% AMS-TeX
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\leftheadtext{Kenneth A. Ribet}
\rightheadtext{component groups}

\newcount\sectCount

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\edef\nonouterhead{\noexpand\head}

\def\section#1\endhead{\advance\sectCount by 1%
\nonouterhead\the\sectCount. #1\endhead\noindent}

% As usual, we automatically number the references

\newcount\refCount
\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}
  
   \newref BLR %not boston-lenstra-ribet, but rather...
   \newref CAR
   \newref CSZ
   \newref DERA
   \newref DIAMONDREF
   \newref EDIX
   \newref EDIXINVENT
   \newref JL
   \newref KATZ
   \newref KATZMAZUR
   \newref EIS %Mazur's Eisenstein ideal paper
   \newref RAY %Raynaud's Asterisque paper
   \newref FERMAT
   \newref BX
   \newref MOTIVES
   \newref DUKE
   \newref SERRELETTER
   \newref SILV %silverman book
   \newref TUNN
   \newref WILES



\def\startproof{\demo{Proof}}


\def\endproof{{\unskip\nobreak\hfil\penalty50\hskip1em
  \hbox{}\nobreak\hfil$\square$%
  \parfillskip=0pt \finalhyphendemerits=0
  \par}\enddemo}

\def\leqno(#1){\eqno{\hbox to .01pt{}\rlap{\rm \hskip
  -\displaywidth(#1)}}}

% a bunch of fairly innocuous math macros here
\def\pmone{\{\pm1\}}
\let\phi\varphi
\def\U#1{(\Z/#1\Z)^\ast}
\def\diamond#1{\langle #1\rangle}
\def\Q{{\bold Q}}
\def\Z{{\bold Z}}
\def\C{{\bold C}}
\def\Qbar{\overline{\Q}}
\def\omegabar{\overline{\omega}}
\let\doublemap\rightrightarrows
\def\sbar{\overline{s}}
\def\Qsbar{Q_{\sbar}}
\def\F{{\bold F}}
\def\Fbar{\overline{\F}}
\def\Fq{\F_q}
\def\T{{\Bbb T}}
\def\m{{\goth m}}
\def\jo#1{J_0(#1)}
\def\go#1{\Gamma_0(#1)}
\def\gammaone#1{\Gamma_1(#1)}
\def\hlinefill{\leaders\hrule height 3pt depth -2.5pt\hfill}
\def\emrule{\thinspace\hbox to 0.75em{\hlinefill}\thinspace}
\def\GalQ{\Gal(\Qbar/\Q)}
\def\seteq{\mathrel{:=}}
\def\SL#1{{\bold{SL}}(2,#1)}
\def\GL#1{{\bold{GL}}(2,#1)}

% this could be toyed with even more
\def\notdivide{\setbox1=\hbox{$|$\llap{%\raise2pt -- nah!
\hbox{/}\kern-0.75pt}}\mathrel{\box1}}

% make the \circ a little smaller
\let\tempcirc=\circ
\def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}

% make a lot of mathops
\def\newop#1
{\expandafter\def\csname #1\endcsname{\mathop{\roman{#1}}\nolimits}}
\newop Aut
\newop Ann
\newop Gal 
\newop End 
\newop Ind
\newop Supp 
\newop Hom
\newop Frob
\newop Pic
\newop Spec
\newop coker
\newop tr
\newop Ver




\topmatter

 \title
Irreducible Galois representations arising from
component groups of Jacobians
 \endtitle

 \author
Kenneth A. Ribet
 \endauthor

 \affil
University of California, Berkeley
 \endaffil
\address{UC Mathematics Department, Berkeley, CA  94720-3840 USA}
\endaddress

\email ribet\@math.berkeley.edu \endemail


\thanks 
This work was partially supported by NSF Grant \#DMS 93-06898.
The author wishes to thank 
B.~Edixhoven,
H.~W.~Lenstra, Jr.,
D.~Lorenzini,
N.~Skoruppa,
D.~Ulmer,
and
A.~Wilkinson
for helpful correspondence and suggestions.
We also thank
the International Scientific Committee
and the local organizers
of the Hong Kong
conference on Fermat's Last Theorem
for their generous
invitation.
\endthanks

\endtopmatter

\document
\section Introduction\endhead
Much has been written about component groups of
N\'eron models of Jacobians of modular curves.
In a variety of contexts, these groups have been
shown to be~``Eisenstein,'' which implies that they
can be neglected in the study of irreducible
two-dimensional representations of~$\GalQ$.

The first theorem to this effect may be extracted
from
Mazur's landmark
article \cite\EIS, which concerns the Jacobian $\jo N$
when $N$ is a prime.  In this article, Mazur 
studies the action of Hecke operators on the ``cuspidal
subgroup'' of the Jacobian, and obtains 
information about
the relevant component group as an
application~\cite{\EIS, p.~98}.
More precisely,
consider the fiber
in characteristic~$N$ of the N\'eron model of~$\jo N$.
This commutative group scheme is an extension of a finite
\'etale group scheme $\Phi$ by an algebraic torus; one
is interested in the functorial action of Hecke operators
on~$\Phi$.  Mazur proves 
for all prime numbers $p\ne N$ that the $p$th Hecke
operator $T_p$ acts on~$\Phi$ by multiplication by~$1+p$,
and also that
$T_N$ is the identity on~$\Phi$.
The ``Eisenstein'' terminology arises from the fact that
$1+p$ is the $p$th coefficient of the standard
Eisenstein series of weight~two on~$\go N$.

In his first article on Serre's conjectures~\cite\FERMAT,
the author generalized
Mazur's result to the Jacobian $\jo N$
where $N$ is the product of a positive
integer $M$ and a prime number $q$ prime to~$M$.
According to
Theorem~3.12 of~\cite\FERMAT,
one again has the
identity $T_p=1+p$ for all prime numbers $p$
prime to~$N$
on the group of connected components of the
N\'eron reduction $\jo N_{/\Fq}$.
(A slightly more refined statement appears as
Theorem~3.22 of~\cite\FERMAT; for the proof,
see~\cite\BX.)
Subsequently,
B.~Edixhoven showed in~\cite\EDIX\
that
an analogous result holds for every reduction $\jo N_{/\Fq}$
with $q>3$; here, the new element is that
$N$ is allowed to be divisible by
an arbitrary power of~$q$.

In another article on Serre's conjectures~\cite\MOTIVES,
the author
discusses
the component group 
$\Phi$
attached
to the mod~$q$ reduction of the Jacobian of the modular
curve derived from $\gammaone N\cap\go q$;
here $N$ is a positive integer and $q$ is a prime number
not dividing~$N$.
The analysis of~\cite\MOTIVES, pp.~672--673 proves the 
identity
$T_p=1+p$ on~$\Phi$ for all prime numbers $p\notdivide qN$
and establishes at the same time that
the ``diamond bracket'' operators $\diamond d$
act as the identity on~$\Phi$.

In this article, we consider the situation in which
$\gammaone N $ is replaced by a subgroup $\Gamma$ 
intermediate between $\go N$ and~$\gammaone N$.
In other words, we study the component group $\Phi_\Gamma$
associated to the mod~$q$ reduction coming from
$\Gamma\cap\go q$.  (We continue to assume that
$q$ is prime to~$N$.)  Since $\Phi_\Gamma$ is
Eisenstein in the two extreme situations $\Gamma=\go N$
and~$\Gamma=\gammaone N$, it is natural to ask whether
$\Phi_\Gamma$ is Eisenstein for intermediate
groups.

In order to rule out trivial counterexamples,
we will
generalize slightly
our
definition of~``Eisenstein.''
Let $\Cal S$ be
the complex vector space of weight-two
cusp forms on~$\Gamma\cap\go q$, and let
 $\widetilde\T$ be the subring of~$\End\Cal S$
generated by
the Hecke operators $T_n$ and $\diamond d$ for $n$
a prime number prime to~$qN$ and $d$ prime to~$N$.
As we shall recall below, there is a natural action of
these operators on~$\Phi_\Gamma$.
We shall be interested in
the set of maximal ideals $\m$ of~$\widetilde\T$ which lie in
the support of~$\Phi_\Gamma$ as a $\widetilde\T$-module, i.e.,
the set of maximal ideals
in the
image of~$\widetilde\T$ in~$\End\Phi_\Gamma$.

To each~$\m$, one associates as usual
 a continuous semisimple representation
$$\rho_{\m}\:\GalQ\to\GL{\widetilde\T/{\m}}.$$  This representation
is characterized up to isomorphism by the fact that the trace
of $\rho_{\m}(\Frob_p)$ is~$T_p$ mod~$\m$ 
and the determinant of this matrix is~$p\diamond p$ mod~$\m$,
for all but finitely many
prime numbers~$p$.  (Here, $\Frob_p$
is an arithmetic Frobenius element for~$p$ in~$\GalQ$.
See, e.g., \cite{\MOTIVES, \S7}
for some of the relevant background.)
It is worth stressing that
$\Phi_\Gamma$ gives rise through this construction
to
a collection of two-dimensional representations
of~$\GalQ$, despite the fact there is no natural action of~$\GalQ$
on the group~$\Phi_\Gamma$.

