%%%% latex file %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,latexsym, amsfonts, amscd, amsthm}
\usepackage{graphicx,epsfig}
%\textwidth=16.5cm
%\textheight=22.7cm
%\oddsidemargin=-0.0cm
%\evensidemargin=-0.0cm
%\voffset=-1in
%\parskip=10pt plus 1pt
%\parindent=0pt
\DeclareMathOperator{\sing}{sing}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\fin}{fin}
\DeclareMathOperator{\Fitt}{Fitt}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\gal}{Gal}
\DeclareMathOperator{\inv}{inv}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{question}[theorem]{Question}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newtheorem{assumptions}[theorem]{Assumptions}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\TT}{{\mathbb T}}
\newcommand{\That}{{\hat \TT}}
\newcommand{\Tate}{{ {\rm Ta}}}
\newcommand{\Sl}{{\rm Sel}}
\newcommand{\ds}{\displaystyle}
\newcommand{\disc}{{\rm Disc}}
\newcommand{\sk}{\vspace{0.02in}}
\newcommand{\lra}{\longrightarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\sym}{\mbox{Sym}}
\newcommand{\red}{\mbox{red}}
\newcommand{\source}{\mbox{source}}
\newcommand{\target}{\mbox{target}}
\newcommand{\invlim}{{\lim_\leftarrow}}
\newcommand{\Pic}{\mbox{Pic}}
\newcommand{\norm}{\mbox{norm}}
\newcommand{\spec}{\mbox{Spec}}
\newcommand{\End}{\mbox{End}}
\newcommand{\ad}{\mbox{ad}}
\newcommand{\Norm}{\mbox{Norm}}
\newcommand{\trace}{\mbox{trace}}
\newcommand{\Ann}{\mbox{Ann}}
\newcommand{\PGL}{{\rm \mathbf{PGL}}}
\newcommand{\GL}{{\rm \mathbf{GL}}}
\newcommand{\PSL}{{\rm {\mathbf PSL}}}
\newcommand{\SL}{{\rm {\mathbf{SL}}}}
\newcommand{\gm}{{\rm {\mathbf G}_m}}
\newcommand{\ga}{{\rm {\mathbf G}_a}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\qbar}{{\bar\Q}}
\newcommand{\A}{{\underline{A}}}
\newcommand{\B}{{\underline{B}}}
\newcommand{\Kbar}{{\overline{K}}}
\newcommand{\Zhat}{{\hat{\Z}}}
\newcommand{\X}{{\underline{X}}}
\newcommand{\sha}{{\underline{III}}}
%\newcommand{\sha}{\mbox{{\cyr{X}}}}
\newcommand{\onsum}{{\oplus}}
\newcommand{\sel}{{\rm Sel}}
\newcommand{\DD}{{\mbox{D}}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cJ}{{\cal J}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cT}{{\cal T}}
\newcommand{\PP}{{\mathbb P}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cO}{{\cal O}}
\newcommand{\frob}{{\mbox{frob}}}
\newcommand{\barrho}{{\bar\rho}}
\newcommand{\bL}{{\mathbb L}}
\newcommand{\barL}{{\overline{\mathbb L}}}
\newcommand{\notdiv}{{\not\!|}}
\newcommand{\Trace}{{\mbox{Tr}}}
\newcommand{\cV}{{\cal{V}}}
\newcommand{\hH}{\widehat{H}}
\newcommand{\hhH}{{\hat H}}
\newcommand{\bvert}{v^\circ}
\newcommand{\bedge}{e^\circ}
\newcommand{\om}{{\omega}}
\newcommand{\tom}{\tilde{\om}}
\newcommand{\ep}{{\epsilon}}
\newcommand{\res}{{\mbox{res}}}
\newcommand{\Ends}{{\cal E}_\infty}
\newcommand{\fm}{{\mathfrak{m}}}
\newcommand{\fa}{{\mathfrak{a}}}
\newcommand{\fp}{{\mathfrak{p}}}
\newcommand{\fr}{{\mathfrak{r}}}
\newcommand{\fn}{{\mathfrak{n}}}
\newcommand{\fc}{{\mathfrak{c}}}
\newcommand{\fd}{{\mathfrak{d}}}
\newcommand{\fq}{{\mathfrak{q}}}
\newcommand{\fo}{{\mathfrak{o}}}
\newcommand{\cC}{{\cal C}}
\newcommand{\Sel}{{\rm Sel}}
%\hsize=9in
%\vsize=14in
%\hoffset=-1.05in
%\voffset=-.85in
\newcommand{\oR}{{{\bar R}}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} {#1} & {#2} \\ {#3} & {#4}
\end{array} \right)}
\newcommand{\edges}{{\stackrel{\rightarrow}{{\cal E}}}}
\newcommand{\vertices}{{\cal V}}
\newcommand{\hE}{\widehat{E}}
\begin{document}
\title{The anticyclotomic Main Conjecture for elliptic curves
at supersingular primes }
\author{Henri Darmon \\ Adrian Iovita}
\maketitle
\begin{abstract}
The Main Conjecture of Iwasawa theory
for an elliptic curve $E$ over $\Q$ and the anticyclotomic
$\Z_p$-extension of an imaginary quadratic field $K$
was studied in
\cite{bertolini_darmon},
in the case where $p$ is a prime of ordinary reduction
for $E$.
Analogous results are formulated, and proved,
in the case where $p$ is a prime of supersingular reduction.
The foundational study of supersingular main conjectures carried out by
Perrin-Riou
\cite{PR93}, \cite{PR01},
Pollack \cite{Pollack02},
Kurihara \cite{Kurihara01},
Kobayashi \cite{Kobayashi02}, and Iovita-Pollack \cite{iovita_pollack}
are required to handle this case in which many of the
simplifying features of the ordinary setting
break down.
\end{abstract}
\tableofcontents
\section{Introduction}
Let $E$ be an elliptic curve over $\Q$ of conductor $N_0$, and
let $K$ be an imaginary quadratic field of discriminant prime to $N_0$.
Choose a rational prime $p$ and let $K_\infty$ denote the anticyclotomic
$\Z_p$-extension of $K$.
\sk\noindent
To the datum $(E,K,p)$ are associated two kinds of invariants:
\sk\noindent
1. The twisted special values $L(E/K,\chi,1)$ of the Hasse-Weil
$L$-series of $E$ over $K$, as $\chi$ ranges over the finite-order
characters of
$G_\infty:=\gal(K_\infty/K)$.
These special values satisfy certain algebraicity and integrality properties.
When $p$ is a prime of {\em ordinary} reduction for $E$,
they can be conveniently packaged into a $p$-adic $L$-function
$L_p(E,K)$ which belongs to the Iwasawa algebra
$\Lambda:= \Z_p[\![G_\infty]\!]$.
\sk\noindent
2. The Selmer group $\Sel(K_\infty,E_{p^\infty})$ consisting of classes in
$H^1(K_\infty,E_{p^\infty})$ which are in the images of the local
Kummer maps at all places of $K_\infty$.
This group is a co-finitely
generated $\Lambda$-module. One of the interesting
features of the anticyclotomic setting is that it
need not be $\Lambda$-cotorsion
in general.
Let $\cC$ denote the characteristic
power series of the Pontryagin dual of the Selmer group,
setting $\cC=0$ if this Pontryagin dual is not torsion over
$\Lambda$. The invariant $\cC$ is well-defined up to multiplication
by units of $\Lambda$.
\sk
It is assumed that the discriminant of $K$ is prime to $N:=pN_0$ so that $K$
determines a factorization
$$ N = p N^+ N^-,$$
where $N^+$ is divisible only by primes which are split in $K$ and $N^-$ by
primes which are inert in $K$.
\sk
Under certain technical assumptions stated in the introduction of
\cite{bertolini_darmon} which
will be recalled below, the article \cite{bertolini_darmon}
proves the following
result in the direction of the anticyclotomic main
conjecture of Iwasawa theory
in the {\em ordinary case}. (Cf.~Theorem 1 of \cite{bertolini_darmon}.)
\begin{theorem}
\label{thm:main_bd}
Assume that $N^-$ is the square-free product of an odd number
of primes.
Assume also that the prime $p$ is {\em ordinary}, and that $(E,K,p)$ satisfies the
technical hypotheses stated in \ref{ass:technical} below. Then
$\cC$ divides the $p$-adic $L$-function
$L_p(E,K)$.
\end{theorem}
\begin{remark}
It follows from results of Vatsal \cite{vatsal} that $L_p(E,K)$ is non-zero
under the hypotheses on $N$ made in
Theorem \ref{thm:main_bd}. In particular,
this theorem implies that the Selmer group of $E$ over $K_\infty$ is
$\Lambda$-co-torsion.
By contrast, when $N^-$ is the square-free product of an
{\em even} number of primes, then $L_p(E,K)$ vanishes identically.
Vatsal's theorem on the non-triviality of
Heegner points and arguments of Kolyvagin
can be used to show that the Selmer group of $E$ over $K_\infty$
has $\Lambda$-corank one
in this case.
\end{remark}
The main goal of the present note is to
formulate and prove an analogous result
in the case where $p$ is a prime satisfying
$$ a_p := p+1-\#E(\Z/p\Z) = 0.$$
This implies that $E$ is supersingular at $p$,
and is in fact equivalent to this statement when $p\ge 5$, in light
of the Hasse bound $|a_p|\le 2\sqrt{p}$.
The foundational study of supersingular main conjectures carried out by
Perrin-Riou
\cite{PR93}, \cite{PR01},
Pollack \cite{Pollack02},
Kurihara \cite{Kurihara01},
Kobayashi \cite{Kobayashi02}, and Iovita-Pollack \cite{iovita_pollack}
are required to handle this case in which many of the
simplifying features of the ordinary setting
break down.
\sk\noindent
1. The special values $L(E/K,\chi,1)$ cannot be interpolated in an obvious way
by an element of $\Lambda$. Section \ref{sec:analytic}
explains how the construction of the $p$-adic $L$-function
$L_p(E,K)$ presented in Section 1 of \cite{bertolini_darmon} can be
modified, following the ideas of \cite{PR93} and \cite{Pollack02},
by removing the
infinitely many ``trivial zeroes" that occur at $p$-power roots of unity.
This process
yields {\em two} $p$-adic $L$-functions
$L_p^+(E,K)$ and $L_p^-(E,K)$ which both belong to $\Lambda$ and
emerge as the appropriate substitutes for the $p$-adic $L$-function in the
supersingular setting.
\sk\noindent
2. In tandem with this analytic complication, the Selmer group
$\Sel(K_\infty,E_{p^\infty})$ is {\em never} a co-torsion $\Lambda$-module when $p$
is supersingular. Following an idea of Kobayashi
\cite{Kobayashi02}, Section \ref{sec:selmer} introduces two restricted Selmer groups
$$\Sel_+(K_\infty,E_{p^\infty}) \quad \mbox{ and }
\quad \Sel_-(K_\infty,E_{p^\infty})$$
defined by imposing more
stringent local conditions at the prime $p$.
Let $\cC^+$ and $\cC^-$ denote the characteristic power series of
the Pontryagin duals of $\Sel_+(K_\infty,E_{p^\infty})$ and
$\Sel_-(K_\infty,E_{p^\infty})$ respectively.
(Here we follow the same conventions as before, whereby the characteristic
power series of a non-torsion $\Lambda$-module is taken to be $0$.)
The main conjecture that we are interested in
is formulated
in terms of the plus/minus $p$-adic $L$-functions and the restricted
Selmer groups, as follows:
\begin{conjecture}
\label{conj:main_bd}
Assume that $a_p=0$.
Then
the characteristic power series $\cC^+$ and $\cC^-$
generate the same ideal of $\Lambda$ as
the $p$-adic $L$-functions
$L_p^+(E,K)$ and $L_p^-(E,K)$ respectively.
\end{conjecture}
Fix an integer $n\ge 1$.
A key ingredient in the proof of Theorem \ref{thm:main_bd}
given in \cite{bertolini_darmon} is the construction of certain
global cohomology classes
$$\kappa(\ell)\in \invlim_m H^1(K_m,E[p^n]),$$
indexed by rational
primes $\ell$ satisfying suitable properties (the
{\em $n$-admissible primes} in the sense of \cite{bertolini_darmon}).
These classes
form a kind of Euler system, as spelled out in
Sections 4 and 7 of \cite{bertolini_darmon}.
Section \ref{sec:euler}
of this paper explains how the
construction of \cite{bertolini_darmon}
can be modified in the supersingular case
to yield classes $\kappa^+(\ell)$ and $\kappa^-(\ell)$ satisfying
properties analogous to the classes $\kappa(\ell)$ of
\cite{bertolini_darmon}.
The strategy of the proof of Theorem \ref{thm:main_bd}
carries over to establish one of the divisibilities
predicted by Conjecture \ref{conj:main_bd},
which is the main result of this paper.
\begin{theorem}
\label{thm:main}
Assume that $a_p=0$ and
that $N^-$ is the square-free product of an odd number of primes.
Assume also
that $(E,K,p)$
satisfies the hypotheses stated in
\ref{ass:technical} and \ref{ass:iovita} below.
Then the characteristic power series $\cC^+$ and $\cC^-$
divide
the $p$-adic $L$-functions $L_p^+(E,K)$ and
$L_p^-(E,K)$ respectively.
\end{theorem}
\begin{remark} When $N^-$ is the square-free product of an even number of
primes, the $p$-adic $L$-functions $L_p^+(E,K)$ and $L_p^-(E,K)$
vanish identically, much as in the ordinary case, and the
corresponding Selmer groups are not $\Lambda$-co-torsion.
\end{remark}
Throughout this article, the following assumptions are made on $(E,K,p)$:
\begin{assumptions}
\label{ass:technical}
\begin{enumerate}
\item The prime $p$ is greater or equal to $5$.
\item The Galois representation attached to $E_p$ has image isomorphic
to $\GL_2(\F_p)$.
\item There is a modular parameterization
$X_0(N_0)\lra E$ whose degree is not divisible by $p$.
\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, where
$I_\ell$ denotes the inertia group at $\ell$.
\end{enumerate}
\end{assumptions}
These assumptions are made mainly to simplify
the arguments and could probably be relaxed at the cost of
complicating the proofs. (See Remark 1 after the statement of
Assumption 6 in the introduction of \cite{bertolini_darmon}.)
The next set of assumptions, which does not
appear in \cite{bertolini_darmon}, is imposed
on us by our lack of understanding of the
local condition to impose at $p$ in defining the appropriate Selmer group when $p$
is a supersingular prime and $p$ is inert in $K$.
\begin{assumptions}
\label{ass:iovita}
\begin{enumerate}
\item The prime $p$ is split in $K$, so that it can be written
$p=\fp\bar\fp$, where $\fp$ is a prime of $K$.
\item The prime $\fp$ is totally ramified in the anticyclotomic $\Z_p$-extension
attached to $K$.
\end{enumerate}
\end{assumptions}
Assumption 2 is automatically satisfied if $p$ does not divide the class number of $K$.
It would be desirable to be able to dispense with assumption 1 and
treat the inert primes on the same footing as the split primes, as is done in
\cite{bertolini_darmon} when $p$ is ordinary.
\sk\sk\noindent
{\bf Acknowledgements:} The authors are grateful to the anonymous referee for a
careful proofreading which led to corrections of some inaccuracies and to
significant improvements in the presentation.
Both authors were supported by grants from NSERC. The first author was
supported by a James McGill Chair, and the second
by a Canada Research Chair, while this research was conducted.
\section{The plus/minus $p$-adic $L$-functions}
\label{sec:analytic}
\subsection{Modular forms on quaternion algebras}
Let $N^+$ and $N^-$ be positive integers such that
$N^+$ is divisible only by primes which
are split in $K$ and $N^-$ by primes which are inert in $K$.
