\Sel_{(\ell_2)}(\Q,W_f^*)$ respectively,
so that $\xi_1,\xi_2$ form a basis for
$\Sel_{(\ell_1\ell_2)}(\Q,W_f)$. A direct calculation shows that
the linear combinations of $\xi_1$ and $\xi_2$ which belong
to $\Sel_{\ell_1\ell_2}(\Q,W_f)$ form a
space of dimension at most one, and that
if this space is one dimensional then
$\xi_2$ necessarily
belongs to $\Sel_{\ell_2}(\Q,W_f)$.
\begin{definition}
A pair $(\ell_1,\ell_2)$ of $n$-admissible
primes is said to be an
$n$-{\em admissible pair}
if
the Selmer group $\Sel_{\ell_1\ell_2}(\Q,W_f)$ is trivial.
\end{definition}
In addition to theorem \ref{thm:chebotarev1}
guaranteeing the existence of a plentiful supply of $n$-admissible
primes sufficient to control the Selmer group $\Sel_{f,1}$,
there arises the need for the somewhat more technical theorems
\ref{thm:chebotarev2} and
\ref{thm:chebotarev2prime}
below guaranteeing the existence of a large
supply of $n$-admissible pairs in certain
favorable circumstances.
\begin{theorem}
\label{thm:chebotarev2}
Suppose that $f$ is a $p$-isolated eigenform in
$S_2(\cT/\Gamma)$, and
let $\ell_1$ be an $n$-admissible prime for $f$.
Let $s$ be a non-zero class in
$H^1(K,A_{f,1})$.
Then
there exist infinitely many $n$-admissible primes
$\ell_2$ such that
\begin{enumerate}
\item $\partial_{\ell_2}(s)=0$ and $v_{\ell_2}(s)\ne 0$.
\item Either $(\ell_1,\ell_2)$ is an admissible pair, or
$\Sel_{\ell_2}(\Q,W_f)$ is one-dimensional.
\end{enumerate}
\end{theorem}
{\em Proof}:
Assume as in the proof of theorem \ref{thm:chebotarev1}
that $s$ belongs to a fixed eigenspace for
complex conjugation, so that
$\tau s = \delta s$ for some $\delta\in \{1,-1\}$.
Write $M=K(A_{f,1})$ and let
$M_s$ be the Galois extension of $M$ cut out by $s$ as
in the proof of theorem \ref{thm:chebotarev1}.
The fact that $f$ is $p$-isolated
implies, by proposition \ref{prop:selmeradjoint}, that
the Selmer groups $\Sel_{(\ell_1)}(\Q,W_f)$ and
and $\Sel_{(\ell_1)}(\Q,W_f^*)$ are
one-dimensional over
$\F_p$. Let $\xi$ and $\xi^*$ denote as before
generators of these spaces.
The image $\bar\xi$, $\bar \xi^*$ of $\xi$, $\xi^*$
in $H^1(M,W_f) =
\Hom(G_{M},W_f)$ and $H^1(M,W_f^*)$ cuts out extensions
$M_{\xi}$ and $M_{\xi^*}$ of $M$ whose
Galois groups are identified, via $\bar\xi$ and
$\bar\xi^*$,
with $W_{f}$ and $W_f^*$ respectively.
Let $M_{s,\xi,\xi^*}$ denote the compositum
of $M_s$, $M_\xi$, and $M_{\xi^*}$ over $M$.
Since the Galois representations
$A_{f,1}$, $W_f$ and $W_f^*$ are absolutely irreducible and
pairwise non-isomorphic, the
Galois group of $M_{s,\xi,\xi^*}$ over $\Q$
is isomorphic to the semi-direct product
$$ \gal(M_{s,\xi,\xi^*}/\Q) = (A_{f,1}\times W_f\times W_f^*)
\rtimes \gal(M/\Q),$$
where the action of the quotient
$\gal(M/\Q)$ on the abelian normal
subgroup $(A_{f,1}\times W_f\times W_f^*)$ is given by
$$(\tau^j,T) (v,w,w^*) = (\delta^j \bar Tv, \bar Tw\bar T^{-1},
\bar Tw^*\bar T^{-1} \det(T)).$$
\sk\noindent
{\bf Case 1}: Suppose that $\xi$ belongs to $\Sel_{\ell_1}(\Q,W_f)$,
so that $\xi^*$ belongs also to $\Sel_{\ell_1}(\Q,W_f^*)$.
The group $\gal(M_{s,\xi,\xi^*}/\Q)$ contains an element
of the form
$(v,w,w^*,\tau, T)$, where
\sk\noindent
1. The transformation $T$ acting on $A_{f,n}$ has eigenvalues $\delta$
and
$\lambda$,
were $\lambda$ is an element of $(\Z/p^n\Z)^\times$ of
order prime to $p$ which is $\ne \pm 1$.
\sk\noindent
2. The vector $v$ belongs to the unique line in $A_{f,1}$
on which $T$ acts by $\delta$.
\sk\noindent
3. The vector $w$ (resp. $w^*$) belongs to the
unique line in $W_f$ (resp. $W_f^*$) which is
fixed by $T$.
\sk\noindent
Let $\ell_2$ be a rational prime satisfying
\begin{equation}
\label{eqn:chebotarev_condition2}
\frob_{\ell_2}(M_{s,\xi,\xi^*}/\Q)= (v,w,w^*,\tau,T).
\end{equation}
There are infinitely such primes $\ell_2$, by the
Chebotarev density theorem.
Now, note that
\sk\noindent
1. By the same reasoning as in the proof of theorem
\ref{thm:chebotarev1}, the prime $\ell_2$ is $n$-admissible and
$v_{\ell_2}(s)\ne 0$.
\sk\noindent
2. A similar argument shows that $v_{\ell_2}(\xi)\ne 0$ and
$v_{\ell_2}(\xi^*)\ne 0$. From this
it follows, by part 1 of proposition \ref{prop:selmertrivadj},
that $\Sel_{\ell_1\ell_2}(\Q,W_f)=0$.
Hence $(\ell_1,\ell_2)$ is an $n$-admissible pair, and theorem
\ref{thm:chebotarev2} follows.
\sk\noindent
{\bf Case 2}: Suppose that $\xi$ does not belong
to $\Sel_{\ell_1}(\Q,W_f)$,
so that $\xi^*$ also does not belong to $\Sel_{\ell_1}(\Q,W_f^*)$.
Keeping the notations of case 1,
let $\ell_2$ be a rational prime satisfying
\begin{equation}
\label{eqn:chebotarev_condition2case2}
\frob_{\ell_2}(M_{s,\xi,\xi^*}/\Q)= (v,0,0,\tau,T).
\end{equation}
There are infinitely such primes $\ell_2$, by the
Chebotarev density theorem.
Note that the prime $\ell_2$ is $n$-admissible and
$v_{\ell_2}(s)\ne 0$, and that $v_{\ell_2}(\xi)$ and
$v_{\ell_2}(\xi^*)$ both vanish.
It follows from part 2 of proposition
\ref{prop:selmertrivadj} that either
$\Sel_{\ell_1\ell_2}(\Q,W_f)$ is trivial -- i.e., $(\ell_1,\ell_2)$ is
an admissible pair -- or that
$\Sel_{\ell_2}(\Q,W_f)$ is one-dimensional.
\begin{theorem}
\label{thm:chebotarev2prime}
Suppose that $f$ is a $p$-isolated eigenform in
$S_2(\cT/\Gamma)$, and
let $\ell_1$ be an $n$-admissible prime for $f$.
Let $s$ be a non-zero class in
$H^1(K,A_{f,1})$.
Suppose further that $\Sel_{\ell_1}(\Q,W_f)$ is one-dimensional
over $\F_p$. Then
there exist infinitely many $n$-admissible primes
$\ell_2$ such that
\begin{enumerate}
\item $\partial_{\ell_2}(s)=0$ and $v_{\ell_2}(s)\ne 0$.
\item $(\ell_1,\ell_2)$ is an admissible pair.
\end{enumerate}
\end{theorem}
{\em Proof}: This follows directly from the analysis of
case 1 in the proof of
theorem \ref{thm:chebotarev2}.
\sk\noindent
{\bf Congruences between modular forms}.
Let $\ell_1,\ell_2$ be $n$-admissible primes relative
to $f$.
Let $B'$ be the definite quaternion algebra
of discriminant $\disc(B)\ell_1\ell_2$,
let $R'$ be an Eichler $\Z[1/p]$-order
of level $N^{+}$ in $B'$ and let $\Gamma' := (R')^\times/\Z[1/p]^\times$.
The theory of congruences between modular forms
yields the following proposition:
\begin{proposition}
\label{prop:congadmissible}
There exists an eigenform $g\in S_2(\cT/\Gamma',\Z/p^n\Z)$
such that
\begin{equation}
\label{eqn:congform}
T_q g \equiv a_q(f) g \pmod{p^n}
\end{equation}
for all $q\ne \ell_1,\ell_2$.
If futhermore the pair $(\ell_1,\ell_2)$ is an admissible pair,
then $g$ can be lifted to an
eigenform with coefficients in
$\Z_p$ satisfying (\ref{eqn:congform}) above. This form is
$p$-isolated.