We shall say that $\Phi_\Gamma$ is {\it Eisenstein\/}
 if all $\m$
in the support of~$\Phi_\Gamma$ give rise to representations
$\rho_\m$ which are {\it reducible}.
We say that $\Phi_\Gamma$ is {\it strongly Eisenstein\/}
if the operators $\diamond d$ are 
trivial on~$\Phi_\Gamma$ and one has $T_p=1+p$ on~$\Phi_\Gamma$
for all but finitely many prime numbers~$p$.
It is easy to show that ``strongly Eisenstein'' implies
``Eisenstein'' by using the Cebotarev Density Theorem
and the Brauer-Nesbitt Theorem~\cite{\FERMAT, 5.2c}.


To be sure, one's guess that $\Phi_\Gamma$
is~Eisenstein
turns out to be not very far
off the mark.  Indeed, $\Phi_\Gamma$ is 
strongly Eisenstein
in most cases, and
the prime-to-6 part of~$\Phi_\Gamma$
is strongly Eisenstein in {\it all\/} cases.
Nonetheless, the blanket assertion that $\Phi_\Gamma$
is Eisenstein is definitely false.
To convince the reader of this fact, we exhibit in
the next section some non-Eisenstein component groups; our
construction is  a digression which is intended to
motivate the more systematic study which~follows.

This article's main contribution is an
analysis of~$\Phi_\Gamma$ in the case of general
$\Gamma$ and~$q$.  
As we indicated above, the $\ell$-primary parts of $\Phi_\Gamma$
are strongly
Eisenstein (and cyclic as abelian groups) for all prime
numbers $\ell\ge5$. After recalling this simple result,
we 
investigate the 2-primary
and the 3-primary components of~$\Phi_\Gamma$.
In case $\Phi_\Gamma$ is non-Eisenstein,
we identify the non-Eisenstein ``pieces'' and
describe on these pieces the action of
the Hecke operators $T_n$ and $\diamond d$.

\kern0pt From the point of view of Galois representations, our
main result is the identification of all irreducible
representations with values in~$\GL{\Fbar_2}$
and~$\GL{\Fbar_3}$ that can be associated with component
groups.  These are induced representations, coming
from certain
characters $\Gal(\Qbar/\Q(\sqrt{-1}))\to\Fbar_2^\ast$
and $\Gal(\Qbar/\Q(\sqrt{-3}))\to\Fbar_3^\ast$, respectively.
For a character $\theta$ to intervene in some component group,
it is necessary and sufficient that 
all residue classes modulo its conductor be represented by
rational integers.

This article is an outgrowth of the author's talks at
the Hong Kong conference on Fermat's Last Theorem.
These talks outlined 
portions of
A.~Wiles's  manuscript~\cite\WILES. 
This manuscript provides ample motivation for
a detailed analysis of mod~3
representations of~$\GalQ$ which arise from
modular forms.  Indeed,
Wiles's attack on
the Taniyama-Shimura conjecture begins with a theorem
of J.~Tunnell~\cite\TUNN\
which implies that 
the mod~3 representation of~$\GalQ$ arising from an
elliptic curve over~$\Q$ is associated to some 
Hecke eigenform. Assuming that
this mod~3 representation is irreducible, one
shows that the eigenform may be chosen
in accordance with the conjectures of~\cite\DUKE.
For this,
attention must be directed to the special
problems posed by mod~3 representations; see F.~Diamond's
article in this volume~\cite\DIAMONDREF, which provides
an update on Serre's conjectures.
In reflecting on mod~3 representations,
the author was forced to abandon his
``axiom'' that component groups are Eisenstein;
this led to the study which is presented below.

\section Some non-Eisenstein component groups\endhead
Consider
the irreducible continuous representation
$\rho\:\GalQ\to\GL{\F_3}$ arising from
the space of forms 
of weight two on~$\gammaone{13}$.
This space has dimension~two, and the two Hecke eigenforms
in the space are complex conjugates of each other.
Their coefficients lie in
the ring of integers of~$\Q(\sqrt{-3})$;
$\rho$ is the mod~$(\sqrt{-3})$ representation of~$\GalQ$
associated with either of them.
The determinant of $\rho$ is the product of the mod~3
cyclotomic character of~$\GalQ$ and the character of
order~two
which corresponds to the quadratic
extension $\Q(\sqrt{13})$ of~$\Q$.
Let
$H$ be the group of squares in~$(\Z/13\Z)^\ast$.
Then
conjecture (3.2.4$_\roman?$) of Serre's article~\cite\DUKE\
predicts that $\rho$ should arise from a Hecke eigenform
in the space of weight-two cusp forms on the
group
$$\Gamma = \Gamma_H(13)\seteq\left\{ \, \pmatrix
a&b \cr c&d\cr
\endpmatrix \in\SL\Z \Bigm| c\equiv0 \hbox{ mod }13,\ d\in H
\,\right\}.$$
However, as Serre pointed out in a letter
to the author~\cite\SERRELETTER,
this space of forms is zero\emrule the conjecture
is false as stated.

One way to deal with this apparent difficulty is to
reformulate Serre's conjecture as a relation between
mod~$p$ Galois representations and mod~$p$ modular
forms in the sense of~Katz~\cite\KATZ.
Such a reformulation is presented as
Conjecture~4.2 of Edixhoven's article~\cite\EDIXINVENT,
which attributes the reformulation to Serre.
The characters which appear in the conjecture are
then naturally $\Fbar_p^\ast$-valued, and the
difficulty becomes invisible.
On the other hand, 
this solution hides a problem that may be genuinely of
interest, since one wants to 
characterize those spaces of forms which give rise to
a
given Galois representation.

One certainly
knows that problems of the sort
exemplified by~$\rho$
occur only for mod~2 and mod~3
representations of~$\GalQ$.
Further,
among all mod~3 representations, only those
which become abelian
on $\Gal(\Qbar/\Q(\sqrt{-3}))$ can give trouble, 
cf.~\cite{\CAR, p.~796}.  Finally, let
$q>2$ be a prime number which is congruent to~2
modulo~3.  Then the difficulty in the case of~$\rho$
``disappears'' when
the level is augmented by an auxiliary $\go q$
structure:
Lemme~1 of~\cite\CAR\
implies that $\rho$ arises from the space of weight-two
cusp forms on~$\Gamma\cap\go q$.
(The author is grateful to N.~Skoruppa for undertaking
a numerical verification of this assertion when $q=5$.)

Let $J$ be the Jacobian of
the modular curve over~$\Q$
associated with $\Gamma\cap\go q$.
(We again write $J$ for the N\'eron model of this
abelian variety.)
As above, we let $\Phi_\Gamma$ be the component group
associated with the reduction of~$J$ mod~$q$.

\proclaim{Theorem 1}
The group $\Phi_\Gamma$ is non-Eisenstein.
\endproclaim

\startproof
Let $V$ be a two-dimensional $\F_3$-vector space affording
the representation~$\rho$, and for each $r\notdivide13q$
let $a_r\in\F_3$ be the trace of~$\rho(\Frob_r)$, where
$\Frob_r$ is a Frobenius element for~$r$ in~$\GalQ$.  
The fact that $\rho$ arises from~$\Gamma\cap\go q$ implies
that there is a $\GalQ$-equivariant embedding
$V\hookrightarrow J(\Qbar)$ with the following property:
For each prime number $r\notdivide13q$, the endomorphism $T_r$
of~$J$ acts on~$V$ as the homothety~$a_r$.
Since $q$ is prime to~39, $V$ is
unramified at~$q$.  Thus (after choosing a place of~$\Qbar$
lying over~$q$), we may view $V$ as embedded in~$J_{/\F_q}(\Fbar_q)$.

We claim now that 
$V$ does not land entirely in the connected component
$J^o_{/\F_q}$ of~$J_{/\F_q}$.  In other
words, we assert that the image $W$ of~$V$ in
the group of connected components of~$J_{/\F_q}$ is non-trivial.
Since the group of connected components in question
is none other
than~$\Phi_\Gamma$, our claim implies
that $\Phi_\Gamma$ contains a non-zero subgroup of exponent~3
on which each
Hecke operator $T_r$ acts as~$a_r$.  That the $a_r$ are traces
of an {\it irreducible\/} representation indicates that this
subgroup is
non-Eisenstein, thereby proving the Theorem.

The claim concerning~$V$ is proved by an appeal
to results of Deligne-Rapoport \cite\DERA\
and general
theorems of Grothendieck and Raynaud.  This body of work
is summarized
in~\cite\RAY, \cite{\FERMAT, \S2}, \cite{\MOTIVES, \S8}, 
and the
discussion which occurs in \S3\ below.
The main point is that by using~\cite{\DERA, Th.~6.9, p.~286},
one sees that $J^o_{/\F_q}$
is a certain extension of an abelian variety by a torus.
The abelian variety in question appears as
the product of two copies of the Jacobian
of the modular curve associated with the group~$\Gamma$.
However, this modular curve has genus~zero, so its Jacobian is~zero.
Thus $J^o_{/\F_q}$ coincides with its ``toric part" $T$.
Therefore, the statement that $V$ falls entirely in
$J^o_{/\F_q}(\Fbar_q)$ is the statement that $V$ is a
subgroup of~$T(\Fbar_q)$.  The argument given at the
conclusion of~\cite{\MOTIVES, \S8} shows that the inclusion
$V\hookrightarrow T(\Fbar_q)$ is possible only when
$q\equiv1$ mod~3; the choice of~$q$ thus guarantees that
$W$ is~non-zero. 
This proves the claim.\endproof


\section A concrete description of $\Phi_\Gamma$\endhead
In this section, we present a nuts-and-bolts description
of~$\Phi_\Gamma$ as an abelian group furnished with a
family of Hecke operators $T_r$ and~$\diamond d$.
This material is now quite well known, at
least in the case where $\Gamma=\go N$~\cite{\BLR, \RAY}.