Assume that $N^-$ is square-free, and
has an odd number of
prime factors. Let $p$ be a prime which does not divide $N_0:=N^+N^-$.
Let $B$ be the definite quaternion algebra of discriminant
$N^-\infty$,
and let $R$ be an Eichler $\Z[\frac{1}{p}]$-order of conductor
$N^+$ in $B$.
Since $p$ does not divide $N^-$,
we may fix an isomorphism
$$\iota:B_p:=B\otimes\Q_p\lra M_2(\Q_p).$$
Let $R_1^\times$ denote the group of elements of reduced norm one in $R$,
and define
$$\Gamma=\iota(R^\times)\subset \GL_2(\Q_p).$$
Let $\cT$ denote the Bruhat-Tits tree
attached to $\PGL_2(\Q_p)$, whose set $\cV(\cT)$ of
vertices is in bijection with the similarity classes
of $\Z_p$-lattices in $\Q_p^2$, two vertices being joined by an
edge if the corresponding classes of lattices admit representatives in
which one
contains the other with index $p$.
The group $\GL_2(\Q_p)$
(and hence, in particular, $\Gamma$)
acts naturally on $\cT$ and
the quotient of
$\cT$ by the action of $\Gamma$ is a finite graph.
\begin{definition}
A modular form of weight $2$ on $\cV(\cT)$ for $\Gamma$ is a $\Gamma$-invariant
$\Z_p$-valued function on
$\cV(\cT)$.
\end{definition}
Denote by $S_2(\cV/\Gamma)$ the space of all such forms; it is a finitely
generated $\Z_p$-module equipped with a natural action of the
Hecke operators $T_\ell$ (with $\ell\not|N$) described as in
\cite{bertolini_darmon}, Section 1.1.
Furthermore, it admits a natural $\Z$-structure
$S_2(\cV/\Gamma)^\Z$ consisting of the $\Z$-valued
functions in $S_2(\cV/\Gamma)$, which is preserved by the action
of the Hecke operators.
Let $f_E\in S_2(\Gamma_0(N_0))^{\rm new}$ be the eigenform
of weight two corresponding to $E$.
For each prime $\ell$ not dividing $N$ we have
$$T_\ell(f_E)=a_\ell(E) f_E, $$
where $T_\ell$ denotes the Hecke operator acting on the space of classical
cusp forms on $\Gamma_0(N_0)$.
\begin{theorem}[Jacquet-Langlands]
There is an eigenform $f\in S_2(\cV/\Gamma)^\Z$
with the same Hecke eigenvalues as those attached to
$f_E$, i.e., such that
$$ T_\ell(f) = a_\ell(E) f, \quad \mbox{ for all } \ell \not| N.$$
This form is unique up to multiplication by a non-zero scalar.
\end{theorem}
In addition, the function $f$ satisfies the following property:
\begin{proposition}
\label{prop:tp}
For all $v\in \cV(\cT)$,
$$ \sum_{v' \leftrightarrow v} f(v') = a_p(E) f(v),$$
where the sum is taken over the $p+1$ vertices $v'$ adjacent to $v$.
\end{proposition}
\begin{proof} This follows directly from the
description of the action of the Hecke
operator $T_p$ on $S_2(\cV/\Gamma)$:
$$ (T_pf)(v)=
\sum_{v' \leftrightarrow v} f(v'),$$
and the fact that $f$ is an
eigenvector for $T_p$ with associated eigenvalue $a_p(E)$.
\end{proof}
\sk
We normalize the form $f$ so that it is not divisible by any integer
in $S_2(\cV/\Gamma)^\Z$. This makes $f$ well-defined up to a sign.
As shall be seen in the next section,
the $p$-adic $L$-function attached to $E$ and $K$ is defined directly in
terms of $f$ rather than in terms of the classical cusp form
$f_E$.
\subsection{Rankin $L$-functions}
The goal of this section is to define a $p$-adic $L$-function
attached to a modular form $f\in S_2(\cV/\Gamma)$ satisfying $a_p(f)=0$
and to a quadratic subfield $K\subset B$, by combining the construction
described in
Section 1.2. of \cite{bertolini_darmon} with the
ideas of Pollack \cite{Pollack02}.
Suppose for simplicity that the discriminant of $K$ is prime to $N$.
Let $\cO_K$ denote the ring of integers of $K$ and let
$\cO:=\cO_K[1/p]$ denote its ring of $\{p\}$-integers.
For the sake of concreteness, we present the construction under the
further assumption that the class number of $\cO_K[1/p]$ is equal to
$1$. The reader may, if she wishes, adapt the construction to general
class number by following the approach described in Section 1.2 of
\cite{bertolini_darmon}.
The advantage of the class number one assumption is that it allows for a
different,
more concrete and geometric---purely $p$-adic, instead
of adelic---presentation
of Section
1.2 of \cite{bertolini_darmon}, enabling the authors to avoid what would
otherwise be a somewhat tedious repetition of the
constructions in that section.
Let $n$ be fixed integer, and let $f$ be an eigenform in
$S_2(\cV/\Gamma)$ satisfying $a_p(f)\equiv 0 \pmod{p^n}$.
Let
$$\Psi:K \lra B $$
be an embedding of algebras, satisfying
$$ \Psi(K) \cap R := \Psi(\cO). $$
Such an embedding exists if and only if all the primes dividing
$N^+$ are split in $K$, while those dividing $N^-$ are inert in $K$.
It is then unique, up to conjugation by the action of $R^\times$,
thanks to the class number one assumption.
Let $K_p:= K\otimes \Q_p$ be the $p$-adic completion of $K$.
The embedding $\Psi$ induces an action of the
$p$-adic
group $\Pi_\infty := K_p^\times/\Q_p^\times$ on $\cT$
by isometries by setting
\begin{equation}
\label{eqn:star}
g\star x := \iota\Psi(g)(x),
\end{equation}
for any $g\in K_p^\times$ and $x$ any vertex or edge of $\cT$.
We begin by defining certain partial $p$-adic
$L$-functions attached to $\Psi$, by studying this action.
It is somewhat clearer to separate the study into two cases:
\sk\noindent
{\em Case 1}: Suppose $p$ is split in $K$.
The choice of a prime $\fp$ of $K$ above $p$ induces a
homomorphism
$$|\ |_p: K_p^\times/\Q_p^\times\lra\Z $$
defined by
$$ |\alpha|_p := \ord_\fp(\alpha/\bar\alpha).$$
Note that replacing $\fp$ by $\bar \fp$ only changes
the resulting homomorphism $|\ |_p$ by a sign, so that the abuse of notation
inherent in the notation $|\ |_p$ is not serious.
The kernel of $|\ \ |_p$ is the maximal compact
subgroup of $\Pi_\infty$, denoted
$U_0$. This group is identified with
$\Z_p^\times$ under the map which sends
$\alpha$ to $\alpha/\bar\alpha$.
Let
\begin{equation}
\label{eqn:filtration}
\ldots\subset U_n\subset \ldots\subset U_1 \subset U_0
\end{equation}
be the
the natural decreasing filtration of the group
$U_0$ by subgroups in which the index of $U_n$ is
$(p-1)p^{n-1}$. In the action
of $\Pi_\infty$ on $\cT$ of (\ref{eqn:star}),
the maximal compact subgroup
$U_0$ fixes a sequence (infinite in both directions)
of consecutive vertices, i.e.,
a geodesic
$J=J_\Psi$ of $\cT$.
The quotient
$\Pi_\infty/U_0\cong \Z$ acts by translation on this geodesic.
The distance between a vertex $v$ of $\cT$ and the geodesic
$J_\Psi$ is defined to be the shortest distance between
$v$ and a vertex of $J_\Psi$.
If the distance from $v$ to
$J_\Psi$ is equal to $n$, then the stabilizer of $v$ in
$\Pi_\infty$ is exactly $U_n$, and the quotient
$\Pi_\infty/U_n$ acts simply transitively
on the set of vertices at distance
$n$ from $J_\Psi$.
Now let us fix a sequence of consecutive
vertices $v_0,v_1,v_2,...$ such that $v_n$ is
at distance
$n$ from $J=J_\Psi$. (See Figure \ref{pic:split} for an illustration
in the case
where $p=2$.)
\begin{figure}[tb]
\centerline{\psfig{figure=split.eps, width = 8cm, angle = 0}}
\caption{Action of $K_p^\times$ on $\cT$ when $p$ is split.}
\label{pic:split}
\end{figure}
We define a sequence of functions
$$ f_{K,n}: \Pi_\infty/U_n \lra \Z_p, \quad \mbox{by the rule }
f_{K,n}(\alpha) = f(\alpha\star v_n).$$
Let $u_p$ be a fundamental $p$-unit of $K$, i.e., a generator of
the group of elements in
$\cO_K[1/p]^\times$ of norm one, modulo torsion. The quotient
$$\tilde G_\infty = \Pi_\infty/u_p^\Z$$
is a compact $p$-adic group.
By abuse of notation, the groups $U_j$
occurring in the filtration (\ref{eqn:filtration}) and their natural images
in $\tilde G_\infty$ will be denoted by the same symbol.
\begin{lemma}
\label{lemma:punit}
The functions $f_{K,n}$ are invariant under translation
by $u_p$, and hence descend to functions on
${\tilde G_\infty}/U_n$.
\end{lemma}
\begin{proof} Note that
$$
f_{K,n}(u_p\alpha) =
f( (u_p\alpha)\star v_n) = f(\iota\Psi(u_p) (\alpha\star v_n)),$$
and that $\iota\Psi(u_p)$ belongs to $\Gamma$.
The result therefore follows from the invariance of
$f$ under translation by elements of $\Gamma$.
\end{proof}
Thanks to Lemma \ref{lemma:punit}, the functions $f_{K,n}$ defined above can be
viewed as functions on the finite quotients
${\tilde G_\infty}/U_n$.
\sk\noindent
{\em Case 2}: Suppose now that $p$ is inert in $K$.
This case is somewhat simpler, because
$\Pi_\infty=K_p^\times/\Q_p^\times$ is already compact, and is equal, therefore,
to its maximal compact subgroup $U_0$.
This group is identified with
the group of elements in $\cO_K\otimes \Z_p$ of norm one
under the map which sends
$\alpha$ to $\alpha/\bar\alpha$.
Let
\begin{equation}
\ldots\subset U_n\subset \ldots\subset U_1 \subset U_0 = \Pi_\infty
\end{equation}
be the
the natural decreasing filtration of the group
$U_0$ by subgroups of index $(p+1)p^{n-1}$.
The group $\tilde G_\infty$
fixes a distinguished vertex $v_0$.
If the distance from a vertex $v$ to
$v_0$ is equal to $n$, then the stabilizer of $v$ in
$\tilde G_\infty$ is exactly $U_n$, and the quotient
$\tilde G_\infty/U_n$ acts simply transitively
on the set of vertices at distance
$n$ from $v_0$.
Fix a sequence of consecutive
vertices $v_0,v_1,v_2,\ldots$ such that $v_n$ is
at distance
$n$ from $v_0$. (See Figure \ref{pic:inert}).
\begin{figure}[tb]
\centerline{\psfig{figure=inert.eps, width = 8cm, angle =
0}} \caption{Action of $K_p^\times$ on $\cT$ when $p$ is inert.}
\label{pic:inert}
\end{figure}
We then define a sequence of functions
$$ f_{K,n}: \tilde G_\infty/U_n \lra \Z_p, \quad \mbox{by the rule }
f_{K,n}(\alpha) = f(\alpha\star v_n).$$
In conclusion, in both the cases where $p$ is split or inert in $K$, we have
associated to $f$ and $\Psi$ a sequence of functions
$$ f_{K,n}: {\tilde G_n} \lra \Z_p, \quad \mbox{ where } \quad
{\tilde G_n} := {\tilde G_\infty}/U_n.$$
We associate to these functions a sequence of elements
$\tilde \cL_n\in \Z_p[\tilde G_n]$
by setting:
$$
\tilde \cL_{n}:=\sum_{\sigma\in \tilde G_n} f_{K,n}(\sigma)\sigma^{-1}
\in \Z_p[\tilde G_n].
$$
\begin{remark}
The assumption that the class number
of $\cO$ is equal to 1 can readily be disposed of
following the treatment given in Section 1.2 of \cite{bertolini_darmon}.
The definitions given there would yield
a sequence $\tilde \cL_n$ as above, belonging to the finite
group rings $\Z_p[{\tilde G_n}]$, where now
${\tilde G_\infty}$ denotes the group
$$ {\tilde G_\infty} := (K\otimes{\hat Z})^\times/\left( (\Q\otimes{\hat Z})^\times
\prod_{\ell \ne p} (\cO\otimes\Z_\ell)^\times K^\times \right),$$
which is identified with the Galois group over $K$ of the union
of all ring class fields of $K$
of $p$-power conductor, an extension which contains the Hilbert
class field of $K$.
\end{remark}
\sk
Denote by
$$\pi_{n+1,n}:\Z_p[\tilde G_{n+1}]\lra \Z_p[\tilde G_{n}] $$
the
ring homomorphism induced by the natural projection
$\tilde G_{n+1}\lra \tilde G_{n}$.
Occasionally we will abuse notation and view ${\tilde \cL}_n$ as an element of
$\Z_p[\tilde G_{n+1}]$ by replacing it by an arbitrary lift to this
ring under
the homomorphism $\pi_{n+1,n}$.
Of course this element is not well-defined, but the product
$$ \tilde{\xi}_n \tilde \cL_n \in \Z_p[\tilde G_{n+1}]$$
is well-defined, where
$$ \tilde{\xi}_n = \sum_{s\in U_n/U_{n+1}} s. $$
\begin{lemma}
\label{lemma:compat}
The elements $\tilde \cL_n$ satisfy the following compatibility relations under
the projections $\pi_{n+1,n}$:
$$\pi_{n+1,n}(\tilde \cL_{n+1})=a_p(E)\tilde \cL_{n}-\xi_{n-1} \tilde \cL_{n-1}.$$
In particular, if $p$ is supersingular for $E$ so that
$a_p(E)=0$,
$$\pi_{n+1,n}(\tilde \cL_{n+1})= -\tilde{\xi}_{n-1}\tilde \cL_{n-1},
\mbox{ for all }
n\ge 1. $$
\end{lemma}
\begin{proof} If $g_n$ is any element of ${\tilde G}_n$,
let $g_{n+1}$ denote an arbitrary lift of this element to
${\tilde G}_{n+1}$.
A direct calculation shows that
\begin{equation}
\label{eqn:compat1}
\pi_{n+1,n}(\tilde \cL_{n+1}) = \sum_{g_n\in {\tilde G}_n}
\left(\sum_{s\in U_n/U_{n+1}} f_{K,n+1}(s g_{n+1})
\right) g_n^{-1}.
\end{equation}
On the other hand,
\begin{equation}
\label{eqn:compat2}
\sum_{s\in U_n/U_{n+1}} f_{K,n+1}(s g_{n+1}) =
\sum_{s\in U_n/U_{n+1}} f((s g_{n+1})\star v_{n+1}).
\end{equation}
The sum on the right corresponds to summing the function $f$ over
the $p$ vertices which are adjacent to the vertex
$g_n \star v_n$ and are different from $ g_n \star v_{n-1}$ (see
figure \ref{pic:edges}).
\begin{figure}[tb]
\centerline{\psfig{figure=edges.eps, width = 8cm, angle =
0}} \caption{The inner sum}
\label{pic:edges}
\end{figure}
It follows from proposition \ref{prop:tp}
satisfied by $f$ that
\begin{eqnarray}
\sum_{s\in U_n/U_{n+1}} f((s g_{n+1})\star v_{n+1}) &= & a_p(E) f(g_n\star v_n)
-f(g_n \star v_{n-1}) \\
&=& a_p(E) f_{K,n}(g_n) - f_{K,n-1}(g_n).
