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\def\endproof{ $\blacksquare$}  %more standard, maybe looks nicer?

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


%  If you uncomment out the following line, the paper looks better
%          \parskip\smallskipamount
%   in my humble opinion


% Numbering references
% This seems to be an efficient semi-automatic numbering machine.

\newcount\refCount

\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}

  \newref AL  %Atkin-Li
  \newref AS  %Ash-Stevens
  \newref BEG %Lucile Begueri's thesis
  \newref BLR % Boston-Lenstra-Ribet
  \newref BOS %Boston's Duke paper
  \newref CAREARLY
  \newref CAR %Carayol
  \newref CERED %Cerednik
  \newref CoV   % Coleman-Voloch
  \newref DEL % Deligne Bourbaki
  \newref DERA % Deligne-Rapoport
  \newref DS %Deligne-Serre
  \newref DIA % Diamond in Asterisque
  \newref DRIN % Drinfeld
  \newref EDIX %Edixhoven
  \newref FJ  % Faltings-Jordan
  \newref BHG % Gross
  \newref SGA % Grothendieck
  \newref HIDAZERO % old Hida
  \newref HIDAONE % Hida at the Ecole
  \newref HIDATWO % Hida Inventiones
  \newref IWA % Iwasawa's book
  \newref JLEARLY % Jordan-Livne compositio paper
  \newref JL  %Jordan-Livne Bulletin paper
  \newref HIGHERC % Katz's Higher congruences
  \newref KATZBONN % katz on theta
  \newref KATZMAZUR
  \newref LANGLANDS
  \newref LI
  \newref LIVNE
  \newref MAZURLETTER % Mazur to Mestre
  \newref MAZR    % Mazur-Ribet on ``dimension two''
  \newref WMC % McCallum in Duke J
  \newref MI % Miyake
  \newref QUEEN
  \newref MYTHESIS % my American Journal article
  \newref BONN %my article in Lecture notes vol 601
  \newref ICM
  \newref BX
  \newref BIMODULES %ribet's bimodules paper in Iwasawa volume
  \newref FERMAT % Ribet's ``Fermat'' article in Invent Math
  \newref DPP % Ribet's DPP article.
  \newref NOMULT % my "duke" article
  \newref SERREINV %Serre Inventiones
  \newref JPS
  \newref ANTWERP % Serre's Antwerp article
  \newref ENS %Serre's ens math paper
  \newref ARCATA  % Serre to Mestre
  \newref DUKE   % Serre's Duke article
  \newref SERRELETTER  % Serre to Ribet
  \newref ST  %Serre-Tate
  \newref SHIM %Shimura's book
  \newref HPF % Swinnerton Dyer a Anvers
  \newref TATECORVALIS %Tate's article at Corvalis


% math macros

 \def\doublemap{\mathrel{\null _{\raise.2ex\hbox{$\textstyle\rightarrow$}}
   ^{\lower.2ex\hbox{$\textstyle\rightarrow$}}}}  %makes a double arrow

 \let\tempcirc=\circ
 \def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}
      % The above two lines make "circ" smaller (I prefer it this way)