Let $N$ be a positive integer.
Subgroups $\Gamma$ between $\gammaone N$ and~$\go N$
are in 1-1 correspondence with
subgroups
of~$\U N$.  Let $H$ be such a subgroup, and set
$$\Gamma = \Gamma_H(N) =
\left\{ \, \pmatrix
a&b \cr c&d\cr
\endpmatrix \in\SL\Z \Bigm| c\equiv0 \hbox{ mod }N,\ d\in H
\,\right\}.$$
Thus $\Gamma=\gammaone N$ in case $H$ is the trivial
subgroup of~$\U N$, while $\Gamma=\go N$
if $H=\U N$.
The modular curve corresponding to~$\Gamma$
will be called~$X_H(N)$; thus, $X_H(N)$ is the
quotient of~$X_1(N)$ by the image of~$H$ in
the Galois group
$\U N/\pmone$ of the covering $X_1(N)\to X_0(N)$.
Postponing the assumption that $q$ is~prime, we let
$q$ be a positive integer prime to~$N$, and put
$$ \Gamma(H,q) = \Gamma_H(N)\cap\go q.$$
(A more precise, but less compact,
 name for this group would have been $\Gamma_H(N,q)$)

The modular curve $X(H,q)$ corresponding
to~$\Gamma(H,q)$ is the quotient of~$X_1(Nq)$ by the
image of $H\times(\Z/q\Z)^\ast$ in $(\Z/Nq\Z)^\ast/\pmone$.
This image is unchanged if we augment $H$ by $\pmone$; thus
we can, and will, assume that $H$ contains~$-1$.
(If $N\le2$, we have $-1=+1$ in~$\U N$, and we
are imposing no condition on~$H$.)
We view $X(H,q)$ as classifying elliptic curves $E$
which are furnished with a subgroup $C$ of order~$q$ and a
point $P$ of order~$N$.   In the classification,
the point $P$ is considered ``mod~$H$" in the sense
that the triples
$(E,C,P)$ and~$(E,C,hP)$ are identified for all $h\in H$.
(Compare \cite{\DERA, IV, \S3}.)  The orbit of~$P$ mod~$H$
will be called~$\alpha_P$; we will say that $X(H,q)$
classifies triples $(E,C,\alpha)$ where $\alpha$ is a point
of order~$N$ on~$E$ which is taken ``mod~$H$."

Define $J_H(N)$ and~$J(H,q)$ to be the Jacobians of
the modular curves $X_H(N)$ and~$X(H,q)$, respectively.

\kern0pt From now on, we assume that $q$ is a {\it prime\/} number.
For convenience, we impose the assumption $q\ge5$ at this point.
(The cases $q=2$ and~$q=3$ presumably could be included with
little modification in what follows.)
Consider the reduction of~$J(H,q)$
modulo~$q$.  
The group of components associated with this reduction
is the group~$\Phi_{\Gamma}$ in the discussion above;
of course, this group depends on~$q$
as well as on~$\Gamma$.

A description of~$\Phi_\Gamma$
can be deduced
in a standard way from the theorem of
Deligne and Rapoport which we cited
above~\cite{\DERA, Th.~6.9, p.~286}. 
The theorem provides
a model $\Cal C$ of~$X(H,q)$ over~$\Z_q$
which is an ``admissible curve" in the sense of
\cite{\JL, \S3} and~\cite{\FERMAT, \S2}.
The special fiber of~$\Cal C$ has two irreducible
components, each isomorphic to the modular curve $X_H(N)$.

The set of singular points of ${\Cal C}_{\Fbar_q}$
is in bijection with the set $S$ of supersingular
$\Fbar_q$-valued
points of~$X_H(N)$, i.e., those points
which arise from pairs $(E, P)$ where $E$
is a supersingular elliptic curve over~$\Fbar_q$
and $P$ is a point of order~$N$ on~$E$.
To each $s\in S$ we associate an integer $e(s)\in\{1,2,3\}$
in the following way: If $s$ is represented
by the pair $(E,P)$, then $2e(s)$ is the number of automorphisms
of~$E$ which map $P$ to some point $hP$ with $h\in H$; in
other words, $2e(s)$ is the number of automorphisms
of~$(E,\alpha_P)$.
(Because $-1$ belongs to~$H$, the automorphism ``$-1$'' of~$E$
induces an automorphism of~$(E,\alpha_P)$.  The number of such
automorphisms is then 2, 4, or~6 because we have 
assumed $q\ge5$.)
The singular point of~$\Cal C$  corresponding to~$s$
is then described in~$\Cal C$
by the local equation $XY=q^{e(s)}$.

Consider now the diagonal pairing on~$\Z^S$ for which
$\langle s , s \rangle$
is the positive integer $e(s)$.
Let $L$ be the group of degree~0 elements in~$\Z^S$, and
let $\iota\: L\hookrightarrow\Hom(L,\Z)$ be the linear
map which describes the restriction of this pairing to~$L$.
The following result is a consequence of
the theorem of Deligne and Rapoport
and
work of Grothendieck and Raynaud.  (See \cite\RAY, and perhaps
the discussion in~\cite{\FERMAT, p.~438}.)

\proclaim{Theorem 2}
The component group
$\Phi_\Gamma$ may be identified with the
cokernel of~$\iota$.
\endproclaim

\noindent
One understands that the
identification between $\Phi_\Gamma$ and~$\coker\iota$ is compatible
with the functorial actions of Hecke
operators and of the Galois group~$\Gal(\Fbar_q/\F_q)$.
We shall discuss the Hecke operators.

As usual, Hecke operators act on $J(H,q)$ in two different
ways, because the Jacobian construction is both covariant
and contravariant.
In what follows we adopt
the contravariant
point of view in which $J(H,q)$ is regarded as
$\Pic^0(X(H,q))$.  We describe the action of
Hecke operators on~$\Phi_\Gamma$ without supplying any
substantial
justification.  However, the 
 reader may consult~\cite\RAY,
which discusses in detail the special case case where $H=\U N$.

First, let $d$ be an integer prime to~$N$.
The association $(E,C,P)\mapsto (E,C,dP)$
defines a ``diamond bracket'' automorphism $\diamond d$
on~$X(H,q)$.  Using the contravariant functoriality, we
obtain automorphisms of~$J(H,q)$ and then $\Phi_\Gamma$,
both of which we shall call simply~$\diamond d$.
In the description of~$\Phi_\Gamma$ as the cokernel of
an injection $L\hookrightarrow\Hom(L,\Z)$,
the map $\diamond d$ on~$\Phi_\Gamma$ is induced
% in a fairly evident way 
from two automorphisms of~$L$.
Namely, consider the
permutation of~$S$ which
takes the class of a pair $(E,P)$ to the class of~$(E,dP)$.
This permutation induces an automorphism $\diamond d_L$
of~$L$,
which we regard as a subgroup of~$\Z^S$.
Let $\diamond d^\prime_L$
be the {\it inverse\/} of this permutation.
Then we have a commutative diagram
$$\matrix
0 &\to& L& \buildrel\iota\over\hookrightarrow &\Hom(L,\Z)& \to& 
   \Phi_\Gamma&\to&0 \cr
  & &\downarrow& &\downarrow&&\downarrow\cr
 0 &\to& L& \buildrel\iota\over\hookrightarrow &\Hom(L,\Z)& \to& 
    \Phi_\Gamma&\to&0 \cr
\endmatrix $$
in which the three vertical arrows are respectively
$\diamond d^\prime_L$, $\Hom(\diamond d_L,\Z)$ and~$\diamond d$.



Similarly,  suppose that $p$ is a prime number which
does not divide~$qN$.
The symbol $T_p$ then denotes a host of objects:
the standard Hecke correspondence $T_p$ on~$X(H,q)$,
the endomorphism of~$J(H,q)$ which this correspondence
induces by contravariant functoriality, and finally
the endomorphism $T_p$ of~$\Phi_\Gamma$ which then
arises from the functoriality of the N\'eron model.
We seek to identify this latter endomorphism.


Fix $p$ for the moment and
let $T$ be the correspondence on~$S$ defined by the rule
$(E,\alpha)\mapsto \sum (E/D, \alpha \hbox{ mod }D)$, where the
sum is taken over the subgroups $D$ of~$E$ having order~$p$. 
In analogy with the situation above, we write $T_L$
for the induced endomorphism of~$L\subset\Z^S$.
The analogue of~$\diamond d^\prime_L$ is then the
endomorphism $T^\prime_L\seteq
\diamond p_L^{-1}\circ T_L$.
We have a commutative diagram like the one above in which
the three vertical arrows are respectively
$T^\prime_L$, $\Hom(T_L,\Z)$ and the endomorphism $T_p$
of~$\Phi_\Gamma$.
Since $p$ may vary in what follows, we restore $p$
to the notation, referring to the first of these operators
as $T_p^\prime$ and the second as $\Hom(T_p,\Z)$.
In other words, we will permit $T_p$ to denote the endomorphism
of~$L$ induced from the correspondence 
$(E,\alpha)\mapsto \sum (E/D, \alpha \hbox{ mod }D)$ on~$S$
by $\Z$-linearity.