\label{eqn:compat3}
\end{eqnarray}
The lemma follows by combining (\ref{eqn:compat1}), (\ref{eqn:compat2}), and
(\ref{eqn:compat3}).
\end{proof}
We may write
$$ \tilde G_\infty = \Delta \times G_\infty,$$
where $\Delta$ is the torsion subgroup of $\tilde G_\infty$ and
$G_\infty$ is its maximal torsion-free quotient, which is
topologically isomorphic to $\Z_p$.
The image of $\Delta$ in $\tilde G_n$ is identified with $\Delta$,
and the group $G_\infty$ can be written as
$$ G_\infty = \invlim \ G_n, \quad \mbox{ where } G_n :=
{\tilde G}_{n+1}/\Delta\simeq \Z/p^n\Z.$$
Let $\pi:\Z_p[\tilde{G}_{n+1}]\lra \Z_p[G_n]$ be the natural homomorphism induced
by the projection ${\tilde G}_{n+1} \lra G_n$. Then
for each $n\ge 0$, we set:
$$ \cL_n := \pi ({\tilde \cL}_{n+1}).$$
By abuse of notation, denote again by
$$\pi_{n+1,n}: \Z_p[G_{n+1}] \lra \Z_p[G_n]$$
the maps induced by the natural group homomorphisms.
The elements $\cL_n$ inherit from the $\tilde\cL_n$ the compatibility
properties
of Lemma \ref{lemma:compat}
under these maps, i.e., when $p$ is supersingular:
\begin{equation}
\label{eqn:compat}
\pi_{n+1,n}(\cL_{n+1})=-\xi_{n} \cL_{n-1}, \quad \mbox{ for all } n\ge 1.
\end{equation}
Here $\xi_n:=\pi(\tilde{\xi}_{n+1})$ is the element of $\Z_p[G_n]$ given by
$\ds \xi_n=\sum_{\sigma\in H_n}\sigma$, where $H_n=\ker(G_n\lra G_{n-1})$.
Let us fix a topological generator
$$\gamma\in \lim_{\leftarrow} G_n\cong \Z_p. $$
This determines the identification sending $\gamma$
to $1+T$
$$ \Lambda=\lim_{\leftarrow}\Z_p[G_n]\cong \Z_p[\![T]\!].$$
In this way
$\Z_p[G_n]$ is identified with $\Z_p[T]/\omega_n\Z_p[T]$, where
$\omega_n=(T+1)^{p^n}-1$, and
the element $\xi_n$ is identified with the $p^n$-power cyclotomic polynomial in
$T+1$. (In other words, the roots of $\xi_n(T)$ are of the form
$\zeta-1$, where $\zeta$ ranges over all
primitive $p^n$-th roots of unity.)
Note that we have
$$ \omega_n(T) = T \prod_{j=1}^n \xi_j(T).$$
Let us also write
$$ \tom_n^+(T) := \prod_{\stackrel{j=2}{j {\rm \ even }}}^n \xi_j(T),
\quad
\tom_n^-(T) := \prod_{\stackrel{j= 1}{j {\rm \ odd }}}^n \xi_j(T),$$
and set
$\om_n^\pm(T) =T\tom_n^\pm(T) $.
\noindent
The following technical lemma is straightforward to derive,
but we note it for better reference.
\begin{lemma}
\label{lemma:div_p^n}
Let $n$ be a positive integer and $\ep$ denote the sign of $(-1)^n$.
\begin{enumerate}
\item
Multiplication by $\tom_n^{-\ep}$ induces a natural isomorphism
$$
\Lambda/(\om_n^{\ep})\lra \tom_n^{-\ep}\Lambda/(\om_n).
$$
\item For all $r\ge 1$,
multiplication by
$\tom_n^{-\ep}$ induces a natural
isomorphism
$$\Lambda/(\om_n^\ep, p^r) \lra
\tom_n^{-\ep}
\Lambda/(\om_n, p^r). $$
\item
If $X$ is a free $\Lambda_{r,n}:= \Lambda/(\om_n, p^r)$-module
and $x\in X$ is annihilated by
$\om_n^\ep$,
then there is a unique $y\in X/\om_n^\ep X$ such that $x=\tom_n^{-\ep}y$.
\end{enumerate}
\end{lemma}
The following proposition is key in the construction of the plus and minus
$p$-adic $L$-functions.
\begin{proposition}
\label{prop:div}
Let $\ep$ denote the sign of $(-1)^n$. Then
\begin{enumerate}
\item
$\om_n^\ep \cL_n=0$
\item
There is a unique element
$L_n^\ep\in \Lambda/\om_n^\ep\Lambda$
such that $\cL_n=\tom_n^{-\ep} L_n^\ep$.
\end{enumerate}
\end{proposition}
\begin{proof}
For the first assertion, suppose first that $n > 2$ is even.
Then
$$ \om_n^+\cL_n = \om_{n-2}^+ \xi_n \cL_n =\om_{n-2}^+ \xi_n\pi_{n,n-1}(\cL_n).$$
But by equation (\ref{eqn:compat}),
$$\om_{n-2}^+ \xi_n\pi_{n,n-1}(\cL_n) = -\om_{n-2}^+
\xi_n
\xi_{n-2}
\cL_{n-2}.$$
This allows the statement to be reduced by induction to the case $n=2$.
For this value of $n$ it
follows from the direct calculation
$$ \omega_2^+ \cL_2 = T \xi_2(T) \cL_2 = T\xi_2 \pi_{2,1}(\cL_2) =
-T \xi_1\xi_2 \cL_0,$$
where the last equality follows from (\ref{eqn:compat}).
But this expression is $0$ because $T\xi_1\xi_2 = 0 $ in
$\Z_p[G_2]$.
The proof when $n$ is odd is identical.
For the second (divisibility) assertion, we invoke
Lemma \ref{lemma:div_p^n},
noting that we have $\om_n=\om_n^{\ep} \tom_n^{-\ep}$.
Therefore an element
of $\Lambda/\om_n\Lambda$ annihilated by $\om_n^\ep$
is divisible by $\tom_n^{-\ep}$
and the result of the division is unique
in $\Lambda/\om^\ep_n\Lambda$.
\end{proof}
Let us now denote by
$$
\left\{
\begin{array}{ll}
\cL_n^+= (-1)^{\frac{n}{2}}L_n^+ & \mbox{ if $n$ is even };\\
\cL_n^-=(-1)^{\frac{n+1}{2}}L_n^- & \mbox{ if $n$ is odd.}
\end{array}
\right.
$$
\begin{lemma}
\label{lemma:norm_compatible}
The
sequence $\{\cL_n^+\}_{n\ {\rm even}}$ is
compatible with respect to the natural projections
$$\Lambda/\om_n^+\lra \Lambda/\om_{n-2}^+,$$
and likewise for the sequence
$\{\cL_n^-\}_{n\ {\rm odd}}$.
\end{lemma}
\begin{proof}
Let us, for all $n\ge 0$, choose lifts of $\cL_n$ and
$L_n^{\pm}$ to
$\Lambda$ and denote them by the same symbols. Suppose first that
$n\ge 2$ is even. Then we have
$$
\cL_n=-\xi_{n-1}\cL_{n-2} \pmod{\om_{n-1}}.
$$
This implies that there exists $F\in \Lambda$ such that
$$
\tom_n^-L_n^+=-\xi_{n-1}\tom^-_{n-2}L_{n-2}^++\om_{n-1}F.
$$
Noting that $\omega_{n-1} = \omega_{n-2}^+ \tom_n^-$,
we can
cancel by $\tom_n^-=\xi_{n-1}\tom_{n-2}$ to obtain
$$
L_n^+=-L_{n-2}^++\om_{n-2}^+F,
$$
which proves the statement when $n$ is even. The case
where $n$ is odd is similar.
\end{proof}
Thanks to this lemma we may denote by
$$\cL_f^+:=\lim_{\leftarrow}
\cL_n^+\in \lim_{\leftarrow}\Lambda/\om_n^+\cong \Lambda,$$
and define $\cL_f^-\in \Lambda$ similarly.
Let $L\mapsto L^*$ denote the involution in $\Lambda$ sending every group-
like element in this completed group ring to its inverse.
We set
$$L_p(f,K)^\pm:=\cL_f^\pm(\cL_f^\pm)^*,$$
following definition 1.6 of \cite{bertolini_darmon}.
\section{Selmer groups}
\label{sec:selmer}
Class field theory identifies ${\tilde G}_\infty$ with
$\gal({\tilde K_\infty}/K)$, where ${\tilde K}_\infty$ is the union of
all the ring class fields of $K$ of $p$-power conductor.
For each integer $m\ge 0$,
the quotient ${\tilde G_m}$ is identified with $\gal({\tilde K_m}/K)$,
where ${\tilde K_m}$ is the ring class field of $K$ of conductor $p^m$.
The subfield of ${\tilde K}_\infty$ fixed by $\Delta$ is the
anticyclotomic $\Z_p$-extension $K_\infty$ of $K$,
so that
$$ G_\infty = \gal(K_\infty/K).$$
Under this identification, the group $G_m$ corresponds to the Galois group
$\gal(K_m/K)$, where $K_m$ is defined to be the
$m$-th layer of the
$\Z_p$-tower $K_\infty$.
Following the notations of Section 2.1 of \cite{bertolini_darmon},
denote by $V_f$ the two-dimensional
Galois representation attached to the modular form
$f$ (with coefficients in $\Q_p$) and
let $T_f$ denote a $G_\Q$ stable $\Z_p$-sub-lattice.
The finite modules
$$T_{f,n}:= T_f/p^n T_f = (V_f/T_f)[p^n] \qquad (n\ge 1)$$
fit naturally both
into a projective and an inductive system, i.e., for all $r\ge n$ there are natural maps
$$ T_{f,r} \twoheadrightarrow T_{f,n}, \qquad T_{f,n}\hookrightarrow T_{f,r},$$
which will be used to take both projective and injective limits of cohomology groups associated
to the $T_{f,n}$.
The main goal of this section is to define certain {\em Selmer groups}
attached to the Galois representations
$T_{f,n}$, and to prove certain basic facts about their structure.
For each $m\ge 0$ and $n\ge 1$,
the Selmer group $\Sel(K_m,T_{f,n})$ is
defined as a subgroup of the (continuous) Galois
cohomology group $H^1(K_m, T_{f,n})$ by imposing
conditions on restrictions to local decomposition groups.
More precisely, for every rational prime $\ell$,
let
$$K_{m,\ell} :=
K_m\otimes_\Q \Q_\ell = \oplus_{\lambda|\ell} K_{m,\lambda}, \quad
H^1(K_{m,\ell},T_{f,n}) :=
\oplus_{\lambda|\ell} H^1(K_{m,\lambda}, T_{f,n}).
$$
There is a natural restriction map
$$ res_\ell: H^1(K_m,T_{f,n}) \lra H^1(K_{m,\ell},T_{f,n}).$$
For each rational prime
$\ell$ we define certain {\em distinguished subgroups}
$$ H^1_{\fin}(K_{m,\ell},T_{f,n}) \subset
H^1(K_{m,\ell},T_{f,n}), $$
referred to as the {\em finite part}
of these local cohomology groups
as follows. Let $A_f$ denote the abelian variety associated to
the modular form $f$, as described in greater detail
in Section \ref{sec:euler}.
Let $V_p(A_f):= T_p(A_f)\otimes\Q_p$
be the $p$-adic Galois representation associated to $A_f$,
and let
$H^1_{\fin}(K_{m,\ell}, V_p(A_f))$ denote
the image
of $A_f(K_{m,\ell})\otimes \Z_p$
in $H^1(K_{m,\ell}, V_p(A_f))$
under the Kummer map.
By construction, the two-dimensional
Galois representation $V_f$ is a quotient of
$V_p(A_f)$. Write
$$\pi_f:V_p(A_f)\lra V_f, \quad H^1(K_{m,\ell},V_p(A_f))\lra
H^1(K_{m,\ell}, V_f)$$
for the associated $G_\Q$-equivariant projection, as well
as for the maps induced by it on the various local cohomology groups.
We define
$$H^1_{\fin}(K_{m,\ell}, V_f) := \pi_f(H^1_{\fin}(K_{m,\ell}, V_p(A_f))), $$
and $H^1_{\fin}(K_{m,\ell}, T_f)$ as the inverse image of
$H^1_{\fin}(K_{m,\ell}, V_f)$ under the natural map
$$H^1(K_{m,\ell},
T_f)\lra H^1(K_{m,\ell},V_f)$$ induced by
the inclusion $T_f\hookrightarrow V_f$.
Finally, for every integer $n\ge 1$ we let
$H^1_{\fin}(K_{m,\ell}, T_{f,n})$ denote
the image of $H^1_{\fin}(K_{m,\ell}, T_f)$
in $H^1(K_{m,\ell}, T_{f,n})$ under the map induced by the canonical projection
$T_f\lra T_{f,n}$.
\begin{definition}
The {\em Selmer group}
attached to $K_m$ and $T_{f,n}$
is the group
of cohomology classes
$s\in H^1(K_m,T_{f,n})$ satisfying
$$\res_\ell(s) \mbox{ belongs to } H^1_{\fin}(K_{m,\ell},T_{f,n}), \quad
\mbox{ for all } \ell. $$
It is denoted $\Sel(K_m,T_{f,n})$.
\end{definition}
We also set
\begin{equation}
\label{eqn:defselmer}
\Sel(K_\infty,T_{f,n}) := \lim_m \Sel(K_m, T_{f,n}),\quad
\Sel(K_\infty,T_{f,\infty}) := \lim_n \Sel(K_\infty, T_{f,n}),
\end{equation}
where the direct limits are taken with respect to the natural maps induced by restriction
and
the inclusions $T_{f,n}\hookrightarrow T_{f,n'}$.
\subsection{Local conditions at $\ell\ne p$}
\label{sec:locall}
We begin by discussing the groups $H^1_{\fin}(K_{m,\ell},T_{f,n})$ in the case
where $\ell\ne p$.
We define (following \cite{bertolini_darmon}) the {\em singular part} of the local
cohomology group $H^1(K_{m,\ell}, T_{f,n})$ to be the quotient
$$ H^1_{\sing}(K_{m,\ell},T_{f,n}) := \frac{H^1(K_{m,\ell},T_{f,n})}{H^1_{\fin}(K_{m,\ell},T_{f,n})}.$$
If $\ell$ does not
divide $N$, then
$$
H^1_{\sing}(K_{m,\ell}, T_{f,n})=H^1(I_{m,\ell}, T_{f,n})^{G_{K_\ell}}:=
\prod_{\lambda}
H^1(I_{m,\lambda}, T_{f,n})^{G_{K_{\ell}}},
$$
where $\lambda$ runs over the primes of $K_m$
over $\ell$ and $I_{m,\lambda}$ is the inertia
subgroup of $G_{K_{m,\lambda}}$.
Restriction defines a so-called {\em residue map}
$$
\partial_\ell:H^1(K_{m,\ell}, T_{f,n})\lra H^1_{\sing}(K_{m,\ell}, T_{f,n})
$$
such that the following sequence is exact
$$
0\lra H^1_{\fin}(K_{m,\ell}, T_{f,n})\lra H^1(K_{m,\ell}, T_{f,n})\lra
H^1_{\sing}(K_{m,\ell}, T_{f,n}).
$$
The following gives a local control theorem for the Selmer group.
\begin{lemma}
\label{lemma:local_control}
For all rational primes $\ell\ne p$,
the natural map induced by restriction
$$H^1_{\sing}(K_{\ell},T_{f,n}) \lra H^1_{\sing}(K_{m,\ell},T_{f,n})$$
is injective.
\end{lemma}
For example, when $\ell$ does not divide $N$, this follows from the fact that
$K_m/K$ is unramified at the primes above $\ell$, so that any class which becomes
unramified over $K_m$ already had to be unramified over $K$.