\end{proposition}
{\em Proof}: The existence of $g$ follows from standard level
raising theorems such as theorem
\ref{thm:raising}
combined with the Cerednik-Drinfeld interchange of invariants.
Its uniqueness, and the fact that $g$ is $p$-isolated when
$(\ell_1,\ell_2)$ is admissible,
follows from the fact that
$\Sel_1(\Q,W_g) = \Sel_{\ell_1\ell_2}(\Q,W_f)=0$.
Likewise it then follows that $g$ has coefficients in $\Z_p$.
\section{The Euler System Argument}
\label{sec:euler}
\subsection{The Euler system}
\label{sec:euler_system}
Section \ref{sec:construction} describes the construction of certain
global cohomology classes
$$\kappa(\ell)\in \hat H^1_\ell(K_\infty,T_{f,n}),$$
indexed by
the $n$-admissible primes $\ell$ attached to $f$.
The proof of theorem \ref{thm:main_conjecture}
relies crucially on the existence of these classes and on their
behaviour under localisation described in theorems
\ref{thm:residue} and \ref{thm:value} below.
Both theorems are instances of explicit reciprocity laws
relating these explicit cohomology classes to special values of
$L$-functions,
and form the technical heart of
the proof of theorem \ref{thm:main_conjecture}.
\begin{theorem}
\label{thm:residue}
The equality $$\partial_\ell(\kappa(\ell)) = \cL_f
\pmod{p^n}$$ holds in
$\hat H^1_\sing(K_{\infty,\ell},T_{f,n})\simeq \Lambda/p^n\Lambda$,
up to
multiplication by elements of $\Z_p^\times$ and
$G_\infty$.
\end{theorem}
Note that the ambiguity in the statement
of theorem \ref{thm:residue}
is unavoidable, since the identification of
$H^1_\sing(K_{\infty,\ell},T_{f,n})$ with
$\Lambda/p^n\Lambda$, and the element $\cL_f$,
are both only defined up to multiplication by elements in
$\Z_p^\times$ and
$G_\infty$.
\sk\noindent
Theorem \ref{thm:residue}
is proved in section
\ref{sec:explicit1}.
\sk\noindent
The second theorem
describes
the localisation of $\kappa(\ell_1)$ at an $n$-admissible
prime $\ell_2$
which is different from $\ell_1$.
Recall the discrete subgroup
$\Gamma'$ of $\PSL_2(\Q_p)$ and the $\Z/p^n\Z$-valued eigenform $g$ in
$S_2(\cT/\Gamma',\Z/p^n\Z)$ attached to $f$ and $(\ell_1,\ell_2)$ in
proposition \ref{prop:congadmissible}.
\begin{theorem}
\label{thm:value}
The equality
$$v_{\ell_2}(\kappa(\ell_1)) = \cL_g$$ holds in
$\hat H^1_{\fin}(K_{\infty,\ell_2},T_{f,n})\simeq
\Lambda/p^n\Lambda$, up to
multiplication by elements of $\Z_p^\times$ and
$ G_\infty$.
\end{theorem}
Theorem \ref{thm:value} is proved in section
\ref{sec:explicit2}.
\sk
Since the definition of $g$ is symmetric
in $\ell_1$ and $\ell_2$, one obtains the following
reciprocity formula for the classes
$\kappa(\ell)$:
\begin{corollary}
\label{cor:reciprocity}
For all pairs of $n$-admissible primes $\ell_1$, $\ell_2$
attached to
$f$,
the equality
$$ v_{\ell_1}(\kappa(\ell_2)) = v_{\ell_2}(\kappa(\ell_1))$$
holds in
$\Lambda/p^n\Lambda$, up to
multiplication by elements of $\Z_p^\times$ and
$ G_\infty$.
\end{corollary}
\subsection{The argument}
To an ordinary eigenform $f\in S_2(\cT/\Gamma)$
with coefficients in $\Z_p$
one has associated two invariants:
the
$p$-adic $L$-function $\cL_p(f,K)\in \Lambda$ (section
\ref{sec:p-adic_L-function})
and the Selmer group $\Sel_{f,n}$
(section \ref{sec:selmer}).
This section explains the proof of theorem
\ref{thm:main_conjecture}.
In our approach based on congruences between modular
forms, it is indispensable to prove the following generalisation
which is stronger
insofar as it applies to all $p$-isolated
modular eigenforms in $S_2(\cT/\Gamma)$
with coefficients in $\Z_p$ satisfying assumption
\ref{ass:selmer}.
\begin{theorem}
\label{thm:mainf}
Let $f$ be an ordinary eigenform in $S_2(\cT/\Gamma)$ with coefficients
in $\Z_p$ which is $p$-isolated, and satisfies assumption
\ref{ass:selmer}.
The characteristic power series of
$\Sel_{f,\infty}^\vee$
divides the $p$-adic $L$-function $L_p(f,K)$.
\end{theorem}
{\em Proof}:
By proposition \ref{prop:Lambda}, it suffices to show that
\begin{equation}
\label{eqn:mainphi}
\varphi(\cL_f)^2 \mbox{ belongs to }
\Fitt_{\cO}(
\Sel_{f,n}^\vee
\otimes_\varphi\cO),
\end{equation}
for all integers $n$ and all homomorphisms
$\varphi$ of $\Lambda$ into a discrete valuation ring
$\cO$. Fix $\cO$, $\varphi$, and $n$. Write $\pi$
for a uniformiser of $\cO$, and let $e:=\ord_\pi(p)$ be the
ramification degree of $\cO$ over $\Z_p$.
Write $$t_f:= \ord_\pi(\varphi(\cL_f)).$$
Assume without loss of
generality that
\sk\noindent
1. $t_f<\infty$. (Otherwise, $\varphi(\cL_f)=0$ and
(\ref{eqn:mainphi}) is trivially verified.)
\sk\noindent
2. The group $\Sel_{f,n}^\vee\otimes\cO$ is non-trivial.
(Otherwise, its Fitting ideal is equal to $\cO$ and
(\ref{eqn:mainphi}) is trivially verified.)
\sk\noindent
Theorem \ref{thm:mainf} (or rather, equation
(\ref{eqn:mainphi})) is proved by induction on
$t_f$.
\sk\noindent
We begin by describing the construction of certain cohomology
classes attached to an admissible prime $\ell$.
Let
$\ell$ be any $(n+t_f)$-admissible prime, and
enlarge $\{\ell\}$ to an $(n+t_f)$-admissible set $S$.
Let $\kappa(\ell)\in \hat H^1_S(K_\infty,T_{f,n+t_f})$ be the
cohomology class attached to $\ell $ as in section
\ref{sec:euler_system}, and denote by
$\kappa_\varphi(\ell)$ the natural image of this class
in
$$ \cM:=\hat H^1_S(K_\infty, T_{f,n+t_f})\otimes_\varphi \cO.$$
Note that this module is free over
$\cO/p^{(n+t_f)}$, by proposition \ref{prop:h1free}.
By theorem \ref{thm:residue},
$$\ord_\pi(\kappa_\varphi(\ell)) \le
\ord_\pi(\partial_\ell\kappa_\varphi(\ell))
= \ord_\pi(\varphi(\cL_f)) = t_f,$$
so that
$t:=\ord_\pi(\kappa_\varphi(\ell)) \le t_f$.
Choose an element $\tilde \kappa_\varphi(\ell) \in \cM$
satisfying
$$ \pi^t \tilde\kappa_\varphi(\ell) =
\kappa_\varphi(\ell).$$
Note that $\tilde \kappa_\varphi(\ell)$ is well defined
modulo the $\pi^t$-torsion subgroup of
$\cM$ which is
contained the kernel of the natural homomorphism
$$\hat H^1_S(K_\infty,T_{f,n+t_f})\otimes_\varphi \cO \lra
\hat H^1_S(K_\infty,T_{f,n})\otimes_\varphi \cO.$$
To remove
this ambiguity, let
$\kappa_\varphi'(\ell)$ be the natural image
of the class $\tilde\kappa_\varphi(\ell)$ in
$\hat H^1_S(K_\infty,T_{f,n})\otimes\cO.$
The key properties of the class $\kappa_\varphi'(\ell)$
are summarised in lemmas \ref{lemma:propertieskappaprime}
and \ref{lemma:relationell} below.
\begin{lemma}
\label{lemma:propertieskappaprime}
The class $\kappa_\varphi'(\ell)$ enjoys the following properties:
\begin{enumerate}
\item
$\ord_\pi(\kappa_\varphi'(\ell))=0$.
\item $\partial_q\kappa_\varphi'(\ell)=0$, for all $q\not| \ell N^-$.
\item $\ord_\pi(\partial_\ell\kappa_\varphi'(\ell)) = t_f-t.$
\end{enumerate}
\end{lemma}
{\em Proof}: The first statement follows from the fact that
$\ord_\pi(\kappa_\varphi(\ell))=t$. The second property is
a direct consequence of part 1 of
theorem \ref{thm:residue}, while the third follows from part
2 of theorem \ref{thm:residue}.
\begin{lemma}
\label{lemma:relationell}
The element $\partial_\ell(\kappa_\varphi'(\ell))$ belongs to the kernel of
the natural homomorphism
$$ \eta_\ell: H^1_\sing(K_{\infty,\ell}, T_{f,n})\otimes_\varphi
\cO \lra \Sel_{f,n}^\vee\otimes_\varphi\cO.$$
\end{lemma}
{\em Proof}:
Let $I_\varphi$ denote the kernel of $\varphi$.