 \def\seteq{\mathrel{:=}}
 \def\GalQ{\Gal(\Qbar/\Q)}
 \def\GalQq{\Gal(\Qqbar/\Qq)}
 \def\Aut{\mathop{\roman{Aut}}\nolimits}
 \def\Sym{\mathop{\roman{Sym}}\nolimits}
 \def\Supp{\mathop{\roman{Supp}}\nolimits}
 \def\Gal{\mathop{\roman{Gal}}\nolimits}
 \def\End{\mathop{\roman{End}}\nolimits}
 \def\Hom{\mathop{\roman{Hom}}\nolimits}
 \def\Tr{\mathop{\roman{Tr}}\nolimits}
 \def\SL{\mathop{\bold{SL}}\nolimits}
 \def\PSL{\mathop{\bold{PSL}}\nolimits}
 \def\GL{\mathop{\bold{GL}}\nolimits}
 \def\ord{\mathop{\roman{ord}}\nolimits}
 \def\GLTWO#1{\GL(2,#1)}
 \def\SLTWO#1{\SL(2,#1)}
 \def\PSLTWO#1{\PSL(2,#1)}
 \def\trace{\mathop{\roman{trace}}\nolimits}
 \def\Pico{\mathop{\roman{Pic}}\nolimits ^{\roman o}}
 \def\ZNZ{(\Z/N\Z)^*}
 \def\Q{{\bold Q}}
 \def\N{{\Cal N}}
 \def\Qp{\Q_p}
 \def\C{{\bold C}}
 \def\Qbar{\overline\Q}
 \def\Qq{\Q_q}
 \def\Zq{\Z_q}
 \def\Qqbar{{\overline\Q}_q}
 \def\F{{\bold F}}
 \def\Fbar{\overline\F}
 \def\Fp{{\bold F}_p}
 \def\Fq{{\bold F}_q}
 \def\Fell{\F_\ell}
 \def\Zell{\Z_\ell}
 \def\Fpbar{\overline{\F}_p}
 \def\Fqbar{\overline{\F}_q}
 \def\Z{{\bold Z}}
 \def\T{{\bold T}}
 \def\Frob{{\roman{Frob}}}
 \def\m{{\frak m}}
 \def\diamond#1{\langle #1\rangle}
 \def\Gammao{\Gamma_o}  % this should really be _0 , but I like this better
 \def\Xo#1{X_o(#1)}  % ditto
 \def\Jo#1{J_o(#1)}  % ditto

\topmatter

 \title
Report on mod~$\ell$ representations of $\GalQ$
 \endtitle

 \author
Kenneth A. Ribet
 \endauthor

 \affil
University of California, Berkeley
 \endaffil
\rightheadtext{Mod~$\ell$ representations of $\GalQ$}
\address{UC Mathematics Department, Berkeley, CA  94720 USA}
\endaddress

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

\dedicatory
In memory of Kenneth F. Ireland
\enddedicatory

  \thanks 
This paper is in final form and no version of it will be submitted
for publication elsewhere.  % Let's face it, I've had it with this paper
The author was supported in part by NSF Grant \#DMS 88-06815.
He wishes to thank both the I.H.E.S. and the Universit\'e de Paris~7
for gracious hospitality during the period when this article was
begun.  
Thanks are due to Fred Diamond, Bas Edixhoven and 
Barry Mazur for helpful comments.  The referee deserves
special thanks for a thorough reading of the
manuscript and for many useful comments
and suggestions.
  \endthanks

\subjclass Primary 11F11, 11R32; Secondary 11F33, 11G18\endsubjclass

\endtopmatter

% Now on with the show
\document

\head 1. Introduction.\endhead \noindent
Let $N\ge1$ and $k\ge2$ be integers.  
Let $\Gamma_1(N)$ be the group
$$ \left\{ \,
\pmatrix %AMS-TeX's matrices are a raw deal!
a&b \cr c&d\cr
\endpmatrix
\in \SLTWO\Z
   \Bigm| c\equiv 0,\, a\equiv d\equiv1 \hbox{ mod } N \right\}$$
and let $S=S_k(\Gamma_1(N))$ be the complex vector space of cusp forms of
weight~$k$ on~$\Gamma_1(N)$.  There is a standard action 
$d\mapsto\diamond d\in\Aut S$
of the group $(\Z/N\Z)^*$ on~$S$.  In addition,
$S$ comes equipped with a family of Hecke operators $T_n$ ($n\ge1$);
the $T_n$ commute with each other and with the ``diamond bracket"
operators $\diamond d$.  These operators are traditionally written
on the right; for example, the $n^{\roman{th}}$ Hecke operator is normally
written $f\mapsto f|T_n$.