The cup product in local Galois cohomology combined with the Weil pairing
$T_{f,n} \times T_{f,n} \lra \mu_{p^n}$ leads to the non-degenerate
{\em local Tate pairing}
$$ H^1(K_{m,\ell},T_{f,n}) \times H^1(K_{m,\ell},T_{f,n}) \lra H^2(K_{m,\ell},\mu_{p^n})
\stackrel{\inv_\ell}{\lra} \Z/p^n\Z,$$
in which the rightmost map is given by
$$ \inv_\ell(\kappa) := \sum_{\lambda|\ell} \inv_\lambda(\kappa),$$
where $\inv_\lambda$ is the standard identification of $H^2(K_{m,\lambda},\mu_{p^n})$
with $\Z/p^n\Z$ given by local class field theory.
It is a standard fact that the groups
$H^1_{\fin}(K_{m,\ell},T_{f,n})$ are {\em maximal isotropic}
for the local Tate pairing, and that this pairing therefore induces a perfect duality
$$ H^1_{\fin}(K_{m,\ell},T_{f,n}) \times H^1_{\sing}(K_{m,\ell},T_{f,n}) \lra \Z/p^n\Z.$$
(Cf.~Proposition 2.3 of \cite{bertolini_darmon}.)
We now recall the definition
of {\em admissible primes} given in Section 2.2 of \cite{bertolini_darmon}.
\begin{definition}
\label{def:admissible}
A rational prime $\ell$ is said to
{\em $n$-admissible} relative to $f$ if it satisfies the following
conditions:
\begin{enumerate}
\item $\ell$ does not divide $pN$;
\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}
\end{definition}
One of the motivations for singling out these primes is the following
freeness result for the local cohomology group
$H^1(K_{m,\ell},T_{f,n})$ when $\ell$ is $n$-admissible.
\begin{lemma}
\label{lemma:localfreel}
If $\ell$ is an $n$-admissible prime, then
\begin{enumerate}
\item
The groups $H^1_{\fin}(K_{m,\ell},T_{f,n})$ and
$H^1_{\sing}(K_{m,\ell},T_{f,n})$ are free of rank one over
$\Lambda_{n,m}= \Lambda/(p^n,\omega_m)$.
\item
The group $H^1(K_{m,\ell},T_{f,n})$ is
free of rank two over
$\Lambda_{n,m}$.
\end{enumerate}
\end{lemma}
\begin{proof}
See Lemma 2.7 of \cite{bertolini_darmon}.
\end{proof}
An important caveat that needs to be noted
is that the finite part at $\ell$ does not just depend on the underlying
Galois representation $T_{f,n}$, but on the Galois representation $V_f$ from
which it arises.
For example, if $T_{f,n}$ comes from the $p$-division points of
an abelian variety $A_f$ with good reduction at $\ell$, then the representation
$T_{f,n}$ is of course unramified, and $H^1_{\fin}(K_{m,\ell},T_{f,n})$
merely consists of the {\em unramified} cohomology classes. This is not the case if
$T_{f,n}$ is unramified but arises from an abelian variety with multiplicative reduction at
$\ell$.
If $\ell$ is a prime which divides $N$ exactly, then
$V_f$ is
is an {\em ordinary}
representation of $G_{\Q_\ell}$: it contains a unique one-dimensional
$\Q_p$-vector subspace which is stable under the action of the decomposition group at $\ell$.
Let $T_{f,n}^{(\ell)}$ denote the
corresponding rank one $(\Z/p^n\Z)$-submodule of
$T_{f,n}$.
Then
$H^1_{\fin}(K_{m,\ell}, T_{f,n})$ is the image in $H^1(K_{m,\ell}, T_{f,n})$
of
$H^1(K_{m,\ell}, T_{f,n}^{(\ell)})$.
\subsection{Local conditions at $p$}
\label{sec:local_p}
In the supersingular setting, the classical definition of the Selmer group
$\Sel(K_\infty,T_{f,\infty})$
given in
(\ref{eqn:defselmer})
suffers from the fact that the resulting object is not a co-torsion module
over the Iwasawa algebra $\Lambda = \Z_p[\![G_\infty]\!]$.
The idea is to cut down the size of this Selmer group by
imposing {\em more stringent} local conditions at the primes above $p$.
We will follow closely Sections 4 and 6 of \cite{iovita_pollack} with
some
adjustments due to the fact that here we work with torsion coefficients.
In order to define the appropriate subgroups of
$H^1_{\fin}(K_{m,p}, T_{f,n})$, we need to make the following
assumptions which are satisfied in our application:
\begin{assumptions}
\label{ass:iovita_local}
\begin{enumerate}
\item The prime $p$ is split in $K$, so that it can be written
$p=\fp\bar\fp$, where $\fp$ is a prime of $K$.
\item The prime $\fp$ is totally ramified in the anticyclotomic $\Z_p$-extension
$K_\infty$
attached to $K$.
\item The Galois representation $T_{f,n}$ is isomorphic (as a representation of
$G_{\Q_p}$) to $E[p^n]$, where $E$ is an elliptic curve over $\Q_p$ with
supersingular reduction at $p$.
\end{enumerate}
\end{assumptions}
Note that this implies, in particular, that
$$a_p(f)\equiv 0 \pmod{p^n}. $$
Recall that $K_m$ denotes the $m$-th layer in the anticyclotomic
$\Z_p$-extension $K_\infty$.
Let $K_{m,p} := K_m\otimes\Q_p = K_{m,\fp} \oplus K_{m,\bar\fp}$
denote the completion of $K_m$ at $p$.
Let $\hE$ denote the formal group of $E_{\Q_p}$.
First we will recall the description of
$$\hE(K_{m,p}):=\hE(K_{m,\fp})\oplus\hE(K_{m,\bar\fp})$$
as a $\Z_p[G_m]=\Lambda_m$-module, for all $m\ge 0$.
Since the discussion in this section is purely local,
we will lighten notations by letting
$\{L_m\}_{m\ge 0}$ denote
either
of the following towers of local fields: $\{K_{m,\fp}\}_{m\ge 0}$ or
$\{K_{m,\bar\fp}\}_{m\ge 0}$.
The following theorem is essential in defining the plus and minus
Selmer groups attached to $T_{f,n}$.
\begin{theorem}
\label{thm:local_points}
For $m\ge 0$ there exist points
$d_m\in \hE(L_m)$ such that
\begin{enumerate}
\item
$\mbox{Tr}^m_{m-1}(d_m)=-d_{m-2}$ \mbox{ for all } $m\ge 2$
\item
$\mbox{Tr}^1_0(d_1)=ud_0$ \mbox{ with } $u\in \Z_p^\times$
\item
$d_m,d_{m-1}$ generate $\hE(L_m)$ as a $\Z_p[G_m]$-module and $d_0$ generates
$\hE(L_0)$ as a $\Z_p$-module.
\end{enumerate}
\end{theorem}
\noindent
This theorem is proved in \cite{iovita_pollack}. (See Theorem 4.5 of
\cite{iovita_pollack}.)
Using the sequence of points $\{d_m\}_{m\ge 0}$ we consider two subsequences
$$
d_m^+= \left\{ \begin{array}{ll}
d_m & \mbox{ if $m$ is even}; \\
d_{m-1} & \mbox{ if $m$ is odd};
\end{array} \right. \quad
d_m^-= \left\{ \begin{array}{ll}
d_{m-1} & \mbox{ if $m\ge 2$ is even};\\
d_m & \mbox{ if $m$ is odd.} \end{array} \right.
$$
Now we define $\ds \hE^\pm(L_m):=\Lambda_m d_m^\pm \subset \hE(L_m)$.
Let us remark that the $\Lambda_m$-submodule $\hE^\pm(L_m)$ thus defined is
independent
of the choice of the sequence of points $\{d_m\}_{m\ge 0}$ as in Theorem
\ref{thm:local_points}. (See
Lemma 4.13
of \cite{iovita_pollack}.)
Let us now fix integers $m,n$.
\begin{lemma}
\label{lemma:injective}
The natural map
$$ j:\hE^\pm(L_m)/p^n\hE^\pm(L_m)\lra \hE(L_m)/p^n\hE(L_m)$$
is
injective for all $m,n$.
\end{lemma}
\begin{proof}
We consider the case where the sign is $+$, the other case being
proved in a similar way.
Let $P\in \hE^+(L_m)$ be a point whose image under $j$ is 0.
Then there exists $Q\in \hE(L_m)$ such that $P=p^nQ$.
Since $\omega_m^+$ is the exact annihilator
of $\hE^+(L_m)$ in $\hE(L_m)$ (cf.~Proposition 4.11 of
\cite{iovita_pollack})
we have
$$ p^n(\omega_m^+ Q)=0, $$
and as there are no non-zero $p$-power torsion points in
$\hE(L_M)$, we conclude that
$$ \omega_m^+ Q=0. $$
Hence
$Q$ itself belongs
to $\hE^+(L_m)$, which proves the lemma.
\end{proof}
\sk
\noindent
Recall from Section \ref{sec:locall}
that local Tate duality induces perfect pairings
$$
\langle \ , \rangle_{m,n}:H^i(L_m, T_{E,n})
\times H^{2-i}(L_m, T_{E,n})\lra \Q_p/\Z_p
$$
and
$$
\langle \ , \rangle_m:H^i(L_m, T_pE)\times
H^{2-i}(L_m, E[p^\infty])\lra \Q_p/\Z_p.$$
Let us define, following Section 4.3 of \cite{iovita_pollack},
\begin{eqnarray*}
H^1_{\fin\pm}(L_m, T_{f,n}) & := & (\hE^\pm(L_m)\otimes \Z/p^n\Z), \\
H^1_{\fin}(L_m, T_{f,n}) & := & (\hE(L_m)\otimes \Z/p^n\Z), \\
H^1_{\pm}(L_m, T_{f,n}) & := & (\hE^\pm(L_m)\otimes \Z/p^n\Z)^\perp,
\end{eqnarray*}
where the orthogonal complement in the last definition is
taken relative to the pairing $\ds \langle \ , \rangle_m$.
We'll also write
$$H^1_{\pm}(L_m, T_{E,n}):=(\hE^\pm(L_m)\otimes \Z/p^n\Z)^\perp,$$
with the orthogonal complement taken under the
pairing $\ds \langle \ , \rangle_{m,n}$.
\begin{lemma}
\label{lemma:46IP}
$H^0(L_m,E[p^n]) = H^2(L_m,E[p^n])=0.$
\end{lemma}
\begin{proof} This follows from
Lemma 4.6 of
\cite{iovita_pollack} and the non-degeneracy of the local
Tate pairing.
\end{proof}
\begin{lemma}
\label{lemma:free}
$H^1_{\pm}(L_m, T_{f,n})$ is a free $\Z/p^n\Z[G_m]$-module of rank 1.
\end{lemma}
\begin{proof}
Taking the $L_m$-cohomology of the exact sequences
$$ 0\rightarrow T_pE\stackrel{p^n}{\lra} T_pE \lra T_{E,n} \rightarrow 0, \quad
0\rightarrow T_{E,n}\lra E[p^\infty] \stackrel{p^n}\lra E[p^\infty] \rightarrow 0$$
and using Lemma \ref{lemma:46IP}
yields the
natural isomorphisms of
$\Z/p^n\Z[G_m]$-modules
$$
H^1(L_m, T_{E,n})\cong H^1(L_m, T_pE)/p^nH^1(L_m, T_pE)
$$
and
$$
H^1(L_m, T_{E,n})\cong H^1(L_m, E[p^\infty])[p^n].
$$
The pairing $\langle \ , \rangle_{m,n}$ is naturally induced from the pairing
$\langle \ , \rangle_m$ under these identifications. Therefore, since
$\hE^\pm(L_m)\otimes \Q_p/\Z_p$ is a $p$-divisible group we immediately obtain
\begin{eqnarray*}
H^1_{\pm}(L_m, T_{E,n}) &\cong& (\hE^\pm(L_m)\otimes \Z/p^n\Z)^\perp \\
&\cong& (\hE^\pm(L_m)
\otimes
\Q_p/\Z_p)^\perp/p^n(\hE^\pm(L_m)\otimes \Q_p/\Z_p)^\perp.
\end{eqnarray*}
This yields the isomorphism
$$\ds H^1_{\pm}(L_m, T_{E,n})\cong
H^1_{\pm}(L_m, T_pE)/p^nH^1_{\pm}(L_m, T_pE).$$
Proposition 4.16 of \cite{iovita_pollack} implies that $H^1_{\pm}(L_m, T_pE)$
is a free
$\Z_p[G_m]$-module of rank 1. Lemma
\ref{lemma:free} follows.
\end{proof}
The following result, which is a consequence of Theorem 10.1 of
\cite{greenberg_iovita_pollack}, shows that
the subgroup $H^1_{\fin}(K_{m,p},T_{f,n})$
depends only on the Galois representation $T_{f,n}$, unlike its counterpart
for $\ell\ne p$ in general, so that in particular
it behaves well
under congruences.
As \cite{greenberg_iovita_pollack} is not yet available we will sketch here
the main arguments of the proof. Let $f_1,f_2\in S_2(\cV/\Gamma)$ and let us denote by
$T_i:=T_{f_i}$ and $T_{i,n}:=T_{f_i,n}$ for $i=1,2$ and some $n\ge 1$.
\begin{theorem}
\label{thm:cont_finite}
Suppose that the $G_{\Q_p}$-representations
$T_1$ and $T_2$ are congruent modulo $p^n$,
i.e. we have an isomorphism
$$\iota:T_{1,n} \cong T_{2,n}$$
as $\Z/p^n\Z[G_{\Q_p}]$-modules. We'll further assume that Assumptions
\ref{ass:iovita_local} hold for $T_{1,n}$ (and
consequently also for $T_{2,n}$)
and that $L$ is one of the local fields $K_{m,\fp}$ or $K_{m,\bar\fp}$ for some $m\ge 0$.
Then $\iota$ induces a natural isomorphism
$$
g_L:H^1_{\fin}(L, T_{1,n})\lra H^1_{\fin}(L, T_{2,n}).$$
\end{theorem}
\begin{proof} First we have natural isomorphisms and inclusions
$$H^1_{\fin}(L, T_i)/p^nH^1_{\fin}(L, T_i)\cong H^1_{\fin}(L, T_{i,n})
\hookrightarrow H^1(L, T_{i,n}), \quad
\mbox{ for }
i=1,2$$
as a consequence of the assumptions and lemma \ref{lemma:46IP}. Therefore the
conclusion of the theorem makes sense.
\noindent
Second, given the totally ramified extension $L/\Q_p$ we perform (see \cite{faltings}
and \cite{breuil}) the following construction. Let $\pi$ be a uniformizer of $L$ and
let
$$E(u)=u^e+a_1u^{e-1}+...+a_e\in \Z_p[u] $$
be the minimal polynomial of $\pi$ over $\Z_p$.
Let $S$ denote the $p$-adic completion of the $\Z_p$-algebra $\ds \Z_p[u, u^{ie}/i!]_{i\in \N}
\subset \Q_p[u]$. It has the following structure:
\begin{enumerate}
\item a continuous $\Z_p$-linear Frobenius $\sigma:S\lra S$ such that $\sigma(u)=u^p$;
\item a natural continuous derivation $d:S\lra \Omega_{S/\Z_p}$;
\item a decreasing filtration $(\mbox{Fil}^iS)_{i\in \N}$, where
Fil$^iS$ is the $p$-adic completion of $\ds \sum_{j\ge i}(E(u)^j/j!)S$. (One checks that
Fil$^iS$ is an ideal of $S$.)