\section Extra automorphisms\endhead
We
will now determine those points $s\in S$
with $e(s)>1$.
A point $s$ in~$S$ is defined by a supersingular
elliptic curve $E$ over~$\Fbar_q$, together with
a point $P$ on~$E$ of order~$N$.
Write $\alpha$ for the orbit of~$P$ under~$H$
and $C$ for the subgroup of~$E$ generated by~$P$.
Let $\sbar$ be the point on~$X_0(N)$ defined
by~$(E,C)$.  

Let $R$ be the subring $\Z[\Aut E]$
of~$\End E$.
If $e(s)$ is different from~1, then $\Aut E$ is different
from~$\pmone$.
Since $\End(E)$ is a definite quaternion algebra over~$\Q$,
$\Aut E$ is different from~$\pmone$
if and only if $E$
has an automorphism of order 4 or~6.
In fact,
the condition $q>3$
implies that the automorphism group of~$E$
can only be cyclic of order~2, order~4, or order~6.
Hence $R$ is either~$\Z$, or else the integer ring in
an imaginary quadratic field of discriminant $-3$ or~$-4$.
As is well known,
$E$ has an automorphism of order~4 if and only if
its $j$-invariant is~1728.  Since $E$ is supersingular,
the prime $q$ must be congruent to~3 mod~4.
Similarly, $E$ has an automorphism of order~3 if and only
if its
$j$-invariant is~zero; if the curve with this
$j$-invariant is supersingular, we have $q\equiv2$ mod~3.
(Compare, for example, \cite\SILV, page~103 and pp.~143--144.)

Suppose that $\Aut E\ne\pmone$.  Then
$(E,C)$ has a non-trivial automorphism if and
only if
$\Aut E$ coincides with its subgroup $\Aut(E,C)$, i.e., if
and only if
$C$ is
stable under~$R$.  If this condition is
satisfied, then
the group $C$ is free of rank~1 over~$R/I$, where
$I$ is the annihilator of~$C$ in~$R$.
Since $C$ is cyclic, this puts a numerical constraint on~$N$:
No prime factor of~$N$ can remain inert in the ring~$R$,
and the prime which ramifies in~$R$ can occur in~$N$ only
to the first power if it occurs at all.
Further, the inclusion of~$\Z$ in~$R$ induces
an identification $R/I\buildrel\sim\over\to\Z/N\Z$.
Thus the pair $(E,C)$ has a non-trivial
automorphism if and only if $C$ is the kernel on~$E$
of an ideal $I$ of~$R$ such that $R/I$ is
isomorphic to~$\Z/N\Z$.  Thus there is a 1-1 correspondence
between cyclic subgroups $C$ of order~$N$ which are stable
by~$R$ and ideals $I$ of~$R$ such that the additive
group of~$R/I$ is cyclic of order~$N$.  One sees that distinct
subgroups $C$ lead to distinct point of~$X_0(N)$.  Indeed,
if $(E,C)$ and~$(E,C')$ are isomorphic, then the isomorphism
between them is induced by an automorphism of~$E$, which
preserves $C$ (and $C'$) by the definition of~$R$.

The conjugation map $r\mapsto\overline{r}$
of~$R$ induces an
an
involution on the set of~$I$.
This Atkin-Lehner style involution 
has no fixed points if~$N>3$.  
Indeed, suppose that $I=\overline{I}$.
For each $r\in R$, there is an integer $n$ such that $r-n$
lies in $I$.  Since $\overline{n}=n$, $r-\overline{r}$
belongs to $I+\overline{I}=I$.  Hence $I$ contains
$\sqrt{-3}$ or~$2$, according as the discriminant
of~$R$ is
$-3$ or~$-4$.  Hence $N$ divides~3 in the former case and
$N$ divides~2 in the latter.


Suppose now that $(E,C)$ has a non-trivial automorphism.
Then $e(s)>1$
if and only if $\Aut(E,\alpha)=\Aut(E,C)$.
This equality 
translates into
the statement that 
$$ \mu \buildrel?\over\subseteq H \subseteq
   \U N,\leqno($\ast$)$$
where $\mu$ is the image in~$(R/I)^\ast$
of the unit group of~$R$.
This condition involves only~$(E,C)$
and~$H$; hence it applies simultaneously to all points
on~$X_H(N)$ which lie above the point $\sbar$ on~$X_0(N)$.
One checks that when ($\ast$)
is satisfied, the number of points on~$X_H(N)$
which lie above~$\sbar$ is precisely
$(\U N:H)$.

In summary, to find those points $s\in S$ with $e(s)>1$,
we first look for the supersingular points $\sbar$ in
$X_0(N)(\Fbar_q)$ which satisfy the analogous condition
$e(\sbar)>1$.  
Mark off those points (if any) which satisfy the supplementary
condition~($\ast$).  Above each marked point, we find 
$(\U N:H)$ points in~$S$ with ``extra automorphisms.''


The
condition ($\ast$) may be true for some points $\sbar$
with $e(\sbar)>1$ that arise
from a given supersingular elliptic
curve and false for others.  
For example, suppose that $E$ has six automorphisms and that
$N=7\cdot13=91$.  
There are four ideals $I$ of~$R$ with $R/I\approx
\Z/91\Z$.  For two of these ideals,
the image of~$R^\ast$ in~$\U{91}$ is the cyclic
group generated by~10; for the other two, the image is
generated by~17.  If $H$ is one of these two groups, then
($\ast$) will be satisfied for two of the $I$, but not for
all.
Notice, however, that the set of~$I$ for which
($\ast$) is satisfied is stable under the natural conjugation
map on~$R$.  Indeed, $R^\ast$ is stable under this conjugation.
Furthermore,  we have observed that the inequality
$N>3$ implies that an ideal $I$ with
$R/I\approx\Z/N\Z$ can never be its own conjugate.
This gives

\proclaim{Lemma 1}
Suppose that $N\ge4$.  Then the set
of points $\sbar$ for which $e(\sbar)>1$ and
for which ($\ast$) is satisfied 
has an even number of elements.  In particular, the number
of such points is different from~1.
\endproclaim

The case $N\le3$ does not play an important role in
the analysis which follows.  In fact, one has

\proclaim{Lemma 2}
If $N\le12$ and there is at least one point $s$ with $e(s)>1$,
then
$H=\U N$.
\endproclaim

\startproof
Assume that $N<13$.  If
$e(s)=3$, then there is an ideal $I$ of $R=\Z[{1+\sqrt{-3}\over2}]$
with $R/I\approx\Z/N\Z$.  This implies that $N$ is 3 or~7.
In both cases, $R^\ast$ maps onto $\U N$.
If $e(s)=2$, there is an ideal $I$ of~$R=Z[\sqrt{-1}]$
with $R/I\approx\Z/N\Z$.  The possibilities for~$N$ are
2, 5, and~10.  Again,  $R^\ast$ maps onto~$\U N$ in all cases.
\endproof

The prime 13 splits both in $\Z[{1+\sqrt{-3}\over2}]$
and in~$Z[\sqrt{-1}]$.  If $R$ is one of these two rings, and
if $I$ is a prime lying over~13 in~$R$, then the image of~$R^\ast$
in $\U{13}$ is strictly smaller than~$\U{13}$.
This circumstance (if not the discussion in~\S2)
shows that in the statement of the~Lemma
12 cannot be replaced by a larger
integer.



\section A canonical subgroup of $\Phi_\Gamma$\endhead
Let $\Phi=\Phi_\Gamma$.
We shall discuss a cyclic subgroup $\Phi_0$ of~$\Phi$
which has already been studied in case $\Gamma=\go N$
(see~\cite\FERMAT\ and~\cite\RAY).

For each element $s$ of~$S$, 
let $\phi_s\:L\to\Z$
be the linear form $\ell\mapsto\langle \ell,s\rangle$
and
let $\omega_s$ be the
linear form on~$L$
$$ \sum_t n_t t \mapsto n_s.$$
Clearly, $\Hom(L,\Z)$ is generated by the~$\omega_s$,
so that
$\Phi$ is generated by their images $\omegabar_s$
in~$\Phi=\Hom(L,\Z)/L$.
One has
$e(s)\omega(s)=\phi_s$.