An {\it eigenform\/} in~$S$ is a non-zero cusp form $f\in S$ which is
an eigenvector for each of the operators $T_n$ and $\langle d\rangle$.
In particular,
if $f$ is an eigenform, then there is a Dirichlet character
$\epsilon$ defined mod~$N$ such that
$f|\diamond d = \epsilon(d) f$ for $d\in \ZNZ$.
The functional equation satisfied by
a weight-$k$ cusp form implies that this ``Nebentypus"
character
has the same
parity as~$k$:  $\epsilon(-1)=(-1)^k$.
The eigenvalues $a_n$ of the $T_n$ acting on~$f$
are  algebraic integers,
and generate together a finite extension of~$\Q$.

Let $\Qbar$ be an algebraic closure of the field of rational
numbers, and consider the Galois group $\GalQ$.  
Suppose that $f\in S$ is an eigenform, and let $E$ be a number field
containing the associated eigenvalues $a_n$ and $\epsilon(d)$.
A construction of Deligne \cite\DEL\
attaches to~$f$ a family of continuous representations
$$\rho_\lambda\:\GalQ\to\GLTWO{E_\lambda};$$ here, $\lambda$ runs
over the set of prime ideals of the
integer ring of~$E$, and $E_\lambda$ denotes the
completion of~$E$ at~$\lambda$.  Each representation $\rho_\lambda$
is irreducible; it is characterized up to isomorphism by the identities 
$$\trace(\rho_\lambda(\Frob_p))=a_p,\qquad
\det(\rho_\lambda(\Frob_p))=\epsilon(p)p^{k-1}$$ 
for prime numbers $p$ which are
prime both to~$N$ and to the norm of~$\lambda$.
The symbol $\Frob_p$ denotes an
arithmetic Frobenius element for~$p$ in~$\GalQ$.
(For a survey
of this material, the reader may wish to consult~\cite\BONN.)

The reduction of $\rho_\lambda$ mod~$\lambda$ is a certain
semisimple representation 
$$\bar\rho_\lambda\:\GalQ\to\GLTWO{\F_\lambda},$$
where
$\F_\lambda$ is the residue field of~$\lambda$.  Its construction 
 involves a choice: one chooses a model of~$\rho_\lambda$ which has
values in~$\GLTWO{{\Cal O}_\lambda}$, where
${\Cal O}_\lambda$ is the ring of integers of~$E_\lambda$, and then forms
the ``naive reduction" of this model, i.e., the composite of
$\rho_\lambda$ and the canonical map $\GLTWO{{\Cal O}_\lambda}\to
\GLTWO{\F_\lambda}$. The {\it semisimplification\/} of this
naive reduction depends only on~$\rho_\lambda$.  It is the desired
representation~$\bar\rho_\lambda$.

Consider the representations $\bar\rho_\lambda$ obtained
for varying $N$, $k$, $f$, but with $\F_\lambda$ having a fixed
characteristic.
  To assemble them, fix a prime number~$\ell$,
and choose a place $v$ dividing~$\ell$ of the field of algebraic
numbers in~$\C$.  Let $\F$ be the residue field of~$v$, so that
$\F$ is an algebraic closure of its prime field~$\F_\ell$.  For each
algebraic number field $E$ contained in~$\C$, $v$ induces a prime
$\lambda$ on~$E$, together with a canonical inclusion 
$\F_\lambda \hookrightarrow \F$.  In particular, if $E$ is the field
generated by the eigenvalues attached to an eigenform~$f$, 
and $\lambda$ is the prime of~$E$ induced by~$v$, the representation
$\bar \rho_\lambda$ may be viewed as taking values in~$\GLTWO\F$.