\end{enumerate}
\noindent
Let us consider the Galois representations $T_1,T_2$ as above, denote by
$D_1,D_2$ the strongly divisible lattices attached to $T_1,T_2$ respectively as in
\cite{fontaine_laffaille} and set $M_i:=D_i\otimes_{\Z_p}S$, for $i=1,2.$
Let finally $M$ be any one of the $S$-modules $M_1$, $M_2$, $M_1/p^nM_1$, or $M_2/p^nM_2$.
Then $M$ is endowed with the following structure:
\begin{enumerate}
\item a one step filtration $M_0\subset M$;
\item $\sigma$-linear Frobenii $\varphi:M\rightarrow M$ and $\varphi_0:M_0\rightarrow M$
such that $\varphi|_{M_0}=p\varphi_0$;
\item a connection (nilpotent modulo $p$) $\ds \nabla:M\lra M\otimes_S\Omega_{S/\Z_p}$
such that $\ds \nabla\circ \varphi_0=(\varphi/p)\circ \nabla|_{M_0}$.
\end{enumerate}
\bigskip
\noindent
Given $M$ as above we define the double complex of $\Z_p$-modules
$$
\begin{array}{cccccc}
&M_0&\stackrel{\nabla}{\lra}&M\otimes_S\Omega_{S/\Z_p}\\
C^{\bullet,\bullet}(M): &\alpha\downarrow&&\downarrow\beta\\
&M&\stackrel{\nabla}{\lra}&M\otimes_S\Omega_{S/\Z_p},
\end{array}
$$
where $\alpha=\varphi_0-1$ and $\beta=-(1-\varphi/p).$ Let us write
$$H^n(M):=H^n(C^{\bullet,\bullet}(M)), \quad \mbox{ for } n\ge 0. $$
\noindent
Then under the conditions of the theorem one can prove (for the details see
\cite{greenberg_iovita_pollack}) that $H^2(M)=0$ and that
there is a canonical and functorial
isomorphism
$$
H^1(M_i)\cong H^1_{\fin}(L, T_i)\mbox{ for } i=1,2.
$$
{From} the exact sequences
$$
0\lra T_i\stackrel{p^n}{\lra}T_i\lra T_{i,n}\lra 0
$$
for $i=1,2$ we deduce the isomorphisms:
$$H^1(M_i/p^nM_i)\cong H^1_{\fin}(L, T_i)/
p^nH^1_{\fin}(L, T_i)\cong H^1_{\fin}(T_{i,n}).$$
The fact that $T_{1,n}$ and $T_{2,n}$ are isomorphic
implies that the same is true
for $M_1/p^nM_1$ and $M_2/p^nM_2$. The result follows.
\end{proof}
We can use Theorem
\ref{thm:cont_finite} to meaningfully
define the
restricted local conditions at $p$ for
$f$. Namely, assume that
the Assumptions \ref{ass:iovita_local} hold.
In particular $T_f$ is congruent
to $T_pE$ modulo $p^n$ so we
define $H^{1,\pm}_{\fin}(K_{m,p}, T_{f,n})$ to
be the image of $\hE^\pm(K_{m,p})/p^n\hE^\pm(K_{m,p})$
under the composition
$$
\hE(K_{m,p})/p^n\hE(K_{m,p})\cong H^1_{\fin}(K_{m,p}, T_{E,n})\cong
H^1_{\fin}(K_{m,p}, T_{f,n})
$$
where the first isomorphism is the Kummer map and the second
is provided by Theorem
\ref{thm:cont_finite} via local duality. We also define the group
$H^1_\pm(K_{m,p}, T_{f,n})$
to be the orthogonal complement of $H^{1,\pm}_{\fin}(K_{m,p}, T_{f,n})$
under local duality.
Clearly this group is the image of $H^1_\pm(K_{m,p}, T_{E,n})$ under
the isomorphism
$H^1(K_{m,p}, T_{E,n})\cong H^1(K_{m,p}, T_{f,n})$ induced by $\iota$.
We recall that $\Lambda_{n,m} := \Z/p^n\Z[G_m]=\Lambda_m/p^n$ denotes
the group ring of $G_m$
with mod $p^n$ coefficients.
From the previous discussion, we have
\begin{corollary}
\label{cor:localfree}
The local cohomology groups
$H^1_\pm(K_{m,p}, T_{f,n})$
are free $\Lambda_{n,m}$-modules
of rank two.
\end{corollary}
\begin{proof} Since $p=\fp\bar\fp$, we have
$$ H^1_\pm(K_{m,p}, T_{f,n}) =
H^1_\pm(K_{m,\fp}, T_{f,n}) \oplus
H^1_\pm(K_{m,\bar\fp}, T_{f,n}).$$
But each of the summands on the right
is a free $\Lambda_{n,m}$-module,
by Lemma \ref{lemma:free}. The result follows.
\end{proof}
\subsection{Generalised Selmer groups}
We are now in a position to define more general Selmer groups
that will play a key role in our
argument.
We retain the notations and assumptions of earlier sections; in particular
Assumption \ref{ass:iovita_local} of Section \ref{sec:local_p}.
Let
$$\Gamma_m=\gal(K_{\infty}/K_m), \quad
\mbox{ so that } G_m=\Gamma/\Gamma_m.$$
If $s$ belongs to $H^1(K_m, T_{f,n})$ and $\ell$ is a rational prime
we denote by $s_\ell:=\res_\ell(s)$
the image of $s$ in $H^1(K_{m,\ell}, T_{f,n})$ under restriction.
\begin{definition}
\label{def:selmer_finite}
Let $m$ and $n$ be non-negative integers.
The {\em unrestricted Selmer group}
attached to $f$, $n$, and $K_m$,
denoted
$\Sel_\Box(K_m,T_{f,n})$, is
the group of classes $s\in H^1(K_m, T_{f,n})$
satisfying
$$ s_\ell \in H^1_{\fin}(K_{m,\ell},T_{f,n}) \mbox{ for all } \ell\ne p.$$
The subgroups
$$\Sel_0(K_m,T_{f,n}) \subset \Sel_{\pm}(K_m,T_{f,n}) \subset \Sel(K_m,T_{f,n}) \subset
\Sel_\Box(K_m,T_{f,n})$$
are defined by the additional conditions:
\begin{enumerate}
\item
$s\in \Sel_0(K_m,T_{f,n})$ if $s_p=0$;
\item
$s\in \Sel_{\pm}(K_m,T_{f,n})$ if $s_p\in H^1_{\fin\pm}(K_{m,p},T_{f,n})$;
\item
$s\in \Sel(K_m,T_{f,n})$ if $s_p\in H^1_{\fin}(K_{m,p},T_{f,n})$.
\end{enumerate}
\end{definition}
We also define
\begin{eqnarray*}
\sel_\sharp(K_\infty,T_{f,n}) &:=& \lim_{\rightarrow, m} \sel_\sharp(K_m,T_{f,n}), \\
\sel_\sharp(K_\infty,T_{f,\infty}) &:=& \lim_{\rightarrow, n} \sel_\sharp(K_\infty,T_{f,n}), \\
\end{eqnarray*}
where the transition maps for the inductive limit are restrictions and the
inclusions $T_{f,n}\hookrightarrow T_{f,n'}$,
and
$\sharp$ is either $0$, $\pm$, or $\Box$.
Let $S$ be a square-free integer prime to $N$.
\begin{definition}
The {\em generalized Selmer group}
$\Sel_{S,\sharp}(K_m,T_{f,n})$ is the set of classes in
$\Sel_{\sharp}(K_m,T_{f,n})$ satisfying
$$ s_\ell = 0 \quad \mbox{ for all } \ell|S.$$
\end{definition}
\begin{definition}
\label{def:compact_sel}
Let $\sharp= 0,\fin, \pm$, or $\Box$.
The {\em dual Selmer group}
attached to $f$, $n$, $K_m$, $S$ and $\sharp$ is defined to be the
subgroup
$H^1_{S,\sharp}(K_m, T_{f,n})$
of classes $\kappa\in
H^1(K_m, T_{f,n})$ satisfying
\begin{enumerate}
\item
$\kappa_\ell\in H^1_{\fin}(K_{m,\ell}, T_{f,n})$ for all rational primes $\ell$ not dividing
$pS$;
\item $\kappa_p$ belongs to $H^{1}_{\sharp}(K_{m,p}, T_{f,n})$ if $\sharp\in \{\fin, \pm\}$,
and $\kappa_p=0$ if $\sharp=0$;
\item
$\kappa_\ell$ is arbitrary if $\ell | S$.
\end{enumerate}
\end{definition}
Note the sequence of inclusions
$$ H^1_{S,0}(K_m,T_{f,n}) \subset
H^1_{S,\fin}(K_m,T_{f,n}) \subset
H^1_{S,\pm}(K_m,T_{f,n}) \subset
H^1_{S,\Box}(K_m,T_{f,n}).$$
As in \cite{bertolini_darmon}, denote by
$$ \hH^1(K_\infty, T_{f,n})=\lim_{\leftarrow, m}H^1(K_m,T_{f,n}),$$
where the transition maps are co-restrictions.
Similar conventions are a\-dop\-ted in defining $\hH^1(K_{\infty,p}, T_{f,n})$.
Both $\Sel_{S,\sharp}(K_m,T_{f,n})$ and $H^1_{S,\sharp}(K_m,T_{f,n})$ are special cases of the general
notion of an {\em abstract Selmer group}: a subgroup of the global cohomology group
$H^1(K_m,T_{f,n})$ defined by local conditions which agree with the unramified classes, for all but finitely
many primes of $K_m$.
The following pairs are {\em dual Selmer groups} in the sense that the local conditions defining them are
orthogonal to each other under the local Tate pairings:
\begin{eqnarray*}
\Sel_{S,0}(K_m, T_{f,n}) \quad &\mbox{and} & \quad H^1_{S,\Box}(K_m, T_{f,n}); \\
\Sel_{S,\pm}(K_m, T_{f,n}) \quad &\mbox{and} & \quad H^1_{S,\pm}(K_m, T_{f,n}); \\
\Sel_{S}(K_m, T_{f,n}) \quad &\mbox{and} & \quad H^1_{S,\fin}(K_m, T_{f,n}); \\
\Sel_{S,\Box}(K_m, T_{f,n}) \quad &\mbox{and} & \quad H^1_{S,0}(K_m, T_{f,n}).
\end{eqnarray*}
Finally we define
$$
\widehat{H}^1_{S,\sharp}(K_\infty,T_{f,n}):=\lim_{\leftarrow, m} H^1_{S,\sharp}(K_m, T_{f,n}),
$$
where the transition maps are co-restrictions.
\subsection{Freeness results for Selmer groups}
\label{sec:algebraic}
The Euler system of this section is constructed (just like the Euler system
of \cite{bertolini_darmon}) from a system of Heegner points on a
collection of Shimura curves indexed by certain admissible primes $\ell$.
The main new difficulty arising in the supersingular setting is that the classes
manufactured directly from these Heegner points are not
compatible under norms (co-restriction). To obtain a norm-compatible family
of cohomology classes it is necessary to divide the classes
obtained ``directly" from Heegner points
by certain
products of $p$-power
cyclotomic polynomials, a process which mirrors the division performed
in the construction of the $p$-adic $L$-function
in Section \ref{sec:analytic} following \cite{Pollack02}.
In order to show that this division can be performed,
a number of results concerning the
structure of generalized Selmer groups as modules over the group
rings $\Lambda_{n,m}$
are required.
Let $\ell$ be an $n$-admissible prime.
Note (cf.~the discussion preceding Theorem 4.1 of \cite{bertolini_darmon})
the {\em canonical} direct sum decomposition:
$$ \hH^1(K_{\ell,\infty},T) = \hH^1_{\rm fin}(K_{\ell,\infty},T) \oplus
\hH^1_{\rm sing}(K_{\ell,\infty},T).$$
Denote (as in \cite{bertolini_darmon}) by $v_\ell$ and
$\partial_\ell$ the projections onto the first and second
factors.
We recall the following proposition from
\cite{bertolini_darmon} that makes it possible
to produce many $n$-admissible primes.
\begin{proposition}
\label{prop:chebotarev}
Let $s$ be any nonzero element of $H^1(K,T_{f,1})$.
There exist infinitely many $n$-admissible primes
$\ell$ relative to $f$ such that
$\partial_\ell(s)=0$ and $v_\ell(s)\ne 0$.
\end{proposition}
\begin{proof} This is
Theorem 3.2 of \cite{bertolini_darmon}, whose proof relies on a careful
application of the Chebotarev density theorem, and makes no use of the local
properties of $T_{f,1}$ at $p$, so that it applies equally well to the
supersingular case.
\end{proof}
Adopting the terminology of Definition 2.22
of \cite{bd_derived_regulators}, we make the following definition.
\begin{definition}
\label{def:admissible_set}
A square-free product $S$ of $n$-admissible
primes
is said to be {\em $n$-admissible} if
the natural map
$$\sel_{\Box}(K,T_{f,n})\lra \oplus_{\ell | S} H^1_{\fin}(K_\ell,
T_{f,n})$$
is injective.
\end{definition}
Note that, if $S$ is $n$-admissible, then
\begin{equation}
\label{eqn:trivS}
\sel_{S,\Box}(K,T_{f,n})=0.
\end{equation}
Recall that $\Lambda_{n,m} =\Z/p^n\Z[G_m] =\Lambda/(\omega_m,p^n)\Lambda$ is the
group ring at level $m$ with $\Z/p^n\Z$ coefficients. Let
$I$ be the augmentation ideal of $\Lambda_{n,m}$ and denote by
$\fm= \langle p,I\rangle$ the maximal ideal of this local ring.
We begin by noting the following ``control theorems" for the
Selmer groups that have been introduced:
\begin{lemma}
\label{lemma:controlS}
If $S$ is an $n$-admissible set, then the natural
maps induced by restriction and the inclusion $T_{f,1}\lra T_{f,n}$
\begin{eqnarray*}
\sel_{S,\Box}(K,T_{f,1}) &\lra& \sel_{S,\Box}(K_m,T_{f,n})[\fm], \\
H^1_{S,0}(K,T_{f,1}) &\lra& H^1_{S,0}(K_m,T_{f,n})[\fm]
\end{eqnarray*}
are isomorphisms.
\end{lemma}
\begin{proof}
Consider the following commutative diagram with exact rows:
\begin{equation}
\label{eqn:comdi}
\begin{array}{ccccccc}
0 & \rightarrow & \sel_{S,\Box}(K,T_{f,n}) & \rightarrow & H^1(K,T_{f,n}) & \rightarrow & \Omega_S(K) \\
& & \downarrow & & \downarrow & & \downarrow \\
0 & \rightarrow& \sel_{S,\Box}(K_m,T_{f,n})^{G_m} & \rightarrow & H^1(K_m,T_{f,n})^{G_m} & \rightarrow &
\Omega_S(K_m),
\end{array}
\end{equation}
where
\begin{eqnarray*}
\Omega_S(K) & := & \left(\oplus_{\ell|S} H^1(K_\ell,T_{f,n})\right) \oplus \left(\oplus_{\ell\nmid pS}
H^1_{\sing}(K_\ell,T_{f,n})\right), \\
\Omega_S(K_m) & := &
\left(\oplus_{\ell|S} H^1(K_{m,\ell},T_{f,n})\right)
\oplus \left(\oplus_{\ell\nmid pS} H^1_{\sing}(K_{m,\ell},T_{f,n})\right).
\end{eqnarray*}
The inflation-restriction sequence for $T_{f,n}$
implies that
the middle vertical map in (\ref{eqn:comdi}) is an isomorphism, because
$T_{f,n}^{G_m}=0$.
The rightmost vertical map is injective, by Lemma \ref{lemma:local_control} and the fact that primes in
$S$ split completely in $K_m/K$.