The class $\overline\phi$
of~$\phi_s$ in $\Phi=\Hom(L,\Z)/L$ is independent of~$s$,
since $\phi_s-\phi_t$ is the image in~$\Hom(L,\Z)$
of~$s-t\in L$, if $s,t\in S$.
Let $\Phi_0$ be the subgroup of~$\Phi$ generated by the
canonically given element $\overline\phi$.
We set
$$ Q=\Phi/\Phi_0.$$
In view of the formula
$e(s)\omega(s)=\phi_s$, we have
$e(s)\omegabar_s = \overline{\phi}$ for each~$s$.
Hence, as noted in~\cite{\FERMAT, \S2}, we have a surjection
$$ \bigoplus_{s\in S} \left(\Z/e(s)\Z\right) 
\longrightarrow Q,\qquad
  (a_s)\mapsto \sum_s a_s\omegabar_s.$$

One may view $\Z^S$ as
embedded in~$\Hom(\Z^S,\Z)$ via the symmetric bilinear
pairing $\langle \, , \, \rangle$ on~$\Z^S$.  Let $L^\perp\subset
\Hom(\Z^S,\Z)$ be the group of linear forms which vanish
on~$L$, and let $U=L^\perp\cap\Z^S$ be group of vectors in~$\Z^S$
which are orthogonal to~$L$ under the pairing.  Since
$\Hom(L,\Z)=\Hom(\Z^S,\Z)/L^\perp$, we have
$\Phi=\Hom(\Z^S,\Z)/(L\oplus L^\perp)$.  The image of~$\Z^S/U$
in~$\Phi$, i.e., the group $\Z^S/(L\oplus U)$, is clearly~$\Phi_0$.
(Indeed, the image of~$s\in\Z^S$ in~$\Hom(\Z^S,\Z)$ is~$\phi_s$.)

Now $U$ is free of rank~one over~$\Z$; it consists of those
multiples of~$\sum {1\over e(s)} s\in \Q^S$ which
lie in~$\Z^S$.  Hence $U$ is generated by $\sum {m\over e(s)} s$,
where $m$ is the least common multiple of the~$e(s)$.  (Thus,
$m$ divides~6.)  It follows that the order
of~$\Phi_0=\Z^S/(L\oplus U)$ is
$m\cdot\sum e(s)^{-1}$, cf.~\cite{\RAY, p.~16}.
Also, $L^\perp/U$ has order~$m$, since
$L^\perp$ is generated by the element
$\sum \omega_s$ and since $m\sum\omega_s=
\sum {m\over e(s)} s$ in~$\Hom(\Z^S,\Z)$.

Consider the commutative diagram with exact rows
$$
\matrix
0 & \to & L^\perp &\to& \Hom(\Z^S,\Z)&\to 
   &\Hom(\Z^S,\Z)/L^\perp &\to& 0\cr
&&\uparrow&&\uparrow&&\uparrow&&\cr
0 & \to & U&\to& \Z^S &\to&\Z^S/U&\to&0\cr
\endmatrix
$$ 
in which the central vertical map is the inclusion given
by~$\langle \, , \, \rangle$; the two other vertical
maps are the evident, related inclusions.
The cokernels of the vertical maps form the exact sequence
$ 0\to\Z/m\Z\to \bigoplus_{s\in S} % can be displayed
      (\Z/e(s)\Z)\to Q\to0$,
where the map from~$\/m\Z$ to the direct sum takes $1\in\Z/m\Z$
to the vector $(1,\ldots,1)$.
Note that $Q$ is now seen to have
order $m^{-1}\prod e(s)$.  Therefore, we recover the formula
$\left(\vphantom{\null^{-1}}\prod_s e(s)\right)
   \cdot\left(\sum_se(s)^{-1}\right)$
for
the order of~$\Phi$, cf.~\cite{\BLR, Ch.~9, Prop.~10}.





\proclaim{Lemma 3}
The group $\Phi_0$ is strongly Eisenstein.
The identities $\diamond d=1$
 and $T_p=1+p$  hold on~$\Phi_0$ for
$d\in\U N$ and $p$
prime to~$qN$.
\endproclaim

\startproof
We first consider the action of~$T_p$ on~$\overline\phi$.
Choose $s\in S$. From
the perspective introduced
at the end of~\S3, we see that
the action of $T_p$ on this element of~$\Phi$
is derived from the action of
$\Hom(T_L,\Z)$ on~$\phi_s\in\Hom(L,\Z)$.  
The map $\phi_s\circ T_L$ is the linear form
mapping $\ell\in L$ to
$\langle T_L\ell,s\rangle = \langle \ell,  T^\prime_L s \rangle
= \langle \ell, \diamond p^{-1} T_L s\rangle$.
Since $\diamond p^{-1} T_L s$ is an element of~$\Z^S$ of
degree~$p{+}1$, the image in~$\Phi$ of $\ell\mapsto
\langle \ell, \diamond p^{-1} T_L s\rangle$
is $(p{+}1)\overline{\phi}$.
This establishes the first identity.  The second is proved by
an analogous computation. \endproof

\proclaim{Corollary}
The group $\Phi$ is Eisenstein in each of the following
situations:
\itemitem{(i)} $e(s)=1$ for all $s\in S$;
\itemitem{(ii)} $H=\U N$;
\itemitem{(iii)} $N<13$;
\itemitem{(iv)} $H=\pmone$.
\endproclaim

\startproof
In the first case, we have $\Phi_0=\Phi$, so that the
assertion to be proved is that given by the lemma.
The assertion in the second case is given as
Theorem~3.12 in~\cite\FERMAT.
In the third case, we are either in case~(i) or case~(ii)
by Lemma~2.  Suppose now that we are in the fourth case,
which corresponds to the equality $X_H(N)=X_1(N)$.
If $N$ is at least~4, we are in case~(i)
by \cite{\KATZMAZUR, Corollary~2.7.4}.  If $N\le4$, then we
are of course in case~(iii).
\endproof



\section The structure of $Q$\endhead
In view of the Corollary above, we will now
impose the condition
$N\ge13$.
\proclaim{Lemma 4}
Suppose that $(E,C)$ represents a supersingular point
of~$X_0(N)_{/\Fbar_q}$.  Let $D$ be a cyclic subgroup
of~$E(\Fbar_q)$ of order prime to~$qN$.
Assume that the automorphism group of the
triple $(E,C,D)$ is larger than $\pmone$. Then the
pair $(E/D, C\hbox{ mod }D)$ is isomorphic to~$(E,C)$.
\endproclaim

\startproof
This statement is proved as
Proposition~2
in~\cite\BX; we recall the proof for the convenience of
the reader.
Let $\epsilon$ be an automorphism
of~$(E,C,D)$ which is different from~$\pm1$, and let
$R$ be the subring of~$\End(E,C,D)$ which is generated
by~$\epsilon$.  Thus $R$
is
isomorphic either to the ring of Gaussian integers or to the
ring of integers of $\Q(\sqrt{-3})$.
If $J=\Ann_R(D)$, then
$J$ is a principal ideal~$(r)$ of~$R$, and
$D$ is the kernel of~$J$ on~$E$.  Thus the map ``multiplication by~$r$''
on~$E$ induces an isomorphism $(E/D, (C\oplus D)/D)
\buildrel\sim\over\rightarrow (E,C)$
as required.
\endproof

Recall now that $Q$ is generated as an abelian group by the
elements $\omegabar_s$ with $s\in S$.
Since $e(s)\omegabar_s=0$, it suffices to consider only
those $\omegabar_s$ with $e(s)>1$.
As we have seen, the inequality $e(s)>1$ means that
$s$ lies over a supersingular point
$\sbar$ of~$X_0(N)(\Fbar_q)$ which satisfies
the numerical condition $e(\sbar)>1$ and the
supplementary condition~($\ast$).

Let $\overline s$ be such a point.
Define $\Qsbar$ to
be the
subgroup of~$Q$ generated by the $\omegabar_s$ with
$s$ lying over $\sbar$.  Thus $\Qsbar$ is a quotient
of the elementary abelian group $\bigoplus(\Z/e(\sbar)\Z)$,
where the sum is extended over the set of points $s$ lying
over~$\sbar$.  As was mentioned above, the number of such
points is the index $(\U N:H)$.  More precisely, the  set of
points $s$ is a principal homogeneous space over $\U N/H$,
the action of this group being given by the ``diamond bracket''
operators.

\proclaim{Proposition 1}
The group $\Qsbar$ is free of rank $(\U N:H)$ over
$\Z/e(\sbar)\Z$.
\endproclaim

\startproof
Let $S_0$ be the inverse image of~$\overline s$ in~$S$.
We must show that the direct sum 
$\bigoplus_{s\in S_0}(\Z/e(s)\Z)$
has trivial intersection with the kernel of the map
$$\bigoplus_{s\in S}(\Z/e(s)\Z)\to Q.$$
As we have seen, this kernel
is the cyclic group 
generated by the vector $(1,\ldots,1)$ in the full direct sum.
Hence it suffices to check that there is a point $s\in S\setminus S_0$
with $e(s)>1$.  This follows from Lemma~1, since $N\ge13$.
\endproof




\proclaim{Proposition 2}
The subgroup $\Qsbar$ of~$Q$
is stable under the Hecke operators $T_p$ and~$\diamond d$.
\endproclaim

\startproof
Let $p$ be a prime number not dividing~$qN$.  As at the 
end of~\S3, we write $T$ for
the correspondence on~$S$
defined by the formula
$(E,\alpha)\mapsto \sum (E/D, \alpha \hbox{ mod }D)$.
This correspondence is summarized by the matrix
of natural numbers
$(a_{tu})$ which one constructs by writing
$Tu = \sum_{t\in S} a_{tu}\cdot t$ for
$u\in S$.
Recall that the action of $T_p$ on~$\Phi$ arises from 
the action of a map labeled
$\Hom(T_L,\Z)$ on~$\Hom(L,\Z)$.
This group is generated by the elements $\omega_s$ for
$s\in S$, and one finds that $\Hom(T_L,\Z)$ maps $\omega_s$
to the sum $\sum_{t\in S} a_{st}\omega_t$.