Suppose that $\rho\:\GalQ\to\GLTWO\F$ is a continuous semisimple
representation.  When can we expect that $\rho$ is isomorphic to one of the
$\bar\rho_\lambda$?   Identify $\epsilon$ with a character
defined on~$\GalQ$, and let $\tilde\chi$
be the $\ell$-adic cyclotomic character.
Recall that the identity
$\det(\rho_\lambda(\Frob_p))=\epsilon(p)p^{k-1}$ 
on Frobenius elements implies
the formula 
$$\det\rho_\lambda=\epsilon\tilde\chi^{k-1}\leqno(1.1)$$
for~$\det\rho_\lambda$.
On reducing this
identity, 
one obtains 
$\det\bar\rho_\lambda = \bar\epsilon\chi^{k-1}$,
where $\chi$ is the mod~$\ell$ cyclotomic character, and $\bar\epsilon$
is the reduction of~$\epsilon$ mod~$\lambda$ (i.e., mod~$v$).
In particular, let $c\in\GalQ$ be a complex conjugation.  Then 
$\det\bar\rho_\lambda(c) = \bar\epsilon(c)\chi(c)^{k-1}$.  Now
$\chi(c)=-1$, and furthermore $\epsilon(c)$ is
another name for~$\epsilon(-1)$.
Using the formula
$\epsilon(-1) = (-1)^k$, we obtain $\det\bar\rho_\lambda(c)=-1$.
In other words, $\bar\rho$ must be  an {\it odd\/} representation
if it is to be isomorphic to some~$\bar\rho_\lambda$.

Serre \cite\DUKE\ has proposed
that this parity condition represents a sufficient, as well
as a necessary, condition for a representation
$\rho\:\GalQ\to\GLTWO\F$ to arise from
some eigenform.
The {\it reducible\/} odd representations $\rho$
arise from Eisenstein series, and will be
neglected in the following discussion.  
For 
{\it irreducible\/} representations,
we have the

\proclaim{(1.2) Serre Conjecture
\cite{\DUKE, (3.2.$3_?$)}}  Let $\rho\:\GalQ\to\GLTWO\F$
be an irreducible, odd representation.  Then $\rho$ is isomorphic to
the representation
$\bar\rho_\lambda$ associated to 
some eigenform in one of the spaces $S_k(\Gamma_1(N))$.
\endproclaim

\noindent
The associated {\bf Refined Conjecture\/}  \cite{\DUKE, (3.2.$4_?$)} 
asserts that
$\rho$ is a representation $\bar\rho_\lambda$ associated to an
eigenform in
a {\it specific\/} space $S_{k(\rho)}(\Gamma_1(N(\rho)))$. The
invariants $k(\rho)$ and~$N(\rho)$ are
defined by local properties of~$\rho$.
More precisely, as we recall in~\S2,
$N(\rho)$ is an integer prime to~$\ell$ which depends only on
the restrictions of~$\rho$ to decomposition
groups in~$\GalQ$ for each prime number $p\ne\ell$.  
Similarly, $k(\rho)$ is an integer $\ge2$ which depends
only on
the restriction of
$\rho$ to a decomposition group for~$\ell$ in~$\GalQ$.
(See \cite{\DUKE, \S2}.)



The formulation of \cite{\DUKE, (3.2.$4_?$)} specifies that
$\rho$ arise from an eigenform $f\in S_{k(\rho)}(\Gamma_1(N(\rho)))$
whose character $\epsilon$ is also predicted in
advance.  This extra requirement
means simply that $\epsilon$ can be chosen to
be of order prime to~$\ell$.
Subsequently, however, Serre found examples 
for~$\ell\le3$
showing that the
requirement cannot always be satisfied \cite\SERRELETTER;
his examples concern the two-dimensional space $S_2(\Gamma_1(13))$.
On the other hand, Carayol \cite{\CAR, Prop.~3} and Serre have each
shown that the situation for $\ell\ge5$ is much more favorable:
the representations $\bar\rho_\lambda$ arising from a
given space $S_k(\Gamma_1(N))$ all arise from eigenforms whose
associated character has prime-to-$\ell$ order.