It follows from the five-lemma that $\sel_{S,\Box}(K,T_{f,n}) = \sel_{S,\Box}(K_m,T_{f,n})^{G_m}$,
and therefore
\begin{eqnarray*}
\sel_{S,\Box}(K_m,T_{f,n})[\fm] &=& \sel_{S,\Box}(K_m,T_{f,n})[I][p]=\\
&=&
\sel_{S,\Box}(K,T_{f,n})[p] =
\Sel_{S,\Box}(K,T_{f,1}),
\end{eqnarray*}
where the last equality follows from the fact that $H^0(K,T_{f,n-1})=0$.
The proof of the second assertion, in which $\Box$ is replaced by $0$, uses in addition
the injectivity of the map $H^1(K_p,T_{f,n}) \lra H^1(K_{m,p},T_{f,n})^{G_m}$ in the
analysis of the diagram analogue to diagram \ref{eqn:comdi}
(which follows from the fact that $T_{f,n}^{G_{K_{m,p}}}=0$), but is otherwise the same.
\end{proof}
\begin{lemma}
\label{lemma:seltrivS}
If $S$ is an $n$-admissible set, then
$\sel_{S,\sharp}(K_m,T_{f,n})=0$, for all $m$ and $\sharp=0,\pm,\fin$, or $\Box$.
\end{lemma}
\begin{proof}
It suffices to show that the finite
$\Lambda_{n,m}$-module
$M:= \sel_{S,\Box}(K_m,T_{f,n})$ is trivial, since this Selmer group contains all
the others.
By Lemma \ref{lemma:controlS} and (\ref{eqn:trivS}),
$$ M[\fm] = \sel_{S,\Box}(K,T_{f,1}) =0,$$
and hence $M=0$.
\end{proof}
The following proposition gives explicit formulae for the cardinality of
the global cohomology groups
$H^1_{S,\sharp}(K_m,T_{f,n})$.
\begin{proposition}
\label{prop:poitou_tate}
Let $t:= \#S-2$, and let $\delta_m := [K_m:K] = p^m$.
For all $n$-admissible sets $S$, and for all $m\ge 0$,
\begin{eqnarray*}
\# H^1_{S,0}(K_m,T_{f,n}) &=& p^{n t \delta_m}, \\
\# H^1_{S,\Box}(K_m,T_{f,n}) &=&
\# H^1_{S,0}(K_m,T_{f,n}) \# H^1(K_{m,p},T_{f,n}).
\end{eqnarray*}
\end{proposition}
\begin{proof}
A general theorem arising from the Poitou-Tate exact sequence in Galois cohomology
(cf.~for example Theorem 2.19 of \cite{ddt}) relates the cardinalities
of a Selmer group and its dual, expressing the ratio of these cardinalities as a product of simple
local terms.
In the present context, Theorem 2.19 of \cite{ddt}
gives:
\begin{eqnarray*}
\frac{\#H^1_{S,0}(K_m,T_{f,n})}{\#\sel_{S,\Box}(K_m,T_{f,n})} &=&
\left(\prod_{\ell|S\infty} \frac{\# H^1(K_{m,\ell},T_{f,n})}{\#H^0(K_{m,\ell},T_{f,n})}\right), \\
\frac{\#H^1_{S,\Box}(K_m,T_{f,n})}{\#\sel_{S,0}(K_m,T_{f,n})} &=&
\left(\prod_{\ell|S\infty} \frac{\# H^1(K_{m,\ell},T_{f,n})}{\#H^0(K_{m,\ell},T_{f,n})}\right)
\times \# H^1(K_{m,p}, T_{f,n}).
\end{eqnarray*}
By Lemma \ref{lemma:seltrivS},
the denominators occurring in the left-hand sides of these formulae are equal to
$1$. Furthermore,
we already know from Lemma \ref{lemma:localfreel} that
$$
\frac{\# H^1(K_{m,\ell},T_{f,n})}{\#H^0(K_{m,\ell},T_{f,n})} = \left\{
\begin{array}{ll}
p^{n\delta_m} & \mbox{ if } v|S; \\
p^{-2 n\delta_m} & \mbox{ if } v=\infty.
\end{array} \right.
$$
The Proposition follows.
\end{proof}
The usefulness of the concept of $n$-admissible
set lies in the
following two propositions concerning the groups
$H^1_{S,0}(K_m,T_{f,n})$ and $H^1_{S,\pm}(K_m,T_{f,n})$. These
propositions can be viewed as global analogues of Lemma
\ref{lemma:localfreel} and Corollary \ref{cor:localfree}.
\begin{proposition}
\label{prop:h1free0}
If $S$ is an $n$-admissible set, then the group
$H^1_{S,0}(K_m,T_{f,n})$ is free
of rank $t:= \#S-2$ over
$\Lambda_{n,m}$.
\end{proposition}
\begin{proof}
Consider the module $M:= (H^1_{S,0}(K_m,T_{f,n}))^\vee$, where
the superscript $\vee$ denotes the Pontryagin dual.
We have
$$
M/\fm M =
(H^1_{S,0}(K_m,T_{f,n})[\fm])^\vee = (H^1_{S,0}(K,T_{f,1}))^\vee \simeq (\Z/p\Z)^t,$$
where the second equality follows from Lemma \ref{lemma:controlS} and the third
isomorphism from Proposition \ref{prop:poitou_tate} with $m=0$ and $n=1$.
Let $\xi_1,\ldots, \xi_t$ be a set of elements of $M$ which map to a basis for
$M/\fm M$. By Nakayama's lemma, these elements
generate $M$ as a $\Lambda_{n,m}$-module, and yield a surjective
map
$\Lambda_{n,m}^t \rightarrow M$
of $\Lambda_{n,m}$-modules.
Proposition \ref{prop:poitou_tate} implies that
$\#M = \#(\Lambda_{n,m}^t)$,
and hence this map is an isomorphism.
It follows that the module $M$ is free of rank $t$, and therefore
$$ H^1_{S,0}(K_m,T_{f,n}) \simeq (\Lambda_{n,m}^\vee)^t.$$
The local ring $\Lambda_m:= \Z_p[G_m]= \Z_p[t]/(t^{p^m}-1)$ is a local complete
intersection in the sense of Definition 5.1 of \cite{ddt}. Hence by Proposition
5.9 of \cite{ddt}, it is Gorenstein in the sense of Definition 5.8
in \cite{ddt}, i.e.,
$$ \Hom_{\Z_p}(\Lambda_m,\Z_p) \simeq \Lambda_m.$$
It follows that
$$ \Hom_{\Z/p^n\Z}(\Lambda_{n,m},\Z/p^n\Z) \simeq \Lambda_{n,m}.$$
This proves Proposition \ref{prop:h1free0}.
\end{proof}
\begin{proposition}
\label{prop:h1free}
If $S$ is an $n$-admissible set, then the group
$H^1_{S,\pm}(K_m,T_{f,n})$ is free
of rank $\#S$ over
$\Lambda_{n,m}$.
\end{proposition}
\begin{proof}
The natural sequence
$$ 0\lra H^1_{S,0}(K_m, T_{f,n}) \lra H^1_{S,\Box}(K_m,T_{f,n}) \lra H^1(K_{m,p},T_{f,n})\lra 0$$
is exact. This assertion
follows from the definition of the objects involved, for all but the penultimate map,
whose surjectivity is a consequence of the second assertion in
Proposition \ref{prop:poitou_tate}.
It follows that the sequence
$$ 0\lra H^1_{S,0}(K_m, T_{f,n}) \lra H^1_{S,\pm}(K_m,T_{f,n}) \lra H^1_\pm(K_{m,p},T_{f,n})\lra 0$$
is exact. Proposition \ref{prop:h1free} now follows from
Corollary \ref{cor:localfree} and Proposition \ref{prop:h1free0}.
\end{proof}
\section{Construction of the Euler system}
\label{sec:euler}
\bigskip
\noindent
We maintain the notations of the previous sections, and
fix an $n$-admissible prime $\ell$ for $f$. Let
$X := X_{N^+,N^-\ell}$ be the Shimura curve introduced in Section
5.1 of \cite{bertolini_darmon}, and let $P_m \in X({\tilde K_m})$ be the
Heegner point of conductor $p^m$ defined in Section 6 of
\cite{bertolini_darmon}, where the integers which are denoted $M^+$ and
$M^-$ in that section are set to be equal to $N^+$ and $N^-\ell$
respectively.
Note that, unlike the setting that is considered in Section 6 of
\cite{bertolini_darmon}, the integer $M^+$ is now assumed to be prime to $p$.
The behaviour of Heegner points under norms (cf. for example
Proposition 3.10 of \cite{darmon_book}) implies that
the Heegner points $P_m$ satisfy the following
compatibilities (expressed as equalities of divisors on $X$) for all
$m\ge 1$:
$$ \Trace_{m}^{m+1}(P_{m+1})= T_p P_m - P_{m-1},$$
where
$\Trace_{m}^{m+1}$ denotes the Galois
trace from the $m+1$-st layer to the $m$-th
layer
(i.e., from $\tilde K_{m+2}$ to ${\tilde K_{m+1}}$, or from
$K_{m+1}$ to $K_m$), and $T_p$ is the $p$-th Hecke operator.
Let $g$ be an eigenform of weight $2$ on $X$ such that
$T_{g,n}\cong T_{f,n}=T$ as Galois modules. Such an eigenform exists by
condition 4 in the definition of $n$-admissible primes: cf. Proposition
3.12 of \cite{bertolini_darmon}.
In order to replace the points $P_m$ on $X$ by degree zero divisors,
we choose a fixed auxiliary prime $q$
which does not divide $N,\ell$, or $p$, and let
$$\tilde{P}_m:=(T_q-(q+1))P_m\in\mbox{Div}^0(X).$$
Denote by the
same symbol the image of this divisor in the Jacobian
$\mbox{Jac}(X)$
of $X$.
Let
$\tilde{\kappa}(\ell)_m$ denote
the image
of $\tilde{P}_m$
in $H^1(\tilde{K}_m, \mbox{Jac}(X)[p^n])$
under the global Kummer map.
Write $\kappa(\ell)_m$ for the image of
$\tilde{\kappa}(\ell)_m$
under the composition
$$
H^1(\tilde{K}_m, \mbox{Jac}(X)[p^n])\lra H^1(\tilde{K}_m, T_{g,n})\lra H^1(K_m, T_{g,n})
\cong H^1(K_m, T).
$$
The first map is induced by ``projection onto the $g$-isotypical component''
$\ds \mbox{Jac}(X)[p^n]\lra T_{g,n}$, the second is co-restriction and the third is
induced by the isomorphism $T_{g,n}\cong T$.
\begin{proposition}
\label{prop:local_behavior}
The element $\kappa(\ell)_m$ belongs to $H^1_{\ell,\fin}(K_m, T)$.
\end{proposition}
\noindent
\begin{proof}
Everything follows from Section 7 and the beginning of Section
8 of \cite{bertolini_darmon} except
the behaviour of the class under localization at primes above $p$. Let $\fp$ be a prime
of $K_m$ above $p$, then by the properties of the Kummer map and of co-restriction
we have $\tilde{\kappa}(\ell)_{m,\fp}\in H^1_{\fin}(K_{m,\fp}, T_{g,n})$. Now apply
Theorem \ref{thm:cont_finite} and deduce that $\tilde{\kappa}(\ell)_{m,\fp}\in
H^1_{\fin}(K_{m,\fp}, T)$. Hence ${\kappa}(\ell)_{m,p}$ belongs to $H^1_{\fin}(K_{m,p}, T)$.
\end{proof}
The classes $\kappa(\ell)_m$ satisfy the compatibility relations under the trace maps
$$\Trace_{m}^{m+1}(\kappa(\ell)_{m+1})=-\kappa(\ell)_{m-1}.$$
Therefore we have
\begin{lemma}
\label{lemma:anih}
Let $\ep$ denote the sign of $(-1)^m$. Then $\om_m^\ep\kappa(\ell)_m=0$.
\end{lemma}
\begin{proof}
Let $\xi_k$ denote the $p^k$-th cyclotomic polynomial
in $T+1$ as in Section \ref{sec:analytic}, and suppose without loss of
generality that $m$ is even. Then
$$ \om_m^+\kappa(\ell)_m= \om_{m-2}^+\xi_m\kappa(\ell)_m
=\om_{m-2}^+\mbox{Tr}_{m-1}^m(\kappa(\ell)_m)
= -\om^+_{m-2}\kappa(\ell)_{m-2}.
$$
The result now follows by induction, using the fact that $T\kappa(\ell)_0=0$.
The proof when $m$ is odd is identical.
\end{proof}
Let $S$ be a square-free
product of primes which is $n$-admissible
in the sense of Definition \ref{def:admissible_set}.
We can view the class $\kappa(\ell)$ as an element of the
larger
$\Lambda_{n,m}$-module
$H^1_{S,\pm}(K_m,T_{f,n})$.
It is useful to do so because of the following proposition:
\begin{proposition}
\label{prop:division}
There exists a unique class
$$\eta(\ell)_m^\ep\in
H^1_{S,\ep}(K_m, T)/\om_m^\ep H^1_{S,\ep}(K_m, T)$$
such that
$$\tom_m^{-\ep}\eta(\ell)_m^\ep=\kappa(\ell)_m.$$
\end{proposition}
\begin{proof}
This follows from the fact that $H^1_{S,\pm}(K_m, T)$ is a free
$\Lambda_{n,m}$-module
by Proposition
\ref{prop:h1free},
using
Lemma \ref{lemma:anih} and
Lemma \ref{lemma:div_p^n}.
\end{proof}
\bigskip
\noindent
Now define the global cohomology classes indexed by the $n$-admissible
primes $\ell$:
\begin{eqnarray}
\kappa (\ell)_m^+ &:=& (-1)^{\frac{m}{2}}\eta(\ell)_m
\in H^1_{S,+}(K_m, T)/\om_m^+ \quad\mbox{ if $m$ is even;} \\
\nonumber
\kappa(\ell)_m^- &:=& (-1)^{\frac{m+1}{2}}\eta(\ell)_m\in H^1_{S,-}(K_m, T)/\om_m^-
\quad\mbox{ if $m$ is odd.}
\end{eqnarray}
An argument identical to the one used in the proof of Lemma
\ref{lemma:norm_compatible} shows that
the sequences $\{\kappa(\ell)_m^+\}_{m\ {\rm even}}$
and $\{\kappa(\ell)_m^-\}_{m\ {\rm odd}}$ are
compatible under co-restriction, so that we can write
\begin{equation}
\label{eqn:defkappaellpm}
\kappa (\ell)^\pm:=\lim_{\leftarrow}\kappa(\ell)_m^\pm.
\end{equation}
This element belongs to
$$
\lim_{\leftarrow}H^1_{S,\pm}(K_m, T)/\om_m^\pm\cong
\lim_{\leftarrow}(H^1_{S,\pm}(K_m, T)\otimes\Lambda/(\om_m^\pm, p^n))
=\hH^1_{S,\pm}(K_\infty, T).
$$
Let $\ell$ be an $n$-admissible prime dividing $S$.
Note that both
$\hH^1_{\rm fin}(K_{\infty,\ell},T)$ and
$\hH^1_{\rm sing}(K_{\infty,\ell},T)$ are
isomorphic to
$\Lambda/p^n$ by
Lemma \ref{lemma:localfreel}.
As in \cite{bertolini_darmon}, the classes $\kappa(\ell)^\pm$ satisfy two
key reciprocity laws relating them to the $p$-adic $L$-functions
$\cL_f^\pm$ defined in Section \ref{sec:analytic}.
The first reciprocity law concerns the properties
of the class $\kappa(\ell)^\pm$ at the prime $\ell$.
\begin{proposition}
\label{prop:reclaw1}
The class $\kappa(\ell)^\pm$ satisfies:
$$v_\ell(\kappa(\ell)^\pm)=0,\qquad
\partial_\ell(\kappa(\ell)^\pm)\equiv\cL_{f}^\pm
\pmod{p^n},$$
where the equality holds in
$\Lambda/p^n$, up to multiplication by elements of $\Z_p^\times$ and
$G_\infty$.