By the global reciprocity law of class field theory,
the class $\tilde\kappa_\varphi(\ell)$ satisfies
\begin{equation}
\label{eqn:sumkappatilde}
\sum_{q|S} \langle \partial_q(\tilde\kappa_\varphi(\ell)),s_q\rangle_q=0,
\end{equation}
for all $s\in \Sel_{f,n+t_f}[I_\varphi]$.
(Here, $s_q$ simply denotes the natural image of $s$ in
$H^1_\fin(K_{q,\infty},A_{f,n+t_f})$.)
On the other hand, $\pi^t \tilde\kappa_\varphi(\ell)=\kappa_\varphi(\ell)$
has
trivial residue at all the primes $q\ne \ell$. Hence, for such primes,
the element $\partial_q(\tilde\kappa_\varphi(\ell))$ annihilates
$\pi^t H^1_\fin(K_{\infty,q}, A_{f,n+t_f})[I_f] \supset
H^1_\fin(K_{\infty,q}, A_{f,n})[I_f].$
Hence, if $s$ belongs to $\Sel_{f,n}[I_\varphi]$,
the terms in the sum (\ref{eqn:sumkappatilde})
corresponding to the primes $q\ne \ell$ are zero. Hence so is the term
corresponding to $\ell$.
It follows that
$\partial_\ell(\kappa_\varphi'(\ell))$
annihilates the image of $\Sel_{f,n}[I_\varphi]$ in
$H^1_\fin(K_{\infty,\ell},A_{f,n})$, as was to be shown.
\sk\noindent
We now turn to the proof of (\ref{eqn:mainphi}),
in the case where $t_f=0$, which provides the basis for the
induction argument.
\begin{proposition}
If $t_f=0$, (i.e., $\cL_f$ is a unit)
then $\Sel_{f,n}^\vee\otimes_\varphi \cO$ is trivial.
\end{proposition}
{\em Proof}: Suppose that the conclusion fails. By Nakayama's lemma, the
group
$$(\Sel_{f,n}^\vee\otimes \cO)/(\pi)=\Sel_{f,n}^\vee/\fm_\Lambda =
(\sel_{f,n}[\fm_\Lambda])^\vee $$
is non-zero.
Let $s$ be a non-trivial element of
$\sel_{f,n}[\fm_\Lambda]$.
By part 1 of theorem \ref{thm:control},
$s$ can be viewed as an element of $H^1(K,A_{f,1})$.
Invoking theorem \ref{thm:chebotarev1},
choose a $1$-admisible prime $\ell$ such that
$v_\ell(s)\ne 0$.
Consider the
cohomology class $\kappa_\varphi'(\ell)=\kappa_\varphi(\ell)$.
The residue $\partial_\ell(\kappa_\varphi(\ell))$ generates
$H^1_\sing(K_\ell,A_{f,1})$ and annihilates the non-zero
class $v_\ell(s)\in H^1_\fin(K_\ell,A_{f,1})$.
This is a contradiction in view of the non-degeneracy of the local Tate
pairing.
Hence $\Sel_{f,n}^\vee$ must be trivial.
\sk
Turning now to the general case of equation (\ref{eqn:mainphi}),
let $\Pi$ be the set of rational primes $\ell$
satisfying the following conditions:
\begin{enumerate}
\item $\ell$ is $(n+t_f)$-admissible.
\item The quantity $t=\ord_\pi(\kappa_\varphi(\ell))$
is minimal,
among all primes satisfying condition 1.
\end{enumerate}
Note that the set $\Pi$ is non-empty,
by theorem \ref{thm:chebotarev1}.
Let $t$ be the common value of $\ord_\pi(\kappa_\varphi(\ell))$ for
$\ell\in\Pi$.
\begin{lemma}
\label{lemma:drop}
One has
$t}
{\rm{ord}}_\ell\circ [\ \ ,\ \ ]=\langle\ \ ,\ \ \rangle.
\end{equation}
In view of theorem \ref{thm:raising}, choose an element $c$ of
$\Phi_\ell/\cI_{f_\ell}$ of order $p^n$, and lift
$c$ to an element $\tilde c$ of $\Phi_\ell$ of $p$-power order $p^{n'}$,
with $n'\geq n$.
In view of proposition \ref{prop:grothendieck},
fix an element $b\in\cX_\ell^\vee$ such that $\tau_\ell(b)=\tilde c$,
and let $a$ be the element of $\cX_\ell$ such that $p^{n'} b=j(a)$.
Formula (\ref{eqn:[]versus<>}) shows that ${\rm{ord}}_\ell$ of
the period $\tilde j(a)\in\cX_\ell^\vee\otimes\Q_\ell^\times$
is divisible by $p^{n'}$.
Thanks to the sequence (\ref{eqn:uniformization}),
the choice of a $p^{n'}$-root of $\tilde j(a)$ determines
an element $\tilde t$ of
$J^{(\ell)}[p^{n'}]$ defined over an unramified extension of $\Q_\ell$,
whose natural image in $\Phi_\ell$ is equal to $\tilde c$.
Writing $t$ for the image of $\tilde t$ in
$J^{(\ell)}[p^{n'}]/\cI_{f_\ell}$,
then the natural image of $t$ in $\Phi_\ell/\cI_{f_\ell}$ is equal to $c$.
Since
$${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}=
J^{(\ell)}[p^{n'}]/\cI_{f_\ell},$$
it follows that $t$ is an element of order $\geq p^n$ in
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$. Since the exponent of
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is at most $p^n$, this proves
lemma \ref{lemma:multiplicity}.
\sk\noindent
\begin{theorem}
\label{thm:multiplicity}
Under the assumptions of theorem \ref{thm:raising},
the Galois representations ${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$
and $T_{f,n}$ are isomorphic.
\end{theorem}
{\em Proof:}
\smallskip\noindent
{\em Step 1.} Let ${\frak m}_{f_\ell}$ be the maximal ideal in $\T_\ell$
containing $\cI_{f_\ell}$. (Thus, $\T_\ell/{\frak m}_{f_\ell}$ is isomorphic
to
$\Z/p\Z$.) This first step shows that
${\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell}$ is isomorphic to $T_{f,1}$.
Taking $p$-torsion in the sequence (\ref{eqn:uniformization})
yields the exact sequence of $\T_\ell[\Gal(\bar
\Q_{\ell^2}/\Q_{\ell^2})]$-modules
\begin{equation}
\label{eqn:tate1}
0\to \cX_\ell^\vee\otimes\mu_{p}\to
J^{(\ell)}[p]\to
\cX_\ell/p\to 0.
\end{equation}
After tensoring (\ref{eqn:tate1}) with $\T_\ell/{\frak m}_{f_\ell}$, one
finds
\begin{equation}
\label{eqn:tate2}
0\to
((\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY)\otimes\mu_{p}
\to
J^{(\ell)}[p]/{\frak m}_{f_\ell}\to
\cX_\ell/{\frak m}_{f_\ell}\to 0,
\end{equation}
where $\cY$ is a certain submodule of $\cX_\ell^\vee/{\frak m}_{f_\ell}$.
Taking Galois cohomology over $\Q_{\ell^2}$
of (\ref{eqn:tate2}) yields an exact sequence
\begin{eqnarray}
\label{eqn:tate3}
\cX_\ell/{\frak m}_{f_\ell}\to
H^1(\Q_{\ell^2},((\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY)\otimes\mu_{p})
& \to &
H^1(\Q_{\ell^2},J^{(\ell)}[p]/{\frak m}_{f_\ell})
\\
& \to &
H^1(\Q_{\ell^2},\cX_\ell/{\frak m}_{f_\ell}).
\end{eqnarray}
Note the identifications
$$
H^1(\Q_{\ell^2},((\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY)\otimes\mu_{p})=
((\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY)\otimes H^1(\Q_{\ell^2},\mu_{p})=
$$
$$
=((\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY)
\otimes \Q_{\ell^2}^\times/(\Q_{\ell^2}^\times)^{p}=
(\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY,
$$
where the last equality follows from the fact that
$\ell$ is an admissible prime and hence $p\not|\ell^2-1$.
Moreover,
$J^{(\ell)}[p]/{\frak m}_{f_\ell}={\rm{Ta}}_p(J^{(\ell)})/{\frak
m}_{f_\ell}$,
and
$$H^1(\Q_{\ell^2},\cX_\ell/{\frak m}_{f_\ell})=\Hom_{\rm{unr}}(\Gal(\bar
\Q_{\ell^2}/\Q_{\ell^2}),\cX_\ell/{\frak m}_{f_\ell})=
\Hom(\Z/p\Z,\cX_\ell/{\frak m}_{f_\ell})
$$
by local class field theory, since $p\not|\ell^2-1$.