More precisely, Carayol proves
the following result \cite{\CAR, Proposition 3}:

\proclaim{(1.3) Theorem}
Assume that $\ell\ge5$ and that $\rho\:\GalQ\to\GLTWO\F$ is
an odd
irreducible representation.  Suppose that $\rho$ arises
    from an eigenform $f\in S_k(\Gamma_1(N))$
and that $\epsilon$ is the Dirichlet character
which is associated to~$f$.  Let $\epsilon'$ be a 
character on $\ZNZ$ which is congruent to~$\epsilon$ mod~$v$.
Then $\rho$ arises from an eigenform $f'\in S_k(\Gamma_1(N))$
whose Nebentypus character is~$\epsilon'$.
\endproclaim

\noindent
In view of this result, and the counterexamples for $\ell=2$
and~$\ell=3$, we will
restrict attention to
the invariants $N(\rho)$ and~$k(\rho)$ in
discussing Serre's conjectures.


The Refined Conjecture
has a number of striking consequences, which are
catalogued in~\S4 of~\cite\DUKE.  Among these
are Fermat's Last ``Theorem," the conjecture of Taniyama 
that all elliptic curves over~$\Q$ are modular, and variants of
Taniyama's conjecture concerning other motives of rank~2.

To bridge the gap between the Serre Conjecture and its
refinement,
we formulate the following

\proclaim{(1.4) Motivating Problem}  Suppose that $\rho\:\GalQ\to\GLTWO\F$
is an irreducible representation which arises from some space
of cusp forms $S_{k}(\Gamma_1(N))$.  Does $\rho$ arise from
an eigenform in~$S_{k(\rho)}(\Gamma_1(N(\rho)))$, 
where $k(\rho)$ and~$N(\rho)$ are the
invariants assigned to~$\rho$ by Serre?
\endproclaim

\noindent
An affirmative solution to (1.4) would imply an equivalence between
Serre's conjecture and its refinement.
At the time of this writing, one is very close to a complete
solution of~(1.4), at least when $\ell$ is an
odd prime.  The solution involves the work of a large
number of mathematicians, including
N.~Boston, H.~Carayol, F.~Diamond, B.~Edixhoven,
G.~Faltings, B.~H.~Gross, B.~Jordan, H.~W.~Lenstra, Jr.,
R.~Livn\'e, B.~Mazur, J-P.~Serre, and the author.

The goal of this article is twofold.  On the one hand, we shall
describe what is known about~(1.4).  On the other, we shall
present the following new result:

\proclaim{(1.5) Theorem}
Let $\ell\ge3$ be a prime, and let $\rho\:\GalQ\to\GLTWO\F$
be irreducible.
Suppose that $\rho$ 
arises from an eigenform $f$ of weight two and trivial character
on~$\Gamma_1(M)\cap\Gammao(p)$,
 where $p$ is prime to~$\ell M$.  Assume that
$\rho$ is unramified at~$p$.  Then $\rho$ arises from a weight-two
eigenform with trivial character on~$\Gamma_1(M)$.
\endproclaim

\noindent  The new feature of this theorem is that the level~$M$
is not required to be prime to~$\ell$.  In fact, $M$ can be
divisible by an arbitrarily high power of~$\ell$.  As it stands,
(1.5) applies only to forms with trivial
character on~$\Gamma_1(M)\cap\Gammao(p)$, i.e., to forms 
on~$\Gammao(Mp)$.  However, it is very likely that the proof
we give for~(1.5) will generalize without difficulty to cover
eigenforms on~$\Gamma_1(M)\cap\Gammao(p)$ whose Nebentypus
characters are {\it arbitrary\/} characters mod~$M$.
The discussion below shows that (1.4) will
be solved for primes $\ell\ge5$ as soon as Theorem~1.5
is generalized to this case. 

One can also contemplate a variant of~(1.5) which applies to forms
of weight~$k$ on~$\Gamma_1(M)\cap\Gammao(p)$, where $k$ satisfies
$2\le k\le \ell{+}1$ and $M$ is required to be
prime to~$\ell$.  Such a variant, which accepts arbitrary
characters mod~$M$, will again lead to a solution of~(1.4).
It seems very likely that the methods of Faltings,
Jordan and Livn\'e \cite\FJ, \cite\JL\ will lead to a variant of this
type.