\end{proposition}
\begin{proof}
Let us fix $m\ge 0$ and
consider the following commutative diagram
$$
\begin{array}{cccccccc}
H^1_{S,\pm}(K_m, T)/\om_m^\pm&\stackrel{\partial_{\ell}}{\lra}&H^1(K_{m,\ell},T)/\om_m^\pm
&=&
\Lambda/(\om_m^\pm, p^n)\\
\uparrow&&\uparrow&&\uparrow\\
H^1_{S,\pm}(K_m, T)&\stackrel{\partial_{\ell}}{\lra}&H^1(K_{m,\ell}, T)&=&\Lambda/(\om_m, p^n)\\
\cup&&||&&||\\
H^1_{S,\fin}(K_m, T)&\stackrel{\partial_{\ell}}{\lra}&H^1(K_{m,\ell}, T)&=&\Lambda/(\om_m, p^n)
\end{array}
$$
The proof of the first explicit reciprocity law given in
Section 8 of \cite{bertolini_darmon} adapts
without change
to the classes $\kappa(\ell)_m$ considered here and
yields
$$\partial_{\ell}(\kappa(\ell)_m)=\cL_m$$
up to units in
$\Lambda_{n,m}$.
Moreover as $$\kappa(\ell)_m=(-1)^{[\frac{m+1}{2}]}
\tom_m^\mp\kappa(\ell)_m^\pm,$$ we have
$$
(-1)^{[\frac{m+1}{2}]}\tom_m^\mp\partial_\ell(\kappa(\ell)_m^\pm)=\cL_m
=(-1)^{[\frac{m+1}{2}]}
\tom_m^\mp \cL_m^\pm
$$
in $\tom_m^\mp\Lambda/(\om_m, p^n)$. Using the isomorphism
of Lemma \ref{lemma:div_p^n}
we conclude that $$\partial_{\ell}(\kappa(\ell)_m^\pm)=\cL_m^\pm$$
in $\Lambda/(\om_m^\pm, p^n)$
up to units in this ring.
\end{proof}
\sk\noindent
Let $\ell_1$ and $\ell_2$ be distinct $n$-admissible primes relative
to $f$,
such that $p^n$ divides $\ell_1+1-\epsilon_1 a_{\ell_1}(f)$
and $\ell_2+1-\epsilon_2 a_{\ell_2}(f)$, for $\epsilon_1$ and
$\epsilon_2$ equal to $\pm 1$.
It is further assumed that the pair $(\ell_1,\ell_2)$ is
{\em rigid} in the sense of Section 3.3 of \cite{bertolini_darmon}.
The second reciprocity law
describes
the localization of $\kappa(\ell_1)$
at
$\ell_2$.
Note that this localization
belongs to the finite part of the local cohomology group at
$\ell_2$.
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 the following equalities modulo $p^n$ hold:
\begin{equation}
\label{eqn:congform}
T_q g \equiv a_q(f) g\ \ \ (q\not|N\ell_1\ell_2),\ \ \ \ \
U_q g \equiv a_q(f) g\ \ \ (q|N),
\end{equation}
$$
U_{\ell_1} g\equiv\epsilon_1 g,\ \ \ U_{\ell_2} g\equiv\epsilon_2 g.$$
Furthermore (because of the assumption that
the pair $(\ell_1,\ell_2)$ is rigid)
the form $g$ can be lifted to an
eigenform with coefficients in
$\Z_p$ satisfying (\ref{eqn:congform}) above. This form is
$p$-isolated.
\end{proposition}
\begin{proof} The existence of the mod $p^n$ eigenform $g$,
which relies on the concepts and notations introduced
in Sections 5 and 9 of \cite{bertolini_darmon}, is proved in Theorem
9.3 of \cite{bertolini_darmon}.
\end{proof}
For any class $\kappa \in \hH^1_{S,\pm}(K_\infty,T)$,
and any $n$-admissible prime $\ell$ which
does not divide $S$, write $v_\ell(\kappa)$ for the natural image of
$\kappa$ in $\hH^1_{\rm fin}(K_{\infty,\ell},T_{f,n})$ under the restriction map
at $\ell$. Note again that the target module for $v_\ell$ is
isomorphic to $\Lambda/p^n\Lambda$ by Lemma 2.7 of \cite{bertolini_darmon}.
With these notations we are ready to state the
second explicit reciprocity law.
\begin{proposition}
\label{prop:reclaw2}
The equality
$$v_{\ell_2}(\kappa(\ell_1)^{\pm}) = \cL_g^{\pm}$$ holds in
$\widehat H^1_{\fin}(K_{\infty,\ell_2},T_{f,n})\simeq
\Lambda/p^n\Lambda$, up to
multiplication by elements of $\Z_p^\times$ and
$ G_\infty$.
\end{proposition}
\begin{proof}
This is essentially Theorem 4.2 of \cite{bertolini_darmon}, whose
proof, explained in Section
9 of that article, adapts to the setting where
$a_p=0$, the class $\kappa(\ell_1)$ is replaced by $\kappa(\ell_1)^\pm$
and $\cL_g$ is replaced by $\cL_g^\pm$.
\end{proof}
We record the following consequence of Propositions
\ref{prop:reclaw1} and \ref{prop:reclaw2}:
\begin{corollary}
\label{cor:reciprocity}
For all pairs of $n$-admissible primes $(\ell_1, \ell_2)$ attached to $f$,
the equality
$$v_{\ell_1}(\kappa(\ell_2)^\pm) = v_{\ell_2}(\kappa(\ell_1)^\pm)$$
holds in $\Lambda/p^n\Lambda$, up to multiplication by elements of
$\Z_p^\times$ and $G_\infty$.
\end{corollary}
\section{Proof of the main result}
Following Section 2.1 of \cite{bertolini_darmon}, we make the following
assumptions on the mod $p$ Galois representation attached to $f$
which correspond to some of the hypotheses made in
Assumption \ref{ass:technical} on $E$.
\begin{assumption}
\label{ass:assforms}
The Galois representation attached to $T_{f,1}$ has image isomorphic to $\GL_2(\F_p)$.
Furthermore, for all $\ell$ dividing $N$ exactly, the
Galois representation
$T_{f,1}$ has a unique
$\gal(\bar \Q_\ell/\Q_\ell)$-stable one-dimensional subspace.
\end{assumption}
Thanks to the reciprocity laws given in proposition \ref{prop:reclaw1} and
\ref{prop:reclaw2} of the previous section,
the classes
$\kappa(\ell)^\pm\in\widehat H^1_{\{\ell,\pm\}}(K_\infty,T_{f,n})$
indexed by the $n$-admissible primes attached to $f$ enjoy exactly the same properties
as the classes $\kappa(\ell)$ used in \cite{bertolini_darmon}
in the study of the main conjecture
in the ordinary case.
They will be used to show:
\begin{theorem}
\label{thm:main_form}
Let $f$ be an eigenform in $S_2(\cV/\Gamma)$ with
coefficients in $\Z_p$ which is
$p$-isolated and satisfies assumption
\ref{ass:assforms} above.
Then the characteristic power series of
$(\Sel_{\pm}(K_\infty,T_{f,\infty}))^\vee$ divides the $p$-adic
$L$-function $L_p^\pm(f,K)$.
\end{theorem}
The proof (as well as the statement!) of this theorem
is identical to that of Theorem 4.4. of \cite{bertolini_darmon},
after replacing
$\Sel(K_\infty,T_{f,\infty})$ by
$\Sel_\pm(K_\infty,T_{f,\infty})$
and $\kappa(\ell)$ by $\kappa(\ell)^\pm$.
Before launching into this proof, let us first make the following
general comments.
\begin{enumerate}
\item
Unlike the approach that is followed in \cite{Kato}, where
$p$-Selmer groups are bounded via global cohomology classes whose
local behaviour at the prime $p$ is related to $p$-adic $L$-functions,
our approach adapts to the supersingular
setting the ideas of \cite{bertolini_darmon}, where
the
$p$-Selmer group is controlled using global classes whose local
behaviour at primes $\ell\ne p$ is related to $p$-adic
$L$-series. The main new difficulty in the supersingular
case lies in the construction of the classes $\kappa(\ell)^\pm$ satisfying
the same relation to the Pollack-style $p$-adic $L$-functions
$\cL_f^\pm$ and $\cL_g^\pm$ as the classes $\kappa(\ell)$ did
with $\cL_f$ and $\cL_g$. With these classes in hand, the
argument of \cite{bertolini_darmon}
never involves the local behaviour of the Galois
representations $T_{f,n}$ and $T_{f,n}$ at $p$,
but only at $n$-admissible primes $\ell\ne p$ which split completely
in $K_\infty/K$.
This is why the Euler
system argument in the proof of Theorem 4.4 of \cite{bertolini_darmon}
extends to the
supersingular setting
without raising new difficulties
or requiring substantial modifications.
\item
Our approach to Theorem \ref{thm:main_form} is
to prove it by induction, reducing the statement about
$f$ to an identical one
about $g$ for an appropriately chosen modular form $g$ which is
congruent to $f$ modulo
$p^n$. It is for this reason that a
more general main conjecture (applying to all modular eigenforms
on definite quaternion algebras with $\Z_p$-coefficients,
and not just those associated to
elliptic curves) is needed
even if one is only interested in establishing
Theorem \ref{thm:main} of the
introduction.
\end{enumerate}
\sk\noindent
{\em Proof of Theorem \ref{thm:main_form}}:
For the convenience of the reader, we recall here the main lines of the
argument, with an emphasis on the aspects that
are specific to the supersingular setting.
Note however that we follow the strategy of the
proof of Theorem 4.4
of \cite{bertolini_darmon} very closely, and that the modifications
that need to be made to this proof are comparatively minor.
Proposition 3.1 of \cite{bertolini_darmon},
implies that it is enough to show that
\begin{equation}
\label{eqn:mainphi_prelim}
\varphi(\cL_f^\pm)^2 \mbox{ belongs to }
\Fitt_{\cO}(
(\Sel_{\pm}(K_\infty,T_{f,n}))^\vee
\otimes_\varphi\cO),
\end{equation}
for 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
absolute
ramification degree of $\cO$.
Write $$t_f:= \ord_\pi(\varphi(\cL_f^\pm)).$$
Assume without loss of
generality that
\sk\noindent
1. $t_f<\infty$. (Otherwise, $\varphi(\cL_f^\pm)=0$ and
(\ref{eqn:mainphi}) is trivially verified.)
\sk\noindent
2. The group $(\Sel_{\pm}(K_\infty,T_{f,n}))^\vee\otimes\cO$ is non-trivial.
(Otherwise, its Fitting ideal is equal to $\cO$ and
(\ref{eqn:mainphi}) is trivially verified.)
We propose to show that
\begin{equation}
\label{eqn:mainphi}
\varphi(\cL_f^\pm)^2 \mbox{ belongs to }
\Fitt_{\cO}(
(\Sel_{\pm}(K_\infty,T_{f,n}))^\vee
\otimes_\varphi\cO)
\end{equation}
by induction on $t_f$.
\sk\noindent
We begin by using the classes $\kappa(\ell)^\pm$ to
construct global cohomology
classes that will be used to bound $\Sel_{\pm}(K_\infty,T_{f,n})$.
Let
$\ell$ be any $(n+t_f)$-admissible prime, and
let $S$ be a square-free product of $(n+t_f)$-admissible primes with
$\ell|S$.
Let
$$\kappa(\ell)^\pm\in
\hH^1_{\{\ell\},\pm}(K_\infty,T_{f,n+t_f})
\subset \hH^1_{S,\pm}(K_\infty,T_{f,n+t_f})$$
be the
cohomology class attached to $\ell $ in
(\ref{eqn:defkappaellpm}),
and denote by
$\kappa_\varphi(\ell)^\pm$ the natural image of this class
in
$$ \cM:=\hH^1_S(K_\infty, T_{f,n+t_f})\otimes_\varphi \cO.$$
Note that this module is free over
$\cO/p^{(n+t_f)}$, by Proposition \ref{prop:h1free}.
By Proposition \ref{prop:reclaw1},
$$\ord_\pi(\kappa_\varphi(\ell)^\pm) \le
\ord_\pi(\partial_\ell\kappa_\varphi(\ell)^\pm)
= \ord_\pi(\varphi(\cL_f^\pm)) = t_f,$$
so that
$t:=\ord_\pi(\kappa_\varphi(\ell)^\pm) \le t_f$.
Choose an element $\tilde \kappa_\varphi(\ell)^\pm \in \cM$
satisfying
\begin{equation}
\label{eqn:divpit}
\pi^t \tilde\kappa_\varphi(\ell)^\pm =
\kappa_\varphi(\ell)^\pm.
\end{equation}
Note that $\tilde \kappa_\varphi(\ell)^\pm$ is well defined
modulo the $\pi^t$-torsion subgroup of
$\cM$, which is
contained in the kernel of the natural homomorphism
$$\hH^1_{S}(K_\infty,T_{f,n+t_f})\otimes_\varphi \cO \lra
\hH^1_{S}(K_\infty,T_{f,n})\otimes_\varphi \cO.$$
Let $\kappa_\varphi'(\ell)^\pm$ denote the image
of $\tilde\kappa_\varphi(\ell)^\pm$ in
$\hH^1_S(K_\infty,T_{f,n})\otimes\cO.$
Note that this class does not depend on the choice of ${\tilde \kappa_\varphi(\ell)^\pm}$
satisfying (\ref{eqn:divpit}).
The key properties of the class $\kappa_\varphi'(\ell)^\pm$
are summarized in the following two Lemmas.
\ref{lemma:propertieskappaprime}
and \ref{lemma:relationell} below.
\begin{lemma}
\label{lemma:propertieskappaprime}
The class $\kappa_\varphi'(\ell)^\pm$ belongs to
$\hH^1_{\ell,\pm}(K_\infty,T_{f,n})\otimes_\varphi \cO,$
and
\begin{enumerate}
\item
$\ord_\pi(\kappa_\varphi'(\ell)^\pm)=0$.
\item $v_\ell(\kappa_\varphi'(\ell)^\pm)=0$, and
$\ord_\pi(\partial_\ell\kappa_\varphi'(\ell)^\pm) = t_f-t$.
\end{enumerate}
\end{lemma}
\begin{proof}
The fact that $\kappa_\varphi'(\ell)^\pm$ belongs to
$\hH^1_{\ell,\pm}(K_\infty,T_{f,n})\otimes_\varphi \cO$
follows from the fact that
$\kappa(\ell)^\pm$ belongs to $\hH^1_{\ell,\pm}(K_\infty,T_{f,n+t_f})$.
Property 1 follows from the construction of
$\kappa_\varphi'(\ell)^\pm$, while property 2
is a direct consequence of
Proposition \ref{prop:reclaw1}.
\end{proof}
\begin{lemma}
\label{lemma:relationell}
The residue $\partial_\ell(\kappa_\varphi'(\ell)^\pm)$ belongs to the kernel of
the natural homomorphism
$$ \eta_\ell: \hH^1_{\sing}(K_{\infty,\ell}, T_{f,n})\otimes_\varphi
\cO \lra (\Sel_{\pm}(K_\infty,T_{f,n}))^\vee\otimes_\varphi\cO.$$
\end{lemma}
\begin{proof}
The proof is the same as that of Lemma 4.6 of \cite{bertolini_darmon}.
\end{proof}
\sk\noindent
We now turn to the proof of (\ref{eqn:mainphi})
when $t_f=0$---the basis for the
induction argument.
\begin{proposition}
If $t_f=0$, (i.e., $\cL_f^\pm$ is a unit)
then $(\Sel_{\pm}(K_\infty,T_{f,n}))^\vee$ is trivial.