Thus, (\ref{eqn:tate3}) can be re-written as
\begin{equation}
\label{eqn:tate4}
\cX_\ell/{\frak m}_{f_\ell}\to (\cX_\ell^\vee/{\frak m}_{f_\ell})/\cY
\to
H^1(\Q_{\ell^2},{\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell})\to
H^1_{\rm{unr}}(\Q_{\ell^2},\cX_\ell/{\frak m}_{f_\ell}),
\end{equation}
where the first map is induced by the monodromy pairing on $\cX_\ell$.
By proposition \ref{prop:grothendieck}, one obtains the exact sequence
\begin{equation}
\label{eqn:tate5}
0\to \bar\Phi_\ell/{\frak m}_{f_\ell}\to
H^1(\Q_{\ell^2},{\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell})\to
H^1_{\rm{unr}}(\Q_{\ell^2},\cX_\ell/{\frak m}_{f_\ell}),
\end{equation}
where $\bar\Phi_\ell/{\frak m}_{f_\ell}$ is a quotient of
$\Phi_\ell/{\frak m}_{f_\ell}$.
By the main result of \cite{BLR}, the module
${\rm{Ta}}_p(J^{(\ell)})/{\frak
m}_{f_\ell}$ is
semisimple over $\F_p[\Gal(\bar \Q/\Q)]$,
and hence is isomorphic to $k\geq 1$ copies of $T_{f,1}$ by
the Eichler-Shimura relations.
Hence, $H^1(\Q_{\ell^2},{\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell})$ is
isomorphic to
$H^1(\Q_{\ell^2},T_{f,1})^k$. The $\F_p$-vector space
$H^1(\Q_{\ell^2},T_{f,1})^k$ is $2k$-dimensional, and it can be decomposed
as
the direct sum of two $k$-dimensional subspaces, one of which generated by
unramified cohomology classes, and the other by ramified cohomology
classes. Since $\Phi_\ell/{\frak m}_{f_\ell}$ is
$1$-dimensional by theorem \ref{thm:raising}, it follows
that $\bar\Phi_\ell/{\frak m}_{f_\ell}$ is equal to
$\Phi_\ell/{\frak m}_{f_\ell}$, and that
$k$ is equal to $1$. Hence
${\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell}$ is isomorphic to $T_{f,1}$.
\smallskip\noindent
{\em Step 2.}
It remains to show that
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is isomorphic to $T_{f,n}$.
There is a natural $G_\Q$-equivariant projection
\begin{equation}
{\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}
\buildrel{\pi}\over\longrightarrow
{\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell}.
\end{equation}
In view of lemma \ref{lemma:multiplicity},
let $t$ be an element of order $p^n$ in
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$, and let
$\bar t=\pi(t)$. Since $T_{f,1}$ is irreducible, there is an element $g\in
G_\Q$ such that $\bar t$ and $\bar t^g$ are a basis for
${\rm{Ta}}_p(J^{(\ell)})/{\frak m}_{f_\ell}$. Nakayama's lemma shows that
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is generated by $t$ and $t^g$.
Moreover, since $g$ acts as an automorphism of
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$,
the order of $t^g$ is equal to the order of $t$.
Finally, one checks directly that the submodules generated by
$t$ and $t^g$ have trivial intersection, so that
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is isomorphic to $(\Z/p^n\Z)^2$.
This implies that ${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is isomorphic to
$T_{f,n}$.
\sk\noindent
Recall the quadratic imaginary field $K$ introduced in the previous
sections. Note that the $n$-admissible prime $\ell$ is inert in $K$ by
definition, so that the completion $K_\ell$ is isomorphic to $\Q_{\ell^2}$.
Given $m\geq 0$, let $\cO_m=\Z+p^{m+1}\cO_K$ be the order of
$K$ of conductor $p^{m+1}$.
Let $\tilde K_m$ be the ring class field of $K$ of conductor
$p^{m+1}$, and let
$\tilde K_\infty$ be the union of the $\tilde K_m$.
(The field $\tilde K_\infty$
was introduced in section \ref{subsec:cohomology}.)
The field $\tilde K_m$ can be constructed by
adjoining to $K$ the value $j(\cO_m)$ of the modular function $j$ (viewed
as a function of lattices) on $\cO_m$.
By the theory of complex multiplication, $\tilde K_m$ is an
abelian extension of $K$, which contains the Hilbert class field $\tilde
K$ of
$K$. The Galois group of
$\tilde K_m$ over $\tilde K$ is cyclic, of order $p^m(p-\epsilon(p))/u$,
where
$\epsilon(p)$ is equal to $1$
or $-1$ depending on whether $p$ is split or inert in $K$, respectively,
and where $u$ denotes
half the order of the group of units of $K$.
Write ${\tilde G}_m$ for the Galois group of $\tilde K_m$ over $K$, and
${\tilde G}_\infty$
for the Galois group of $\tilde K_\infty$ over $K$.
By class field theory, the Galois group $\tilde G_\infty$ is
identified with
the group $\tilde G_\infty$ defined in section
\ref{subsec:p-adic_L-functions}
in terms of the ideles of $K$.
Write $\Phi_{\ell,m}$, respectively, $\Phi'_{\ell,m}$ for
$\oplus_{\lambda |\ell}\Phi_\lambda$, where
the sum is taken over the primes $\lambda$ of $K_m$, respectively, $\tilde
K_m$
dividing $\ell$, and
$\Phi_{\lambda}$ denotes the group
of connected components of $J^{(\ell)}$ at $\lambda$.
Define $\hat\Phi_\ell$, respectively, $\tilde\Phi_\ell$
to be the inverse limit of the groups $\Phi_{\ell,m}$, respectively,
$\Phi_{\ell,m}'$
with respect to the norm maps.
Since the prime $\ell$ is inert in $K$, it splits completely in
$\tilde K_\infty/K$.
Hence, the choice of a prime of $\tilde K_\infty$ above $\ell$ identifies
$\hat\Phi_\ell$ with $\Phi_\ell\otimes\Z[\![G_\infty]\!]$, and
$\tilde\Phi_\ell$ with $\Phi_\ell\otimes\Z[\![\tilde G_\infty]\!]$.
It follows that
the identification of $\Phi_\ell/\cI_{f_\ell}$ with $\Z/p^n\Z$
of theorem \ref{thm:raising}
yields isomorphisms
$$\hat\Phi_\ell/\cI_{f_\ell}\simeq\Lambda/p^n\Lambda,\ \ \ \ \
\tilde\Phi_\ell/\cI_{f_\ell}\simeq\Z/p^n[\![\tilde G_\infty]\!].$$
\begin{corollary}
\label{cor:multiplicity}
\begin{enumerate}
\item
There is an isomorphism
$$\Phi_\ell/\cI_{f_\ell}\to H^1_{\rm{sing}}(K_{\ell},T_{f,n}),$$
which is canonical up to the choice of an identification of
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ with $T_{f,n}$.
\item
There is an isomorphism
$$\hat\Phi_\ell/\cI_{f_\ell}\to\hat
H^1_{\rm{sing}}(K_{\infty,\ell},T_{f,n}),
$$
which is canonical up to the choice of an identification of
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ with $T_{f,n}$.
\end{enumerate}
\end{corollary}
{\em Proof:} An argument similar to the proof of the first step of theorem
\ref{thm:multiplicity}, with the ideal $\cI_{f_\ell}$ replacing
${\frak m}_{f_\ell}$, yields the exact sequence (analogue of
(\ref{eqn:tate5}))
\begin{equation}
\label{eqn:tate6}
0\to \bar\Phi_\ell/{\cI}_{f_\ell}\to
H^1(K_\ell,{\rm{Ta}}_p(J^{(\ell)})/{\cI}_{f_\ell})\to
H^1_{\rm{unr}}(K_\ell,\cX_\ell/{\cI}_{f_\ell}),
\end{equation}
where $\bar\Phi_\ell/{\cI}_{f_\ell}$ is a quotient of
$\Phi_\ell/{\cI}_{f_\ell}$.
By theorem \ref{thm:multiplicity},
${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$ is isomorphic to $T_{f,n}$.
Hence
$H^1(K_\ell,{\rm{Ta}}_p(J^{(\ell)})/{\cI}_{f_\ell})$ is free of rank two
over $\Z/p^n\Z$, and it can be decomposed as the direct sum of two rank one
submodules, one generated by an unramified class, and the other by a
ramified class. Thus, $\bar\Phi_\ell/{\cI}_{f_\ell}$ is equal to
$\Phi_\ell/{\cI}_{f_\ell}$, and the first statement follows.
The second statement is a formal consequence of the first.
\sk\noindent
\section{The theory of complex multiplication}
\label{sec:heegner}
\sk\noindent
(Reference: \cite{BD1}, section 2.)\newline
Fix a positive integer $M$, a rational prime $p$, and
an imaginary quadratic field $K$ of discriminant $-D$, such that $p\mid M$
but $p^2\nmid M$, and
$(M,D)=1$. Define an integer decomposition
$$M=M^+M^-$$
such that $(M^+,M^-)=1$, and $M^+$ is divisible by $p$ and by
the primes divisors of
$M/p$ which are split in $K$.