To prove that $\Qsbar$ is stable under~$T_p$,
it suffices to show
that $a_{st}$ is divisible
by~$e(t)$ whenever $s$ maps to~$\sbar$ and
$t$ does {\it not}.
In other words, we wish to show that $a_{tu}$ is
divisible by~$e(u)$ whenever $t$ and~$u$ have
distinct images on~$X_0(N)$.

This divisibility can be established by the method 
used to prove~\cite{\FERMAT, Th.~3.12}.
Indeed,
suppose that $u$ is represented by~$(E,\alpha)$.
Let $C$ be the cyclic subgroup of order~$N$ on~$E$
which is associated with~$\alpha$.
The group $\Aut(E,\alpha)/\pmone$ operates on the set of
cyclic subgroups of~$E$ of order~$p$.
Clearly, if $D$ and~$D'$ are such subgroups which
are equivalent under this action, then
$(E/D,\alpha\hbox{ mod }D)$ and
$(E/D',\alpha\hbox{ mod }D')$
are isomorphic.
Moreover, suppose that the cyclic subgroup $D$ has
a non-trivial stabilizer under this action.
Then Lemma~4 shows that $(E,C)$ and~$(E/D,(C\oplus D)/D)$
are isomorphic, i.e.,  that $(E,\alpha)$
and $(E/D,\alpha \hbox{ mod }D)$ map down to the
same point on~$X_0(N)$.
Now let $t$ be a point of~$S$ whose image on~$X_0(N)$
is distinct from that of~$s$.  Then the set of~$D$
for which $(E/D,\alpha\hbox{ mod }D)$ represents~$t$
is a union of copies of the group $\Aut(E,\alpha)/\pmone$,
whose cardinality is~$e(u)$.  Hence $a_{tu}$ is
divisible by~$e(u)$, as was claimed.



Now let $d$ be an integer prime to~$N$.
The action of~$\diamond d$ on~$\Phi$ is deduced from
the automorphism $\Hom(\diamond d_L,\Z)$ introduced
at the end of~\S3.  This automorphism maps a given
linear form $\omega_s$ to the linear form 
$\omega_{\diamond d^{-1}s}$.
Here, the automorphism labeled $\diamond d^{-1}$ sends the
class of $(E,P)$ to the class of $(E,d^{-1}P)$; the quantity $d^{-1}$
is computed mod~$N$.  Since $P$ and $d^{-1}P$ generated the same
subgroup of~$E$, it is clear that $\diamond d$ permutes
the $\omegabar_s$ with $s$ having a given image on~$X_0(N)$.
Therefore, $\diamond d$ preserves $\Qsbar$.
\endproof

Let $S_0$ again be the inverse image of~$\overline s$
in~$S$.  We observed above that the diamond bracket operation
of~$\U N$ on~$S_0$ makes $S_0$ a principal homogeneous space
over the group $\Delta\seteq\U N/H$.  Let us identify
$\Qsbar$ with the direct sum 
$\bigoplus_{s\in S_0}(\Z/e(\sbar)\Z)$,
i.e., with the space of functions from~$S_0$ 
to~$\Z/e(\sbar)\Z$.
Then the diamond bracket operation
discussed in Proposition~2, which involves an inverse,
may be viewed as the natural action coming from
the  diamond bracket
action of~$\Delta$ on~$S_0$.


Let $\ell=e(\sbar)$; thus
$\ell=2$ or~$\ell=3$.
Let
$\T=\F_\ell[\Delta]$ be the group algebra consisting of
sums $\sum_{\delta\in\Delta} n_\delta [\delta]$
where the $n_\delta$ are integers mod~$\ell$.
Since $\Qsbar$ is an $\F_\ell$-vector space with an
action of~$\Delta$, $\Qsbar$ may be viewed as a $\T$-module
in a natural way.

\proclaim{Lemma 5}
The $\T$-module $\Qsbar$ is free of rank one.
\endproclaim

\startproof
Proposition~1 shows that $\Qsbar$ is an $\F_\ell$ vector
space of dimension $\#(\Delta)=\dim_{\F_\ell}\T$.
Further, it is clear that $\Qsbar$ is a cyclic $\T$-module,
since $\Delta$ permutes the generators $\omega_s$ of~$\Qsbar$.
\endproof


Now choose
a supersingular elliptic curve $E$ over~$\Fbar_q$
and a cyclic subgroup $C$ on~$E$ so that the pair $(E,C)$
defines the point $\sbar$ of~$X_0(N)$.
As usual, let $R$ be the ring~$\Z[\Aut E]$, and let $I$
be the annihilator of~$C$ in~$R$.
The isomorphism
$R/I\approx\U N$ gives a meaning to~$[r]$
whenever $r\in R$ is prime to~$I$.
The element $[r]$
depends only on the ideal generated by~$r$ in~$R$,
since $H$ contains the image of~$R^\ast$ in~$\U N$.

In fact, this might be
a good time to relate $\Delta$ to the Galois group $\GalQ$.
Let $K$ be the imaginary quadratic field $\Q[\Aut E]=R\otimes\Q$.
Since all fractional ideals of~$K$ are
principal, the quotient of~$(R/I)^\ast$ by $R^\ast$
is the ray class group of~$K$ with conductor~$I$.
Via class field theory, this class group corresponds with
an abelian extension $L$ of~$K$.  It will be useful to
fix an embedding $K\hookrightarrow\Qbar$ and to view $L$
as a subfield of~$\Qbar$.
Since $\Delta$ is then a quotient of~$\Gal(L/K)$,
$\Delta$ becomes the Galois group of an abelian extension of~$K$
in~$\Qbar$.  The conductor of this extension
divides~$I$.


For each prime $p\notdivide qN$, we define an element $\tau_p$
of~$\T$ by the formula:
$$\tau_p =
\cases
0                             &\text{if  $p$ is inert in~$R$,}\\
[\pi]+[\pi']  &\text{if $p=(\pi)(\pi')$ is split in~$R$,}\\
[\pi]  &\text{if $p=(\pi)^2$ is ramified in~$R$.}
\endcases
$$
The third case occurs only when $p=\ell$.

\proclaim{Theorem 3}
For each $p\notdivide qN$, the Hecke operator $T_p$ acts on~$\Qsbar$
by multiplication by~$\tau_p$.
\endproclaim

\startproof
Let $(a_{st})$ be the Brandt matrix which was introduced in the
course of
the
proof of Proposition~2.  Recall that if
$t$ is the isomorphism class of the pair $(E,P)$, where $P$ is
a point of order~$N$ lying in~$C$, then $a_{st}$ is
the number of subgroups $D$ of order~$p$ in~$E$ such that
$(E/D, P\hbox{ mod }D)$ defines~$s$.
We have seen that
$a_{st}\equiv0$ mod~$\ell$ if $s$ and~$t$ 
do not
have the same image on~$X_0(N)$.
We must now calculate $a_{st}$ mod~$\ell$
in the case where $s$ and~$t$ have the same image,
namely~$\overline s$.

Suppose first that $p$ is inert in~$R$.  Then there is no
subgroup of order~$p$ in~$E$ which is stable under~$\Aut E$.
Accordingly, the proof of Proposition~2 shows that $a_{st}$
is divisible by~$\ell$, since the group $(\Aut E)/\pmone$
acts freely on the set of~$D$ for which
$(E/D, P\hbox{ mod }D)$ defines~$s$.

Next, let $p=(\pi)^2$ be ramified in~$R$.
Then $E[\pi]$ is the unique subgroup of order~$p$ on~$E$
which is stable under~$R$.  The pair $(E/D, P\hbox{ mod }D)$
is isomorphic to~$(E,\pi\cdot P)$ via~$\pi$.
With the obvious meaning of~$\diamond\pi$, we may write
$a_{st}\equiv0$ mod~$\ell$ for $s\ne\diamond\pi t$,
while $a_{st}\equiv1$ mod~$\ell$ for $s=\diamond\pi t$.
Tracing through the definitions, one emerges with
the desired assertion
that $T_p$ operates
on~$\Qsbar$ as~$[\pi]$.