In light of the 
substantial progress made in relating Serre's conjecture to
its refinement,
it is striking that there is little to
report concerning the conjecture itself
\cite{\DUKE, 3.2.$3_?$}, aside from
the numerical evidence presented in~\S5 of~\cite\DUKE.
% This conjecture still has the status of an optimistic question.



\head 2. Stripping powers of $\ell$ from the level.\endhead \noindent
Let $\rho\:\GalQ\to\GLTWO\F$ be an irreducible odd
representation, as above, and let $N(\rho)$ and~$k(\rho)$ be
the invariants assigned to~$\rho$ by~Serre.
Recall \cite{\DUKE, \S1} that $N(\rho)$ is a product
$\prod p^{n(p,\rho)}$ extended over the set of prime numbers
$p\ne\ell$.  The integer $n(p,\rho)$ is defined by 
restricting $\rho$ to a decomposition group $D_p$ for~$p$
in~$\GalQ$.  Consider the ramification subgroup $G_0$ of~$D_p$,
along with the higher ramification groups $G_i$ ($i>0$). Let
$V$ be a two-dimensional $\F$-vector space affording the 
representation~$\rho$. For each $i\ge0$, let $V_i$ be
the subspace of $V$ consisting of those $v\in V$
which are fixed by all elements of~$G_i$. Then
$$ n(p,\rho) = \sum_{i\ge0} {1\over (G_0:G_i)} \dim V/V_i.$$

In particular, $N(\rho)$ is prime to~$\ell$, so that Serre's
refined conjecture implies the weaker statement
that $\rho$ arises from some space of cusp forms $S_k(\Gamma_1(N))$,
with $N$ prime to~$\ell$.
Let us say simply
that $\rho$ arises from a subgroup $\Gamma$
of~$\SLTWO\Z$
if it arises from the space of weight-$k$ cusp forms on~$\Gamma$,
for {\it some\/}~$k\ge2$.
Then the weak statement implied by
Serre's conjecture is that $\rho$ arises from the
group $\Gamma_1(N)$ for some $N$ which is prime to~$\ell$.
We shall now show (at least when $\ell\ge3$)
that any $\rho$ which is modular in the sense
that it arises from some $\Gamma_1(N)$ with $N$ not necessarily
prime to~$\ell$ is automatically modular in the apparently stronger
sense that it arises from $\Gamma_1(N)$ for some $N$ which 
is prime to~$\ell$.  This is certainly a ``well known" fact, 
which is close to results which are already in the literature,
cf.~\cite{\DUKE, {\it Remarque}, p.~195}.

\proclaim {(2.1) Theorem}
 %
Assume $\ell\ge3$.
Suppose that $\rho$ arises from  
  $\Gamma_1(M)$, where
  $M$ is the product
$N\ell^\alpha$ with $(N,\ell)=1$.
Then $\rho$ arises from  $\Gamma_1(N)$. 
\endproclaim


\noindent
We  prove (2.1) using the concrete techniques of Serre \cite{\ANTWERP, \S3},
\cite{\ENS, Theorem~5.4} and Queen \cite{\QUEEN, \S3}.
An alternative method would be based on results of N.~Katz:
see the appendices of \cite{\HIGHERC} and the discussions in
\cite{\HIDAONE, \S1} and \cite{\HIDATWO, \S1}.
(Thanks are due to Fred Diamond for
pointing out these references.)
Note that the assumption $\ell\ge3$ is made principally
for convenience; it should be possible to analyze the case  $\ell=2$
 without great difficulty.

Our proof consists of several independent 
steps.  Before beginning it, we shall normalize the eigenforms which
occur in the
proof.  Recall that if $f$ satisfies $f|T_n=a_nf$ for all $n\ge1$,
then we may scale $f$ by a constant so that its Fourier coefficients
are the eigenvalues $a_n$.  We will assume that all eigenforms
are scaled in this way, i.e., that they are ``normalized eigenforms."
We represent these eigenforms as series $\sum_{n\ge1} a_nq^n$; the
variable $q$ initially represents $e^{2\pi i\tau}$, where $\tau$ is
an element of the complex upper half plane, but rapidly becomes
a formal variable.  In particular, we say that two forms are congruent
mod~$\ell$ or mod~$v$ if their Fourier series are formally congruent.
Similarly, the mod~$v$ reduction of an eigenform~$f$ is meant to
be the formal power series $\sum \bar a_nq^n$, where 
``$\bar{\phantom a}$"
denotes the reduction map mod~$v$.