Assume that:
\begin{enumerate}
\item
$M^-$ is squarefree,
\item
$M^-$ is the product of an even number of primes.
\end{enumerate}
Let $X=X_{M^+,M^-}$ be the Shimura curve attached in section
\ref{subsec:moduli} to an
Eichler order $\cR$ of level $M^+$ in the indefinite quaternion algebra
$\cB$ of discriminant
$M^-$. For all $m\geq 0$, there is a point $P_m$ on $X$, which has complex
multiplication by the
order $\cO_m$. More precisely, let $(A_m,\iota_m,C_m)$ be a triple
corresponding to $P_m$ via the
moduli interpretation of $X$ given in section \ref{subsec:moduli}. Write
$\End(P_m)$ for the
ring of endomorphisms of $A_m$ which commute with the action $\iota_m$ and
respect the level
structure $C_m$. Then,
$$\End(P_m)\simeq\cO_m.$$
The point $P_m$ is called a {\em Heegner point} of level $m$. By the theory
of complex multiplication,
$P_m$ is defined over $\tilde K_m$. Choose the $P_m$ so that they
are {\em
compatible}, in the sense
that for all $m\geq 0$, there is an isogeny of degree $p^2$ from $A_m$ to
$A_{m+1}$, which is
equivariant for the actions $\iota_m$ and $\iota_{m+1}$, and preserves the
level structures $C_m$ and
$C_{m+1}$.
One has the following interpretation of the sequence $\{P_m\}$ in terms of
the
Bruhat-Tits tree at $p$. Given a point $P$ on the Shimura curve
$X_{M^+/p,M^-}$, corresponding to a
triple $(A,\iota,C)$, let
$\cT^{(P)}$ denote the tree of $p$-isogenies of $P$.
The vertices of $\cT^{(P)}$
correspond to points of $X_{M^+/p,M^-}$ representing moduli related
to $(A,\iota,C)$ by an
isogeny of $p$-power degree. Two
vertices of $\cT^{(P)}$ are adjacent if the corresponding moduli are
related by an isogeny of degree $p^2$. Thus, the oriented
edges of $\cT^{(P)}$ are
naturally identified
with points on $X$.
The tree $\cT^{(P)}$ is
isomorphic to the Bruhat-Tits tree $\cT$, and has a distinguished vertex
$v_P$
corresponding to $P$.
There is a unique point
$P$ on $X_{M^+/p,M^-}$ such that:
\begin{enumerate}
\item
$\End(P)\simeq\cO_K$,
\item
$P_0$ corresponds to an edge of the tree $\cT^{(P)}$, with origin in $P$.
\end{enumerate}
Then, the points $P_m$ determine a half line of $\cT^{(P)}$ originating
from $P$, with no back-trackings.
\section{Construction of the Euler System}
\label{sec:construction}
\sk\noindent
Notations and assumptions being as in section \ref{sec:raising}, the
construction of section \ref{sec:heegner} yields a compatible family of
Heegner points
$$P_m\in X^{(\ell)}(\tilde K_m),$$ for all $m\geq 0$.
View $P_m$ as an element of the Picard group $\Pic(X^{(\ell)})(\tilde K_m)$.
Since the ideal $\cI_{f_\ell}$ is
not Eisenstein, the natural inclusion
$$J^{(\ell)}(\tilde K_m)/\cI_{f_\ell}\to
\Pic(X^{(\ell)})(\tilde K_m)/\cI_{f_\ell}$$
is an isomorphism.
Let $\alpha_p$ be the unit root
of Frobenius at $p$.
Write $P_m^\ast$ for the image of
$\alpha_p^{-m} P_m$ in $J^{(\ell)}(\tilde K_m)/\cI_{f_\ell}$.
The points $ P_m^\ast$ are norm-compatible.
Hence their images by the coboundary maps
$$J^{(\ell)}(\tilde K_m)/\cI_{f_\ell}\to H^1(\tilde
K_m,{\rm{Ta}}_p(J^{(\ell)})/
\cI_{f_\ell})$$
yield a sequence of cohomology classes which are compatible under the
corestriction maps. The choice of an
isomorphism of ${\rm{Ta}}_p(J^{(\ell)})/
\cI_{f_\ell}$ with $T_{f,n}$, which exists by
theorem \ref{thm:multiplicity}, gives a
class $\tilde\kappa(\ell)$ in $\hat H^1(\tilde K_\infty,T_{f,n})$, where
$\hat H^1(\tilde K_\infty,T_{f,n})$ denotes the inverse limit under the
corestriction maps of the
groups $H^1(\tilde K_m,T_{f,n})$. Define
$\kappa(\ell)$ in $\hat H^1(K_\infty,T_{f,n})$ to be the corestriction
from $\tilde K_\infty$ to $K_\infty$ of $\tilde\kappa(\ell)$.
In other words, writing $Q_m$ for the
norm from
$\tilde K_\infty$ to $K_\infty$ of
$P_m^\ast$, the class
$\kappa(\ell)$ is the natural image in $\hat H^1(K_\infty,T_{f,n})$
of the sequence $Q_m$ via the coboundary maps.
\sk\noindent
\section{The first explicit reciprocity law}
\label{sec:explicit1}
\sk\noindent
This section is devoted to the proof of theorem \ref{thm:residue},
notations being as in the previous sections. Recall that
the class $\kappa(\ell)$ is constructed from a family of points on the
Shimura curve $X^{(\ell)}$, and hence it can be viewed as an element of the
usual (compactified) Selmer group of $J^{(\ell)}$ over $K_\infty$ relative
to
the Galois module ${\rm{Ta}}_p(J^{(\ell)})/\cI_{f_\ell}$.
This shows that $\kappa(\ell)$ belongs to
$\hat H^1_\ell(K_\infty,T_{f,n})$.
Write $\hat J^{(\ell)}(K_\infty)/\cI_{f_\ell}$, respectively,
$\hat J^{(\ell)}(\tilde K_\infty)/\cI_{f_\ell}$ for the inverse limit
of the modules $J^{(\ell)}(K_{m,\ell})/\cI_{f_\ell}$, respectively,
$J^{(\ell)}(\tilde K_{m,\ell})/\cI_{f_\ell}$ with respect to the
norm maps.
The inverse limit of the maps of specialization to connected components
yields the maps
$$
\hat\partial_{\ell}:\hat J^{(\ell)}(K_\infty)/\cI_{f_\ell}
\to\hat\Phi_{\ell}/\cI_{f_\ell}\simeq \Lambda/p^n,$$
$$\tilde\partial_{\ell}:\hat J^{(\ell)}(\tilde K_\infty)/\cI_{f_\ell}
\to\tilde\Phi_{\ell}/\cI_{f_\ell}\simeq \Z/p^n[\![{\tilde G}_\infty]\!].$$
(The groups $\hat\Phi_\ell$ and $\tilde\Phi_\ell$ were defined in section
\ref{sec:raising}.)
\begin{lemma}
\label{lemma:residue_reformulation}
Theorem \ref{thm:residue} is implied by the
equality
$$\tilde\partial_\ell(\{P_m^\ast\})=\tilde\cL_f\pmod {p^n}.$$
\end{lemma}
{\em Remark:} Note that the elements appearing in the equality of lemma
\ref{lemma:residue_reformulation} are well defined only up
to multiplication by elements of $\Z_p^\times$ and of $\tilde G_\infty $.
\sk\noindent
{\em Proof:} The element $\hat\partial_\ell(\{Q_m\})$ is mapped
to $\partial_\ell(\kappa(\ell))$ by the isomorphism of corollary
\ref{cor:multiplicity}. Since $\hat\partial_\ell(\{Q_m\})$ is the norm
of $\tilde\partial_\ell(\{P_m^\ast\})$, and $\cL_f$ is the natural image of
$\tilde\cL_f$ in $\Lambda$, the claim follows.