\end{proposition}
\begin{proof}
The proof is the same as that of Proposition 4.7 of \cite{bertolini_darmon},
which makes no use of the hypothesis that $p$ is ordinary.
\end{proof}
\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 $t=\ord_\pi(\kappa_\varphi(\ell)^\pm)$
is minimal,
among all primes satisfying condition 1.
\end{enumerate}
Proposition \ref{prop:chebotarev} implies that
$\Pi$ is non-empty.
Let $t$ be the common value of $\ord_\pi(\kappa_\varphi(\ell)^\pm)$ for all
$\ell\in\Pi$. Note that $t\le t_f$ by definition.
\begin{lemma}
\label{lemma:drop}
The integer $t$ is strictly less than $t_f$.
\end{lemma}
\begin{proof}
See Proposition 4.8 of \cite{bertolini_darmon}.
\end{proof}
Before stating the next lemma, we need to recall the notion of {\em rigid pairs} of
$n$-admissible primes that is defined in Section 3.3 of \cite{bertolini_darmon}.
This notion relies on the Selmer group
$\Sel_S(\Q,W_f)$ attached to the $3$-dimensional
mod $p$ representation $W_f:= \ad_0(T_{f,1})$
consisting of trace $0$ endomorphisms of $T_{f,1}$, and to a square-free product
$S$ of $1$-admissible primes.
The definition of this Selmer group is the same as in Definition 3.5
of \cite{bertolini_darmon}, except that, since $W_f$ is not ordinary at $p$,
but is crystalline,
the group
$H^1_{\rm fin}(\Q_p,W_f)$ is defined to be the set of cohomology classes that
are crystalline at $p$.
With this change, it is still true that $f$ is $p$-isolated precisely when
$\Sel_1(\Q,W_f)=0$. (This is just Proposition 3.6. of
\cite{bertolini_darmon} whose proof applies just as well
to the case where $f$ is non-ordinary
at $p$.) Following Definition 3.9 of \cite{bertolini_darmon}, we say that a pair
$(\ell_1,\ell_2)$ of $n$-admissible primes is a {\em rigid pair}
if $\Sel_{\ell_1\ell_2}(\Q,W_f)$ is trivial.
\begin{lemma}
\label{lemma:crucial}
There exist primes $\ell_1,\ell_2\in \Pi$ such that
$(\ell_1,\ell_2)$ is a rigid pair.
\end{lemma}
\begin{proof}
See Lemma 4.9 of \cite{bertolini_darmon} whose proof adapts
without change to the supersingular setting.
\end{proof}
\sk
Let $(\ell_1,\ell_2)$ be a rigid pair of $(n+t_f)$-admissible
primes in $\Pi$, whose existence is guaranteed by Lemma
\ref{lemma:crucial}.
By Proposition \ref{prop:reclaw2}, note that
$t=t_g=\ord_\pi(\varphi(\cL_g))$, where
$g$ is the $p$-isolated eigenform in
$S_2(\cT/\Gamma')$ attached to $f$ and $(\ell_1,\ell_2)$ through
proposition \ref{prop:congadmissible}.
Recall the Selmer group
$$\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{f,n})
\subset
\Sel_{\pm}(K_\infty,T_{f,n}) $$
consisting of classes which are locally trivial
at the primes dividing $\ell_1$ and $\ell_2$.
By definition,
there is a natural
exact sequence of
$\Lambda$-modules
\begin{equation}
\label{eqn:esselmerduals}
0\lra S_{\ell_1\ell_2}^f
\lra (\Sel_{\pm}(K_\infty,T_{f,n}))^\vee
\lra (\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{f,n}))^\vee
\lra 0,
\end{equation}
where $S^f_{\ell_1\ell_2}$ denotes the kernel of the natural surjection
of duals of Selmer groups.
Note the natural surjection given by local
Tate duality:
$$ \eta_{f}:
(\hH^1_{\sing}(K_{\infty,\ell_1},T_{f,n}) \oplus
\hH^1_{\sing}(K_{\infty,\ell_2},T_{f,n})) \lra
S_{\ell_1\ell_2}^f$$
induced from the inclusion
$$(S_{\ell_1\ell_2}^f)^\vee \subset H^1_{\fin}(K_{\infty,\ell_1}, T_{f,n})
\oplus H^1_{\fin}(K_{\infty,\ell_2}, T_{f,n}).$$
The domain of $\eta_{f}$ is isomorphic to
$(\Lambda/p^n\Lambda)^2$, by Lemma
2.7 of \cite{bertolini_darmon}.
Let $\eta_{f}^\varphi$ denote the map induced from
$\eta_{f}$ after tensoring by $\cO$ via $\varphi$.
The domain of $\eta_{f}^\varphi$
is isomorphic to $(\cO/p^n\cO)^2$.
By Lemma \ref{lemma:relationell},
the kernel of $\eta^\varphi_{f}$
contains the vectors
$(\partial_{\ell_1}\kappa_\varphi'(\ell_1)^\pm,0)$ and
$ (0,\partial_{\ell_2}\kappa_\varphi'(\ell_2)^\pm)$
in
$$ \left(\hH^1_{\sing}(K_{\infty,\ell_1},T_{f,n}) \oplus
\hH^1_{\sing}(K_{\infty,\ell_2},T_{f,n}) \right) \otimes_\varphi\cO
\simeq
(\cO/p^n\cO)^2.$$
By part 3 of Lemma \ref{lemma:propertieskappaprime},
$$t_f-t_g=\ord_\pi(\partial_{\ell_1}\kappa_\varphi'(\ell_1)^\pm)=
\ord_\pi(\partial_{\ell_2}\kappa_\varphi'(\ell_2)^\pm).$$
Hence
\begin{equation}
\label{eqn:starpi}
\pi^{2(t_f-t_g)} \mbox{ belongs to the Fitting ideal of }
S_{\ell_1\ell_2}^f\otimes_\varphi \cO.
\end{equation}
One may repeat the same argument with the modular form $g$. Thus we have
an exact sequence similar to (\ref{eqn:esselmerduals}) but involving $g$
instead
of $f$:
\begin{equation}
\label{eqn:esselmerdualsg}
0\lra S_{\ell_1\ell_2}^g
\lra (\Sel_\pm(K_\infty,T_{g,n}))^\vee
\lra (\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{g,n})))^\vee
\lra 0,
\end{equation}
as well as a surjection given by local Tate duality:
$$ \eta_{g}:
(\hH^1_{\fin}(K_{\infty,\ell_1},T_{f,n}) \oplus
\hH^1_{\fin}(K_{\infty,\ell_2},T_{f,n})) \lra
S_{\ell_1\ell_2}^g.$$
By global reciprocity, the kernel of the map
$\eta_{g}^\varphi$ obtained from $\eta_g$ after tensoring
by $\cO$ via $\varphi$ contains the elements
$$ (v_{\ell_1}\kappa_\varphi'(\ell_1)^\pm,
v_{\ell_2}\kappa_\varphi'(\ell_1)^\pm) =
(0,v_{\ell_2}\kappa_\varphi'(\ell_1)^\pm) $$
as well as $(v_{\ell_1}\kappa_\varphi'(\ell_2)^\pm, 0)$. But
$$
\ord_\pi(v_{\ell_2}\kappa_\varphi'(\ell_1)^\pm) =
\ord_\pi(v_{\ell_1}\kappa_\varphi'(\ell_2)^\pm) = t_g-t = 0.$$
It follows that the module $S_{\ell_1\ell_2}^g\otimes_\varphi
\cO$ is trivial, and the natural surjection
\begin{equation}
\label{eqn:isomg}
(\Sel_{\pm}(K_\infty,T_{g,n}))^\vee\otimes_\varphi\cO \lra
(\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{g,n}))^\vee\otimes_\varphi\cO \mbox{ is an isomorphism}.
\end{equation}
Recall that, by Lemma \ref{lemma:drop},
$$t_g < t_f, $$
and that the eigenform $g$ satisfies all the hypotheses of Theorem
\ref{thm:main_form}. (The fact that $g$ is $p$-isolated follows from the fact
that
$(\ell_1,\ell_2)$ is a rigid pair of admissible primes.)
By the
induction hypothesis,
\begin{equation}
\label{eqn:starstar}
\varphi(\cL_g)^2 \mbox{ belongs to the
Fitting ideal of } (\Sel_{\pm}(K_\infty,T_{g,n}))^\vee\otimes_\varphi
\cO.
\end{equation}
The theory of Fitting ideals
implies that
\begin{eqnarray*}
\pi^{2t_f} &=& \pi^{2(t_f-t_g)} \pi^{2t_g} \\ &\in&
\Fitt_\cO(S^f_{\ell_1\ell_2}\otimes\cO)\Fitt_\cO((\Sel_{\pm}(K_\infty,T_{g,n}))^\vee\otimes\cO),
\mbox{ \ by } (\ref{eqn:starpi}) \mbox{ and } (\ref{eqn:starstar}) \\
&= &
\Fitt_\cO(S^f_{\ell_1\ell_2}\otimes\cO)
\Fitt_\cO((\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{g,n}))^\vee\otimes\cO), \quad\mbox{by }
(\ref{eqn:isomg}).
\end{eqnarray*}
But note that $\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{g,n}) = \Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{f,n})$
by definition, in light of the fact that the Galois modules
$T_{f,n}$ and
$T_{g,n}$ are isomorphic, and that the local conditions used to define the
associated
Selmer groups are the same outside of
$\ell_1$ and $\ell_2$.
It follows that
\begin{eqnarray*}
\pi^{2t_f} & \in&
\Fitt_\cO(S^f_{\ell_1\ell_2}\otimes\cO)
\Fitt_\cO((\Sel_{\ell_1\ell_2,\pm}(K_\infty,T_{f,n}))^\vee\otimes\cO) \\
&\subset& \Fitt_\cO((\Sel_{\pm}(K_\infty,T_{f,n}))^\vee\otimes\cO), \mbox{ by }
(\ref{eqn:esselmerduals}).
\end{eqnarray*}
Hence
$\varphi(\cL_f^\pm)^2$ belongs to the
Fitting ideal of $(\Sel_{\pm}(K_\infty,T_{f,n}))^\vee\otimes_\varphi\cO$,
and
(\ref{eqn:mainphi}) is therefore proved.
Theorem
\ref{thm:main_form} follows.
Note finally that Theorem \ref{thm:main} follows from Theorem
\ref{thm:main_form} specialized to the case where $f$ has integer Hecke
eigenvalues, and hence corresponds to a modular
elliptic curve $E$ via the Eichler-Shimura construction
combined with the Jacquet-Langlands correspondence.
\begin{thebibliography}{9}
\bibitem[BD1]{bd_derived_regulators}
M.~Bertolini and H.~Darmon.
{\em Derived heights and generalized Mazur-Tate regulators}.
Duke Math. J. {\bf 76} (1994) 75--111.
\bibitem[BD2]{bertolini_darmon}
M.~Bertolini and H.~Darmon. {\em Iwasawa's Main
Conjecture for elliptic curves
over anticyclotomic
$\Z_p$-extensions}. Annals of Mathematics {\bf 162} (2005) 1-64.
\bibitem[BK]{bloch_kato}
S.~Bloch and K.~Kato, {\em L-functions and Tamagawa numbers of motives}, The Grothendieck
Festschrift, I, (1990), 333-401, Birkh\"auser.
\bibitem[Br]{breuil}
C.~Breuil, {\em Repr\'esentations $p$-adiques semi-stables et transversalit\'e
de Griffith},
Math.Ann.{\bf 307} (1997), 191-224
\bibitem[Da]{darmon_book}
H.~Darmon. Rational points on modular elliptic curves.
CBMS Regional Conference Series in Mathematics, {\bf 101}.
Published for the Conference Board of the Mathematical Sciences,
Washington, DC; by the American Mathematical Society, Providence, RI, 2004.
\bibitem[DDT]{ddt}
H.~Darmon, F.~Diamond, and R.~Taylor, {\em Fermat's Last Theorem},
Current Developments in Mathematics 1, 1995, International Press, pp. 1-157.
Reprinted in
Elliptic curves, modular forms \& Fermat's last theorem (Hong Kong, 1993), 2--140,
International Press, Cambridge, MA, 1997.
\bibitem[Fa]{faltings}
G.~Faltings, {\em Integral crystalline cohomology over very ramified valuation rings},
Journal of AMS, {\bf 12}, No 1, (1999), 117-144
\bibitem[FL]{fontaine_laffaille}
J.-M.~Fontaine and G.~Laffaille, {\em Constructions de repr\'esentations
$p$-adiques}, Ann.Sci.\'Ec.Norm.Sup., {\bf 15}, (1988), 547-608
\bibitem[GIP]{greenberg_iovita_pollack}
R.~Greenberg, A.~Iovita, R.~Pollack, {\em On Iwasawa Invariants of
Elliptic Curves at Supersingular Primes}, preprint
\bibitem[IP]{iovita_pollack}
A.~Iovita and R.~Pollack. {\em
Iwasawa theory of elliptic curves at supersingular
primes over $\Z_p$-extensions of number fields.}
Crelle, to appear.
\bibitem[Ka]{Kato}
K.~Kato,
{\em $p$-adic Hodge theory and values of zeta functions of
modular forms}. in
Cohomologies $p$-adiques et applications arithm\'etiques. III.
Ast\'erisque No. 295 (2004), ix, 117--290.
\bibitem[Kob]{Kobayashi02}
S. Kobayashi, {\em Iwasawa theory for
elliptic curves at supersingular primes},
Invent. Math. 152 (2003), no. 1, 1--36.
\bibitem[Kol]{kolyvagin}
V.A.~Kolyvagin,
{\em Finiteness of $E(\Q)$ and $\sha(E,\Q)$
for a subclass of Weil curves}.
Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3,
522--540, 670--671; translation in Math. USSR-Izv. 32 (1989),
no. 3, 523--541.
\bibitem[Ku]{Kurihara01}
M. Kurihara, On the Tate Shafarevich groups over cyclotomic fields
of an elliptic curve with supersingular reduction I,
Invent. Math. {\bf 149} (2002), 195--224.
\bibitem[PR1]{PR90}
B. Perrin-Riou, {\em Th\'eorie d'Iwasawa $p$-adique locale et globale}.
Invent. Math. {\bf 99} (1990), no.~2, 247--292.
\bibitem[PR2]{PR93}
B. Perrin-Riou, {\em Fonctions $L$ $p$-adiques d'une courbe elliptique et
points rationnels}.
Ann. Inst. Fourier (Grenoble) {\bf 43} (1993), no.~4, 945--995.
\bibitem[PR3]{PR94}
B. Perrin-Riou, {\em Th\'eorie d'Iwasawa des repr\'esentations $p$-adiques sur
un corps local}.
Invent. Math. {\bf 115} (1994), no.~1, 81--161.
\bibitem[PR4]{PR01}
B. Perrin-Riou, {\em Arithm\'etique des courbes elliptiques \`a r\'eduction
supersinguli\`ere en $p$},
preprint.
\bibitem[Po1]{Pollack02}
R.~Pollack. {\em On the $p$-adic $L$-function of a modular form at a
supersingular prime}.
Duke Mathematical Journal {\bf 118} (2003) no. 3, 523--558.
\bibitem[Po2]{Pollack03}
R. Pollack, An algebraic version of a theorem of Kurihara,
Journal of Number Theory {\bf 110/1} (2004), 164--177.
\bibitem[Ta]{Tate}
J. Tate, Duality theorems in Galois cohomology over number fields,
in {\it Proc. Internat. Congr. Mathematicians (Stockholm, 1962)},
288--295, Inst. Mittag-Leffler, Djursholm, 1963.
\bibitem[Va]{vatsal}
V. Vatsal, Uniform distribution of Heegner points, Invent. Math. {\bf 148}
(2002), no.~1, 1--46.
\end{thebibliography}
\end{document}