In the final case where $p=(\pi)(\pi')$, there are two
distinct subgroups $D$ which are stable under~$\Aut E$.
The corresponding quotients of~$(E,\alpha)$
are represented by~$(E,\pi\cdot P)$ and~$(E,\pi'\cdot P)$.
Let $s_1=\diamond\pi t$ and~$s_2=\diamond{\pi'} t$.  If
the two $s_i$ are~distinct, the numbers
$a_{st}$~mod~$\ell$ are~1 for $s=s_1, s_2$ and zero otherwise.
If $s_1=s_2$, then $a_{st}=2$ when $s=s_1=s_2$ and $a_{st}=0$
otherwise.  This leads to the required formula $T_p=[\pi]+[\pi']$.
\endproof

Let $\widetilde\T$ be the subring of~$\End J(H,q)$
generated by the endomorphisms $T_p$ and~$\diamond d$ of~$J(H,q)$.
(There is a functorial, faithful action of~$\widetilde\T$
on the space of weight-two cusp forms for $\Gamma\cap\go q$;
thus $\widetilde\T$ may be defined alternatively as in the
Introduction.)
Theorem~3 states that the action of~$\widetilde\T$ on~$\Qsbar$
factors through the action of~$\T$ on~$\Qsbar$, in such a 
way that $\diamond d\in\widetilde\T$ acts as~$[d]\in\T$ for
all $d\in\U N$ that $T_p\in\widetilde\T$ acts as~$\tau_p$.
Equivalently, there is a commutative triangle
$$\matrix
\widetilde\T&&\relbar\joinrel\longrightarrow&&
\T\cr\cr
&\searrow&&\swarrow&\cr\cr
&&\End\Qsbar&&
\endmatrix$$
in which 
the left-hand diagonal arrow describes the action of
Hecke operators on~$\Qsbar$ and the right-hand diagonal
arrow is the structural map making $\Qsbar$ into a $\T$-module.
The horizontal arrow is visibly surjective, since
$\T=\F_\ell[\Delta]$ is generated by the  various~$[d]$.
By Lemma~5, $\Qsbar$ is a free rank-one
$\T$-module.  Therefore, the map $\T\to\End\Qsbar$
is injective.  Accordingly, the image of~$\widetilde\T$
in~$\End\Qsbar$ may be identified with~$\T$.  
Via this identification,
the set of
maximal ideals of~$\widetilde\T$ in the support 
of~$\Qsbar$ becomes the set of maximal ideals
of~$\T=\F_\ell[\Delta]$.

This latter set may be viewed as a set of conjugacy classes
of characters.  Indeed,
if
$\m$ is a maximal ideal of~$\T$, then $\T/\m$ is a finite
field of characteristic~$\ell$.  Hence $\m$ may be obtained
as the kernel of some ring homomorphism $\T\to\Fbar_\ell$.
On the other hand, restriction to~$\Delta$ yields a 1-1
correspondence between
ring homomorphisms $\T\to\Fbar_\ell$
and group homomorphisms $\theta\:\Delta\to\Fbar^\ast_\ell$.
By Galois theory, two ring homomorphisms $\T\doublemap\Fbar_\ell$
have the same kernel if and only if they are conjugate 
under~$\Gal(\Fbar_\ell/\F_\ell)$.  It follows that the set of
maximal ideals of~$\T$ is in correspondence with the set of 
$\Gal(\Fbar_\ell/\F_\ell)$-conjugacy classes 
of characters~$\theta$.

As was noted above, $\Delta$ may be viewed as the Galois group
of an abelian extension of~$K$ in~$\Qbar$.
Accordingly,
the characters $\theta$ become homomorphisms
$\Gal(\Qbar/K)\to\Fbar_\ell^\ast$ which factor through the
Galois group we called~$\Gal(L/K)$.
For each such homomorphism, we let $\theta'$ be the map
$\Gal(\Qbar/K)\to\Fbar_\ell^\ast$ defined by
$$ \theta'(x) = \theta(gxg^{-1}),$$
where $g$ is an element of~$\GalQ$  which does not belong
to~$\Gal(\Qbar/K)$.

\proclaim{Lemma 6}
We have $\theta=\theta'$ if and only if $\theta$ is
identically~1.
\endproclaim

\startproof
If $\theta$ is identically~1, it certainly
coincides with~$\theta'$.  Conversely, suppose that
$\theta=\theta'$.
Let $\goth f$ be the conductor of~$\theta$.
As is well known, the conductor of~$\theta'$ is the
image of~$\goth f$ under the non-trivial involution
$K\to K$.  Because $\theta=\theta'$,
$\goth f$ is invariant under this involution.  It is
in any case a divisor of~$I$.  It follows that $\goth f$
divides $(\sqrt{-3})$ in the case where $K=\Q(\sqrt{-3})$
and that $\goth f$ divides $(1+\sqrt{-1})$ in case
$K=\Q(\sqrt{-1})$.  In the former case, $\theta$
is a map $\F_3^\ast\to\Fbar_3^\ast$ which is trivial on~$\pmone$.
In the latter case, $\theta$ is a map
$\F_2^\ast\to\Fbar_2^\ast$.  Hence $\theta$ is the trivial
map in either case.
\endproof

Now let $\Ind\theta$ be the induced representation
$$ \Ind_{\Gal(\Qbar/K)}^{\GalQ}
  \theta\:\GalQ\to\GL{\Fbar_\ell}.$$
We can make an explicit
 model for~$\Ind\theta$ by choosing $g\not\in
\Gal(\Qbar/K)$ as above and defining
$$\Ind\theta(x)=\cases
\pmatrix
\theta(x)&0\cr
0&\theta'(x)\cr
\endpmatrix
&
\text{for $x\in\Gal(\Qbar/K)$,}\cr
\pmatrix
0 & \theta(xg) \cr
\theta(g^{-1}x)   &0\cr
\endpmatrix &
\text{for $x\not\in\Gal(\Qbar/K)$.}\cr
\endcases$$
In particular, one
reads from this model the well known formula
$$
\tr(\Ind\theta(x)) =
\cases
\theta(x)+\theta'(x)          &\text{if  $x\in\Gal(\Qbar/K)$,}\\
0 &\text{if  $x\not\in\Gal(\Qbar/K)$}
\endcases
$$
for the trace of~$\Ind\theta$.
The determinant of~$\Ind\theta$ is the product of two
characters, the first of which is the homomorphism
$\GalQ\to\pmone$ which corresponds to the quadratic
subfield $K$ of~$\Qbar$.  This homomorphism may
be described alternatively as the mod~$\ell$ cyclotomic
character $\chi_\ell$ which gives the action of~$\GalQ$
on the group of $\ell$th roots of~1 in~$\Qbar$.  It
has order~two if $\ell=3$, while it is trivial if~$\ell=2$.
The second of the two
characters is $\theta\circ\Ver$, where
$\Ver$ is the Verlagerung map from~$\GalQ$ to the
abelianization of~$\Gal(\Qbar/K)$.  It is obtained
by composing: the mod~$N$ cyclotomic character $\GalQ\to\U N$,
the identification $\U N\buildrel\sim\over\to (R/I)^\ast$,
and the
character $\theta\:(R/I)^\ast\to\Fbar_\ell^\ast$.

\proclaim{Lemma 7}
If $\theta$ is non-trivial, the representation $\Ind\theta$
is irreducible.  
Suppose instead that $\theta$ is~trivial.  Then the image
of~$\Ind\theta$ has order~two; 
the kernel of~$\Ind\theta$ is~$\Gal(\Qbar/K)$.
When $\ell=3$,
$\Ind\theta$ is the direct sum of the trivial representation
and the one-dimensional representation with character~$\chi_3$.
When $\ell=2$, $\Ind\theta$
is indecomposable and its semisimplification is the direct sum
of two copies of the trivial representation.
\endproclaim

\startproof
Suppose that $\theta$ is non-trivial.  Then by Lemma~6,
$\theta$ and~$\theta'$ are distinct.  The restriction of~$\Ind\theta$
to~$\Gal(\Qbar/K)$ is thus the direct sum of two {\it distinct\/}
characters.  It follows that there are precisely two lines in
the representation space of~$\Ind\theta$ which are
invariant under~$\Gal(\Qbar/K)$.  These lines are permuted
by the element~$g$; in particular, no line is stable under
the full Galois group~$\GalQ$.
If $\theta$ is~trivial, then the required assertions
are clear from the model of~$\Ind\theta$ presented above.
\endproof

Note that,
in the case where $\ell=2$ and $\theta=1$,
the representation
$\Ind\theta$ is that given by the unique non-trivial
extension of~$\Z/2\Z$ by~$\mu_2$ in the category of
commutative group schemes of type $(2,\ldots,2)$ over~$\Spec\Z$.
(See \cite{\EIS, Ch.~II, \S12}.)

Let $\m$ be the maximal ideal of~$\widetilde\T$ arising
from~$\theta$.
The quotient $\widetilde\T/\m$ becomes
a subfield of~$\Fbar_\ell$ via the embedding
$\widetilde\T/\m\hookrightarrow\Fbar_\ell$ induced by~$\theta$.
To $\theta$ we associate the representation
$$\rho_\theta=\rho_\m \otimes_{\widetilde\T/\m}\Fbar_\ell
  \:\GalQ\to\GL{\Fbar_\ell}$$
obtained by composing $\rho_\m$ with the inclusion 
$\GL{\widetilde\T/\m}\hookrightarrow\GL{\Fbar_\ell}$.
By varying $\theta$ over the set of
characters
$\Delta\to\Fbar_\ell^\ast$, we obtain from~$\Qsbar$
a family of
representations  $\GalQ\to\Fbar_\ell^*$.
We seek to describe this family.