\sk\noindent
In view of lemma \ref{lemma:residue_reformulation}, one is reduced to
studying the specialization of the Heegner points $P_m$ to connected
components. To begin with, it is necessary to recall
the $\ell$-adic description of the point $P_m$
given in section $5$ of \cite{BD3}, and based on Drinfeld's moduli
interpretation of the
$\ell$-adic upper half plane. Let $\bar P_m$ denote the reduction of $P_m$
modulo a fixed prime above $\ell$, and let
$$\End(P_m)\to\End(\bar P_m)$$ be the map obtained by reduction of
endomorphisms. Recall that
$\End(P_m)$ is isomorphic to the quadratic order $\cO_m$. Moreover, the
ring $\End(\bar
P_m)[\frac{1}{\ell}]$ is isomorphic to $\uR[\frac{1}{\ell}]$, where $\uR$
denotes an Eichler
order of level $N^+$ in the definite quaternion algebra $B$ of discriminant
$N^-$, which can be chosen to be independent of $m$. Extending
scalars by $\Z[\frac{1}{\ell}]$, one obtains an injection
$$\Psi_m^0:\cO_m[\frac{1}{\ell}]\to\uR[\frac{1}{\ell}].$$
It can be shown that $\Psi_m^0$ is well defined up to conjugation by
elements in
$\uR[\frac{1}{\ell}]^\times_1$. Therefore, $\Psi_m^0$ can be
identified with
an element of the space
$$(\Hom(K,B)\times\cV(\cT_\ell))/\Gamma_{\ell,1},$$
by mapping $\Psi_m^0$ to the pair $(\Psi_m,v_m)$, where $\Psi_m$ is the
extension of scalars of
$\Psi_m^0$, and $v_m$ is the vertex of $\cT_\ell$ corresponding to the
unique
maximal order of
$B_\ell$ containing $\Psi_m(\cO_m)$. The embedding $\Psi_m$ induces an
action
of $K_\ell^\times$ on
$\cH_\ell$, having two fixed points which belong to $K_\ell-\Q_\ell$, and
which are conjugate by the
generator of $\Gal(K_\ell/\Q_\ell)$. Then, $P_m$ is identified with the
image in
$\cH_\ell/\Gamma_{\ell,1}$ of one of these two points, via the
isomorphism of
theorem
\ref{thm:cerednik}. (A suitable condition of normalization specifies which
point
corresponds to $P_m$, but this will not be of use here.) This description
of $P_m$ in terms of
an $\ell$-adic argument in $\cH_\ell$ shows, in particular, that the
reduction of $P_m$
modulo
$\ell$ lands in a single irreducible component of the fiber
$X^{(\ell)}_{\F_{\ell^2}}$, defined by the
vertex $v_m\pmod{\Gamma_{\ell,1}}$ of the dual graph $\cG_\ell$. By
proposition
\ref{prop:specialization}, $v_m\pmod{\Gamma_{\ell,1}}$ computes
the image of
$P_m$ in the group of
connected components $\Phi_{\ell,m}/\cI_{f_\ell}$.
On the other hand, recalling that $p$ divides the level of the order $\uR$,
strong approximation
gives an identification
$$(\Hom(K,B)\times\cV(\cT_\ell))/\Gamma_{\ell,1}=
(\Hom(K,B)\times\oedge(\cT_p))/\Gamma_p,$$
where $\cT_p=\cT$ and $\Gamma_p=\Gamma$.
The compatibility condition on the Heegner points $P_m$ translates in the
condition
that they may be represented by pairs $(\Psi,e_m)$ in
$\Hom(K,B)\times\oedge(\cT_p)$, where $\Psi$ arises from an
embedding
$$\Psi^0:\cO_K[1/p]\to \uR[1/p]$$
which {\em does not depend} on $m$,
and where the $e_m$ determine a half line with no back-trackings in
$\cT_p$. The results of \cite{BD3}, section $5$ show that the action
of ${\tilde G}_\infty$ on the $P_m$ is compatible with the action of $\tilde
G_\infty$
on the edges $e_m$, which was defined in section
\ref{sec:p-adic_L-function} using the
embedding $\Psi$.
Fix a prime $\lambda_\infty$ of $\tilde K_\infty$ above $\ell$, and set
$\lambda_m=\lambda_\infty\cap \tilde K_m$. For
$\sigma\in G_\infty$, write $\partial_{\lambda_m}(P_m^\sigma)$ for the
natural image of $P_m^\sigma$ in the group of connected components
$\Phi_{\lambda_m}/\cI_{f_\ell}$.
With notations as in equation (\ref{eqn:tildenu}), it follows that the
equality
$$\partial_{\lambda_m}(\sigma P_m)=[\sigma,e_m]_f \pmod {p^n}$$
holds for a suitable choice of $\lambda_\infty$.
Hence,
$$\partial_{\lambda_m}(\sigma P_m^\ast)=
\alpha_p^{-m}[\sigma,e_m]_f \pmod {p^n}.$$
In view of the definition of the
$p$-adic $L$-function given in section \ref{subsec:p-adic_L-functions}, this
concludes the proof.
\section{The second explicit reciprocity law}
\label{sec:explicit2}
\sk\noindent
This section is devoted to the proof of theorem \ref{thm:value}.
Consider the map
\begin{equation}
\label{eqn:kummer}
J^{(\ell_1)}(K_{\ell_2})/\cI_{f_{\ell_1}}\to
H^1(K_{\ell_2},
{\rm{Ta}}_p(J^{(\ell_1)})/\cI_{f_{\ell_1}}),
\end{equation}
arising from Kummer theory.
Since $f$ is $p$-isolated,
the assumptions of theorem \ref{thm:raising} are satisfied
in the current setting. Thus,
by theorem \ref{thm:multiplicity},
${\rm{Ta}}_p(J^{(\ell_1)})/\cI_{f_{\ell_1}}$
is isomorphic to $T_{f,n}$ as a Galois module. Hence the map
(\ref{eqn:kummer})
yields a map
\begin{equation}
\label{eqn:kummer2}
J^{(\ell_1)}(K_{\ell_2})/\cI_{f_{\ell_1}}\to
H^1(K_{\ell_2},
T_{f,n}).
\end{equation}
The image of (\ref{eqn:kummer2}) is equal to
$H^1_{\rm{fin}}(K_{\ell_2}, T_{f,n})$, since
$T_{f,n}$ is unramified at $\ell_2$, and
$\ell_2$ is a prime of good reduction for
$J^{(\ell_1)}$.
Since $p\not=\ell_2$,
the map induced by reduction modulo $\ell_2$
\begin{equation}
\label{eqn:isomodell2}
J^{(\ell_1)}(K_{\ell_2})/\cI_{f_{\ell_1}}
\to J^{(\ell_1)}(\F_{\ell_2^2})/
\cI_{f_{\ell_1}}
\end{equation}
is an isomorphism.
Hence, by composing the inverse of (\ref{eqn:isomodell2})
with (\ref{eqn:kummer2}), and fixing an identification
of $H^1_{\rm{fin}}(K_{\ell_2},
T_{f,n})$ with $\Z/p^n\Z$,
one obtains a surjective map
\begin{equation}
\label{eqn:kummer3}
J^{(\ell_1)}(\F_{\ell_2^2})/\cI_{f_{\ell_1}}\to
\Z/p^n\Z.
\end{equation}
Let $\cS_{\ell_2}\subset X^{(\ell_1)}(\F_{\ell_2^2})$
denote the set of supersingular points
of $X^{(\ell_1)}$ in characteristic $\ell_2$, and
let $\Div(\cS_{\ell_2})$, respectively,
$\Div^0(\cS_{\ell_2})$ be the module
of formal divisors, respectively, degree zero formal divisors
with
$\Z$-coefficients supported on $\cS_{\ell_2}$.
Since the inclusion of $\Div^0(\cS_{\ell_2})$ in $\Div(\cS_{\ell_2})$
induces an identification of
$\Div^0(\cS_{\ell_2})/\cI_{f_{\ell_1}}$ with
$\Div(\cS_{\ell_2})/\cI_{f_{\ell_1}}$, one obtains a natural map
\begin{equation}
\label{eqn:jacobi}
\Div(\cS_{\ell_2})\to
J^{(\ell_1)}(\F_{\ell_2^2})/\cI_{f_{\ell_1}}.
\end{equation}
The composition of (\ref{eqn:jacobi})
with the surjection (\ref{eqn:kummer3})
yields a map
$$\gamma:\Div(\cS_{\ell_2})\to\Z/p^n\Z.$$
Let $\T_{\ell_1}$ be the Hecke algebra
acting on $X^{(\ell_1)}$. Write $T_k$ for the $k$-th Hecke operator
in $\T_{\ell_1}$ (thus the notation $T_k$ is used here also for
the operators denoted by $U_\ell$ elsewhere), and
$\bar T_k$ for the natural image of $T_k$
in $\T_{\ell_1}/\cI_{f_{\ell_1}}=\Z/p^n\Z$.
Let $\epsilon=\pm 1$ be such that $p^n$
divides $\ell_2+1 -\epsilon f_{\ell_1}(T_{\ell_2})$.
\begin{lemma}
\label{lemma:ihara}
The relations
\begin{equation}
\label{eqn:equivariance}
\gamma(T_k x)=\bar T_k\gamma(x), \ \ \ \ \
\gamma({\rm{Frob}}_{\ell_2}x)=\epsilon\gamma(x)
\end{equation}
hold for $(k,\ell_2)=1$ and for $x\in\Div(\cS_{\ell_2})$.
\end{lemma}
{\em Proof:} A direct calculation.
\sk\noindent
\begin{proposition}
\label{prop:ihara}
The map $\gamma$ is surjective.
\end{proposition}
{\em Proof:} The Shimura curve $X^{(\ell_1)}$ considered here
was defined in section \ref{subsec:moduli} in term of an
(indefinite) Eichler order $\cR^{(\ell_1)}$.
The constructions of Ihara's paper \cite{ihara2}
view the curve $X^{(\ell_1)}$ over the finite field $\F_{\ell_2^2}$
as being canonically attached to the group $\Gamma_{\ell_1,\ell_2}$
of norm one elements in $\cR^{(\ell_1)}[1/\ell_2]^\times/\{\pm 1\}$.
Let $\tilde X^{(\ell_1)}$ be the Shimura curve analogue of
$X^{(\ell_1)}$ defined by imposing a $\Gamma_1$-level structure
at $\ell_1$. The curve $\tilde X^{(\ell_1)}$
over $\F_{\ell_2^2}$ is attached by Ihara's theory to the
finite index subgroup $\tilde\Gamma_{\ell_1,\ell_2}$ of
$\Gamma_{\ell_1,\ell_2}$
of elements which are congruent to the standard unipotent matrices
modulo $\ell_1$.
Write $\tilde J^{(\ell_1)}(\F_{\ell_2^2})$ for the
$\F_{\ell_2^2}$-points of
the jacobian of $\tilde X^{(\ell_1)}$, and
$\tilde J^{(\ell_1)}(\F_{\ell_2^2})^{ss}$
for the subgroup generated by divisors supported on
supersingular points.
Since $\tilde\Gamma_{\ell_1,\ell_2}$ is torsion-free
the results of \cite{ihara2} (see in particular
remark G, page 19) establish a canonical isomorphism
\begin{equation}
\label{eqn:jss}
\tilde J^{(\ell_1)}(\F_{\ell_2^2})/
\tilde J^{(\ell_1)}(\F_{\ell_2^2})^{ss}
\simeq
\tilde\Gamma_{\ell_1,\ell_2}^{ab},
\end{equation}
where $\tilde\Gamma_{\ell_1,\ell_2}^{ab}$ denotes the abelianization
of $\tilde\Gamma_{\ell_1,\ell_2}$.
The paper \cite{ihara1}, pages 148-150, proves that the quotient in
(\ref{eqn:jss}) is annihilated by
the operator $(T_{\ell_2}-\ell_2^2-\ell_2)^2$,
where $T_{\ell_2}$ is the Hecke operator acting naturally on
$\tilde X^{(\ell_1)}$.
By the results of chapter 7 of \cite{cornut},
the cokernel of the natural map
\begin{equation}
\tilde J^{(\ell_1)}(\F_{\ell_2^2})\to
J^{(\ell_1)}(\F_{\ell_2^2})
\end{equation}
can be identified with an abelian quotient of
the (finite) image of $\Gamma_0(\ell_1)$ in $\SL_2(\Z/\ell_1\Z)$.
The claim follows from the fact that $\ell_1$ and $\ell_2$
are $n$-admissible primes.
\sk\noindent
Following the notations of section \ref{sec:admissible_pairs},
let $B'$ be the definite quaternion algebra of discriminant
$N^-\ell_1\ell_2$, $R'$ an Eichler $\Z[1/p]$-order
of level $N^+$ in $B$, and $\Gamma'$ the group
$(R')^\times/\Z[1/p]^\times$.
According to proposition \ref{prop:congadmissible},
the above data define an eigenform
$$g:\oedge(\cT)/\Gamma'\to\Z/p^n\Z,$$ congruent
to $f$ modulo $p^n$.
The results of \cite{waterhouse} on the endomorphisms
of supersingular abelian surfaces, combined with
strong approximation, identify the set $\cS_{\ell_2}$ with
$\oedge(\cT)/\Gamma'$. Thus $g$ can also be viewed as a
$\Z/p^n\Z$-valued map on $\cS_{\ell_2}$.
\begin{lemma}
\label{lemma:gamma=g}
The map $\gamma$ is equal to the map obtained by
extending $g$ by linearity,
up to possibly rescaling $g$ by a unit in $\Z/p^n\Z$.
\end{lemma}
{\em Proof:}
The action of the $k$-th Hecke operators on $g$ and on
$\gamma$ is the same when $(k,\ell_2)=1$. The raising
the level result shows that
$$g(T_{\ell_2}x)=\epsilon g(x),$$ since $p^n$
divides $\ell_2+1 -\epsilon f_{\ell_1}(T_{\ell_2})$.
Moreover, it is known that
$T_{\ell_2}x={\rm{Frob}}_{\ell_2}x$. (See \cite{ribet}, proposition 3.8.)
The result follows from proposition \ref{prop:ihara} and
lemma \ref{lemma:ihara}.
\sk\noindent
Consider the sequence $\{P_m\}$ of Heegner points $P_m\in
X^{(\ell_1)}(\tilde K_m)$, constructed in section \ref{sec:heegner}.
Fix a prime $\lambda_\infty$ of $\tilde K_\infty$ above $\ell_2$, and
let $\lambda_m=\lambda_\infty\cap \tilde K_m$.
Since $\ell_2$ is inert in $K$, the point $P_m$ reduces modulo $\lambda_m$
to a supersingular point $\bar P_m\in X^{(\ell_1)}(\F_{\lambda_m})$.
Identifying $\F_{\lambda_m}$ with $\F_{\ell_2^2}$,
$\bar P_m$ can be viewed as an element of $\cS_{\ell_2}$.
Identifying $\cS_{\ell_2}$ with $\oedge(\cT)/\Gamma'$, the sequence
$\{\bar P_m\}$ can be described by a sequence of consecutive edges
$\{e_m\}$ in $\oedge(\cT)$, modulo $\Gamma'$, in such a way that
the map
$\End(P_m)\to \End(\bar P_m)$
of reduction of endomorphisms modulo $\lambda_m$ induces
by extension of scalars an embedding
$$\Psi:K\to B',$$ which is independent of $m$.
Then, the natural Galois action
of $\tilde G_\infty$ on the $P_m$ is compatible with the action of
$\tilde G_\infty$ on the $e_m$ via $\Psi$, which was defined in section
\ref{sec:p-adic_L-function}. Writing
$$\tilde\cL_{g,m}:=\alpha_p^{-m}\sum_{\sigma\in \tilde G_m}g(
\overline{\sigma P}_m)
\cdot \sigma^{-1}
\in\Z/p^n\Z[\tilde G_m],$$
it follows that the sequence $\{\tilde\cL_{g,m}\}$ defines an element of
$\Z/p^n\Z[\![\tilde G_\infty]\!]$, equal to $\tilde\cL_g$.
Define the local cohomology groups
$$H^1_{\rm{fin}}(\tilde K_{m,\ell_2},T_{f,n}):=\oplus_{\lambda|\ell_2}
H^1_{\rm{fin}}((\tilde K_m)_\lambda,T_{f,n}),$$
where the sum is taken over all the primes of $\tilde K_m$ dividing
$\ell_2$,
and
$$\hat H^1_{\rm{fin}}(\tilde K_{\infty,\ell_2},T_{f,n}):=
\lim_{\stackrel{\longleftarrow}{m}}
H^1_{\rm{fin}}(\tilde K_{m,\ell_2},T_{f,n}),$$ where the inverse limit is
taken
with respect to
the natural corestriction maps.
The identification of $H^1_{\rm{fin}}(K_{\ell_2},T_{f,n})$ with
$\Z/p^n\Z$, together with the choice of the prime $\lambda_\infty$, yields
the identifications
$$H^1_{\rm{fin}}(\tilde K_{m,\ell_2},T_{f,n})=\Z/p^n[\tilde G_m], \ \ \ \
\hat H^1_{\rm{fin}}(\tilde K_{\infty,\ell_2},T_{f,n})=\Z/p^n[\![\tilde
G_\infty]\!].$$
In view of lemma \ref{lemma:gamma=g} and the definition of the map $\gamma$,
the image of $P_m^\ast$ in
$H^1_{\rm{fin}}(\tilde K_{m,\ell_2},T_{f,n})$ corresponds to
$\tilde\cL_{g,m}
\pmod {p^n}$, and the image of the compatible sequence $\{P_m^\ast\}$
in $\hat H^1_{\rm{fin}}(\tilde K_{\infty,\ell_2},T_{f,n})$ corresponds to
$\tilde\cL_g\pmod{p^n}$, under the above identifications.
Recall the class $\tilde\kappa(\ell_1)$, defined in
section \ref{sec:construction}
as the image of the sequence $\{P_m^\ast\}$ in
$\hat H^1(\tilde K_{\infty},T_{f,n})$ by the coboundary map.
The value $v_{\ell_2}(\tilde\kappa(\ell_1))$ at $\ell_2$ of
$\tilde\kappa(\ell_1)$ is naturally an element of
$\hat H^1_{\rm{fin}}(\tilde K_{\infty,\ell_2},T_{f,n})$,
and is equal to the image of $\{P_m^\ast\}$, and hence
to $\tilde\cL_g\pmod{p^n}$.
Since $\cL_g$ is the image in $\Lambda$ of $\tilde\cL_g$,
and $\kappa(\ell_1)$ is the corestriction from
$\tilde K_\infty$ to $K_\infty$ of
$\tilde\kappa(\ell_1)$, theorem \ref{thm:value} follows.
\sk\noindent
{\em Remark:} The result proved in this section can be viewed as a
generalization of the main result of \cite{BD5}. The proof given here
follows closely the approach in \cite{vatsal2}, avoiding the study of
certain groups of connected components which was involved in the methods
of \cite{BD5}.
\begin{thebibliography}{XXXX}
\bibitem[BC]{boutot-carayol}
J-F. Boutot, H. Carayol, {\em Uniformisation $p$-adique des courbes de
Shimura:
les th\'eor\`emes de Cerednik et de Drinfeld}, Ast\'erisque 196-197 (1991)
pp.\ 45-158.