\proclaim{Proposition 3}
Let $p$ be a prime number which does not divide~$qN\ell$.
Then the trace of~$\rho_\theta(\Frob_p)$ is
$$
\cases
0                             &\text{if  $p$ is inert in~$K$,}\\
\theta(\pi)+\theta(\pi')  &\text{if $p=(\pi)(\pi')$ is split in~$K$,}
\endcases
$$
while the determinant of~$\rho_\theta(\Frob_p)$
is~$p\theta(p)=\pm\theta(p)$.
\endproclaim

\startproof
The matrix $\rho_\m(\Frob_p)$
has trace $T_p$~mod~$\m$ and determinant~$p\diamond p$.
Viewing $\widetilde\T/\m$ as a quotient of~$\T$, we can replace
$\diamond p$ by~$[p]$ and~$T_p$ by~$\tau_p$.
The map $\theta$ sends each $[\delta]\in\T$ to~$\theta(\delta)$.
The desired formulas now follow.
\endproof



\proclaim{Theorem 4}
For each~$\theta$, $\rho_\theta$ is the semisimplification
of the
induced representation
$ \Ind \theta$.
\endproclaim

\startproof This follows easily from Proposition~3, the
Cebotarev Density Theorem, the Brauer-Nesbitt Theorem
and the formulas given above for the trace and determinant
of~$\Ind\theta$.
\endproof

It is time now to recapitulate.
The group $Q=\Phi/\Phi_0$ is the sum of subgroups $\Qsbar$
belonging to the
supersingular points $\sbar$ of $X_0(N)(\Fbar_q)$
which have extra automorphisms and which satisfy
the condition~$(\ast)$ of~\S4.
To each such point is associated one of the
quadratic fields $\Q(\sqrt{-3})$,
$\Q(\sqrt{-1})$; call this field~$K$.
The point $\sbar$ and the group~$H$
determine an abelian extension $L/K$.
We induce those characters
$\theta\:\Gal(\Qbar/K)\to\Fbar_\ell^\ast$ which factor
through $\Gal(L/K)$ to obtain two-dimensional $\rho$
representations of~$\GalQ$.  According to Lemma~7,
the representations $\rho$ are
irreducible when $\theta$ is non-trivial.
The semisimplifications of the induced representations
$\rho$ are the 
$\Fbar_\ell$-linear two-dimensional representations
of~$\GalQ$ which arise from $\Qsbar$.

\kern0pt From another point of view, let $K$ be one of the two
fields $\Q(\sqrt{-3})$, $\Q(\sqrt{-1})$.  To fix ideas,
take the former field  and embed it in~$\Qbar$.
Let $R$ be the
ring of integers of~$K$, and let $\goth f$ be a non-zero integral
ideal of~$R$ for which $R/\goth f$ is cyclic as an
abelian group.  Consider the group $\Cal C_\goth f$ of
ideal classes of~$K$ modulo~$\goth f$, and let $\theta$ be
a non-trivial homomorphism $\Cal C_\goth f\to\Fbar_3^\ast$.
Using class field theory, regard $\theta$ as being defined
on~$\Gal(\Qbar/K)$ and induce $\theta$ to obtain a
two-dimensional representation $\rho$ of~$\GalQ$.
This representation is irreducible.
Let $N$ be the norm of~$\goth f$, and let $H$ be the image of
$R^\ast$ in~$(R/\goth f)^\ast=\U N$.

\proclaim{Theorem 5}
The representation $\rho$ arises from the component group
in the mod~$q$ reduction
of~$J(H,q)$ whenever $q$ is a prime number
which does not divide
$2N$ and which is congruent to~2 modulo~3.
\endproclaim

\startproof
Since $q$ is 2 modulo~3, we may choose a supersingular elliptic
curve $E$ over~$\Fbar_q$ whose automorphism group is
cyclic of order~six.
Identify $\Z(\Aut E)$ with~$R$ and let $C$ be the kernel
of~$\goth f$ on~$E$.  The pair $(E,C)$ then
represents a point
$\sbar$ on~$X_0(N)$ which 
has extra automorphisms and
satisfies condition~$(\ast)$.
We have seen that $\rho$ arises from the subquotient $\Qsbar$
of the component group associated with~$J(H,q)$.
\endproof

As an application of the Theorem, we obtain the following
statement, which could presumably be proved much more
directly.

\proclaim{Corollary}
The representation $\rho$ arises from the space of weight-two
cusp forms on $\gammaone N$.
\endproclaim

\startproof
Let $q$ be a prime number as in the statement of Theorem~5.
The conclusion of the Theorem implies that
$\rho$ arises from the space of weight-two cusp forms
on $\Gamma_H(N)\cap\Gamma_0(q)$, and hence from the
space of cusp forms on~$\Gamma_1(N)\cap\Gamma_0(q)$.
Since $\rho$ is unramified at~$q$, the desired conclusion
now follows from ``Mazur's Principle" \cite{\MOTIVES, Theorem~8.1}.
\endproof

In the situation of Theorem~5,
one can ask whether or
not $\rho$ arises from the space of weight-two cusp
forms on~$\Gamma_H(N)$.
In the example treated in~\S2, this space is~zero,
so the response is negative.
It might be interesting to answer this
question in general.


\Refs
\catcode`\?=\active
\def?{.\hskip 0.1667em\relax}

\ref\no\BLR\by S. Bosch, W. L\"utkebohmert and M. Raynaud
\book N\'eron models
%\bookinfo  Ergebnisse der Mathematik und ihrer Grenzgebiete
   \publ  Springer-Verlag
   \publaddr Berlin and New York \yr 1990\endref


\ref\no\CAR\by H. Carayol
   \paper Sur les repr\'esentations galoisiennes
  modulo $\ell$ attach\'ees aux formes modulaires
\jour
   Duke Math. J. \vol 59 \yr1989 \pages 785--801\endref

\ref\no\CSZ\by H. Cohen, N. Skoruppa and D. Zagier
\book Tables of coefficients of modular forms\toappear\endref

\ref\no\DERA
\by P. Deligne and M. Rapoport
\paper Les sch\'emas de modules de  courbes elliptiques
\inbook Lecture Notes in Math., vol. 349
\publ   Springer-Verlag \publaddr Berlin and New York
\yr     1973 \pages 143--316
\endref

\ref\no\DIAMONDREF
\by F. Diamond
\paper The refined Serre conjecture
\paperinfo This volume
\endref

\ref\no\EDIX\by B. Edixhoven
\paper L'action de l'alg\`ebre de Hecke sur les groupes
de composantes des jacobiennes des courbes modulaires
est ``Eisenstein"
\jour Ast\'erisque \vol196--197\yr1991\pages159--170\endref

\ref\no\EDIXINVENT\bysame
\paper The weight in Serre's conjectures on modular forms
\jour Invent. Math \vol109  \yr1992\pages563--594
\endref

\ref\no\JL\by B. Jordan and R. Livn\'e
\paper Local diophantine properties of Shimura curves
\jour Math. Ann. \vol270\yr1985\pages235--248\endref

\ref\no\KATZ\by N?M. Katz
\paper
$p$-adic properties of modular schemes and modular forms
\inbook Lecture Notes in Math., vol. 350
\publ   Springer-Verlag \publaddr Berlin and New York
\yr     1973 \pages 69--190
\endref


\ref\no\KATZMAZUR
\by\hbox to3em{\hrulefill}\thinspace and B. Mazur
\book Arithmetic moduli of
   elliptic curves\bookinfo  Annals of Math. Studies {\bf 108}
   \publ  Princeton University Press
   \publaddr Princeton \yr 1985\endref


\ref\no\EIS\by B. Mazur
\paper
Modular curves and the Eisenstein ideal  \jour
Publ. Math. IHES \vol 47\yr1977\pages  33--186\endref

\ref\no\RAY\by M. Raynaud
\paper
Jacobienne des courbes modulaires et op\'erateurs de Hecke
\jour Ast\'erisque
\vol196--197\yr1991\pages9--25
\endref


\ref\no\FERMAT\by K?A. Ribet
\paper
 On modular representations of
$\GalQ$
 arising from modular forms
\jour Invent. Math \vol100  \yr1990\pages431--476
\endref

\ref\no\BX
\bysame
\paper
On the Component Groups and the Shimura Subgroup
        of $J_o(N)$
\jour
S\'em.  Th. des Nombres
        de l'Universit\'e de Bordeaux
\yr1987--88
\paperinfo
expos\'e 6
\endref

\ref\no\MOTIVES\bysame
\paper
Report on mod~$\ell$ representations of
$\GalQ$
\jour
Proceedings of Symposia in Pure Mathematics
\toappear
\vol55 (2)
\yr1994
\pages639--676
\endref

\ref\no\DUKE\by J-P. Serre
 \paper  Sur les repr\'esentations modulaires de degr\'e  
   2 de $\GalQ$\jour  Duke Math. J.\vol 54\yr1987\pages 179--230\endref

\ref\no\SERRELETTER\bysame
\paper Letter to K. Ribet (15 April 1987)
\paperinfo unpublished\endref

\ref\no\SILV\by J. H. Silverman
\book The arithmetic of elliptic curves
\bookinfo Graduate Texts in Math., vol. 106
\publ
   Springer-Verlag\publaddr Berlin and New York \yr1986
   \endref


\ref\no\TUNN\by
J. Tunnell
\paper
Artin's conjecture for representations
of octahedral type
\jour
  Bull. AMS
(new series)
\vol5
\yr1981
\pages173--175
\endref

\ref\no\WILES\by A. Wiles
\paper Modular elliptic curves and Fermat's Last Theorem
\paperinfo uncirculated manuscript
\endref

\endRefs
\enddocument