\bibitem[BD0]{bd_derived_regulators}
M. Bertolini, H. Darmon, {\em Derived heights and generalized Mazur-Tate
regulators} Duke Math. J. {\bf 76} (1994), no. 1, 75--111.
\bibitem[BD$\frac{1}{2}$]{bd_derived_heights}
M. Bertolini, H. Darmon, {\em Derived
$p$-adic heights}. Amer. J. Math. {\bf 117} (1995),
no. 6, 1517--1554.
\bibitem[BD1]{BD1}
M. Bertolini and H. Darmon, {\em Heegner points on Mumford-Tate curves}.
Invent.\ Math {\bf 126} 413--456 (1996).
\bibitem[BD2]{BD2}
M.\ Bertolini and H.\ Darmon, {\em A rigid-analytic Gross-Zagier formula and
arithmetic applications}. Annals of Math {\bf 146} (1997) 111-147.
\bibitem[BD3]{BD3}
M.\ Bertolini and H.\ Darmon, {\em Heegner points, $p$-adic $L$-functions,
and
the Cerednik-Drinfeld uniformization}. Invent.\ Math, {\bf 131} (1998), no.
3,
453--491.
\bibitem[BD4]{BD4}
M.\ Bertolini and H.\ Darmon, {\em $p$-adic periods, $p$-adic
$L$-functions and the
$p$-adic uniformization of Shimura curves}, Duke Math J., to appear.
\bibitem[BD5]{BD5}
M.~Bertolini and H.~Darmon, {\em Euler systems and Jochnowitz congruences},
Amer.\ J.\ Math.\ {\bf 121}, n.\ 2 (1999) 259-281.
\bibitem[BLR]{bosch}
S.\ Bosch, W.\ L\"utkebohmert, and M.\ Raynaud, N\'eron Models,
Ergebnisse der
Mathematik und ihrer Grenzgebiete, 3 Folge - Band 21, Springer-Verlag,
1990.
\bibitem[BoLeRi]{BLR}
N.~Boston, H.~Lenstra, K.~Ribet, {\em Quotients of
groups rings arising form two-dimensional representations,} C.\ R.\ Acad.\
Sci.\ Paris {\bf 312}, S\'erie I (1991) 323--328.
\bibitem[Bu]{buzzard}
K.\ Buzzard, {\em Integral models of certain Shimura curves,} Duke Math.\
J.\
{\bf 87}, no.\ 3
(1998), 591--612.
\bibitem[Co]{cornut}
C.\ Cornut, {\em R\'eduction de familles de points CM,} PhD Thesis,
Universit\'e Louis Pasteur, Strasbourg, 2000.
\bibitem[Dag]{daghigh}
H.~Daghigh, {\em Modular forms, quaternion algebras,
and special values of $L$-functions}, McGill University PhD thesis, 1997.
\bibitem[Da]{darmon_thesis}
H.~Darmon, {\em A refined conjecture of Mazur-Tate
type for Heegner points}. Invent. Math. {\bf 110} (1992), no. 1, 123--146.
\bibitem[DDT]{ddt}
H.~Darmon, F.~Diamond, and R.~Taylor, {\em Fermat's Last Theorem},
Current Developments in Mathematics Vol. {\bf 1}, International Press, 1995,
pp.~1--154.
\bibitem[DR]{deligne-rapoport} P.\ Deligne and M.\ Rapoport, {\em Les
sch\'emas de modules des
courbes elliptiques,} LNM {\bf 349}, Springer-Verlag, New York, 1973,
143-316.
\bibitem[Dr]{drinfeld}
V.G.\ Drinfeld, {\em Coverings of $p$-adic symmetric regions}, (in
Russian),
Funkts.\ Anal.\ Prilozn.\ 10, 29-40, 1976. Transl.\ in Funct.\ Anal.\
Appl.\ 10,
107-115, 1976.
\bibitem[Ed]{bas.appendix}
B.\ Edixhoven, Appendix in [BD2].
\bibitem[Ei]{eichler}
M.~Eichler, {\em The basis problem for modular
forms and the traces of the Hecke operators.}
Modular functions of one variable, I
(Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp.
75--151. Lecture Notes in Math., Vol. 320, Springer, Berlin, 1973.
\bibitem[Gr1]{gross_montreal}
B.H.~Gross, {\em Heights and the special values of $L$-series}.
Number theory (Montreal, Que., 1985), 115--187, CMS Conf. Proc., 7, Amer.
Math.
Soc., Providence, RI, 1987.
\bibitem[Gr2]{gross_zagier}
B.H.~Gross, D.B.~Zagier, {\em Heegner points and derivatives
of
$L$-series}. Invent. Math. {\bf 84} (1986), no. 2, 225--320.
\bibitem[Groth]{grothendieck} A.~Grothendieck, {\em Groupes de
Mo\-no\-dro\-mie
en G\'eom\-\'e\-trie
Al\-g\'e\-bri\-que,} SGA VII, LNM {\bf 288}, Springer-Verlag, New York,
1972.
\bibitem[I1]{ihara1}
Y.\ Ihara, {\em On congruence monodromy problems,} Lect.\ Notes Univ.\
Tokyo {\bf 1} (1968).
\bibitem[I2]{ihara2}
Y.\ Ihara, {\em Shimura curves over finite fields and
their rational points,} Contemporary Math.\
{\bf 245} (1999) 15-23.
\bibitem[JoLi1]{joli.mathann}
B.W.\ Jordan, R.\ Livn\'e, {\em Local diophantine properties of
Shimura curves,} Math.\ Ann.\ {\bf 270} (1985) 235-248.
\bibitem[JoLi2]{joli.compositio}
B.W.\ Jordan, R.\ Livn\'e, {\em On the N\'eron mdel of Jacobians of
Shimura curves,} Compositio Math.\ {\bf 60} (1986) 227-236.
\bibitem[JoLi3]{joli.duke}
B.W.\ Jordan, R.\ Livn\'e, {\em Integral Hodge theory and congruences
between modular forms,}
Duke Math.\ J.\ {\bf 80} (1995) 419-484.
\bibitem[KM]{katz-mazur}
N.\ Katz and B.\ Mazur, {\em Arithmetic moduli of elliptic curves,}
Annals of Math.\ Studies 108, Princeton University Press, Princeton, NJ
(USA), 1985.
\bibitem[MTT]{mtt}
B.~Mazur, J.~Tate, J.~Teitelbaum, {\em On $p$-adic analogues of
the conjectures of Birch and Swinnerton-Dyer}. Invent. Math. {\bf 84}
(1986), no.
1, 1--48.
\bibitem[Ma1]{mazur_special}
B.~Mazur, {\em On the arithmetic of special values of $L$ functions},
Invent.
Math. {\bf 55} (1979), no. 3, 207--240.
\bibitem[Ram]{ramakrishna}
R.~Ramakrishna, {\em Lifting Galois representations}.
Invent. Math. {\bf 138} (1999), no. 3,
537--562.
\bibitem[Ray]{raynaud.picard} M.\ Raynaud, {\em Sp\'ecialization du
foncteur de Picard,} Publ.\
Math., Inst.\ Hautes Etud.\ Sc.\ {\bf 38} (1970) 27-76.
\bibitem[Ri1]{ribet1} K.\ Ribet, {\em Bimodules and abelian surfaces,}
Adv.\ Stud.\ Pure Math.\
{\bf 17} (1989) 359-407.
\bibitem[Ri2]{ribet} K.\ Ribet, {\em On modular representation of
$\Gal(\bar\Q/\Q)$ arising from
modular forms,} Invent.\ Math. {\bf 100} (1990) 431-476.
\bibitem[Ro]{roberts}
D.\ Roberts, Shimura curves analogous to $X_0(N)$, Harvard PhD.\ Thesis,
1989.
\bibitem[Sh]{shimura_book}
G.~Shimura, Introduction to the arithmetic theory
of automorphic functions. Reprint
of the 1971 original.
Publications of the Mathematical Society of Japan, 11.
Kan{\^o} Memorial Lectures, 1.
Princeton University Press, Princeton, NJ, 1994.
\bibitem[Va1]{vatsal1}
V.~Vatsal, {\em Uniform distribution of Heegner points,} preprint.
\bibitem[Va2]{vatsal2}
V.~Vatsal, {\em Special values of anticyclotomic $L$-functions,} preprint.
\bibitem[Vi]{vigneras} M-F.\
M-F. Vigneras, {\em Arithm\'etique des alg\`ebres des quaternions,}
LNM 800, Springer.
\bibitem[Wa]{waterhouse}
W.~ Waterhouse, {\em Abelian varieties over finite fields,} Ann.\ Sci.\
Ec.\ Nor.\ Sup., S\'erie {\bf 4} (1969) 521-560.
\bibitem[W]{wiles} A.\ Wiles, {\em Modular elliptic curves and Fermat's
Last Theorem,} Ann.\ Math.
{\bf 141} (1995) 443-551.
\bibitem[Zh]{zhang}
S.~Zhang, {\em Gross-Zagier formula for $GL_2$}, manuscript, to appear.
\end{thebibliography}
\enddocument