\proclaim{Step 1}
The representation
$\rho$ arises from   $\Gammao(\ell^r) \cap \Gamma_1(\ell N)$
for some $r\ge0$.
\endproclaim

\noindent
When $\ell\ge5$, the assertion follows
\kern0pt from~(1.3).  In that case, we can take $r=\alpha$. 
It can be
proved in an elementary manner, for all
$\ell\ge3$, by the following
argument.

Suppose that $\rho$ arises from an eigenform $f$ in $S_k(\Gamma_1(M))$
and that $\kappa$ is the associated Dirichlet character mod~$M$.
We assume that $k\ge2$ and that
  $\alpha$ is positive (i.e., that $M$ is divisible by~$\ell$).
Decompose $\kappa$ as a product $\epsilon\eta\omega^i$,
where: $\epsilon$ has conductor dividing~$N$, $\eta$ has $\ell$-power
order and $\ell$-power conductor, and $\omega$ is the ``Teichm\"uller"
character, i.e., that character of conductor $\ell$ and order $\ell{-}1$
which is congruent to the identity function mod~$v$.  The exponent
$i$ is naturally an integer mod~$\ell{-}1$.
Because $\eta$ has, in particular,
odd order, we may write it in the form $\xi^{-2}$, where $\xi$ is a character
of $\ell$-power order.  The cusp form $f\otimes\xi$ is 
a form of type $(k,\epsilon\omega^i)$ 
(i.e., weight~$k$ and character~$\epsilon\omega^i$)
which gives rise to~$\rho$.
If $r\ge\alpha$ is large enough so that $\ell^r$ is divisible by the square
of the conductor of~$\xi$, then $f\otimes\xi$ is again of level~$N\ell^r$.

In what follows, we fix an~$r>0$ so that
$\rho$ arises from   $\Gammao(\ell^r) \cap \Gamma_1(\ell N)$.

\proclaim{Step 2}
  $\rho$ arises from  
$\Gammao(\ell^r) \cap \Gamma_1(N)$.
\endproclaim

\noindent
We assume as above that $\rho$ arises from an eigenform
of type $(k,\epsilon\omega^i)$ on
the group $\Gammao(\ell^r) \cap \Gamma_1(\ell N)$.  Take the
exponent $i$ to be a positive integer.  The Eisenstein series
$$G \seteq L(1{-}i,\omega^{-i})/2 + \sum_{n=1\vphantom{d|n}}^\infty
  \left(\sum_{d|n} \omega^{-i}(d) d^{i-1}\right)q^n $$
is then of type $(i,\omega^{-i})$ on $\Gammao(\ell)$, as proved
(for example) in \cite{\ANTWERP, Lemme~10}.
Thus
$fG$ is on the group~$\Gammao(\ell^r) \cap \Gamma_1(N)$, 
if $r$ is positive.  On the other hand, it is well known that the order at~$v$
of $L(1-i,\omega^{-i})$ is negative: this follows from the Kummer congruences
and the Von Staudt theorem.  (See, e.g., 
\cite{\IWA, \S3.4}.)  In other
words, if $c$ is the constant term of~$G$, then $E\seteq c^{-1}G$
is a form with $v$-integral coefficients such that $E\equiv1$ mod~$v$.
Consequently, the product $fE$, viewed mod~$v$, is a
non-zero eigenform with the
same eigenvalues as~$f$.  At the same time, $fE$ is on the group
$\Gammao(\ell^r) \cap \Gamma_1(N)$.
By a well known lemma 
\cite{\DS, Lemme~6.11}, we may find an eigenform
on $\Gammao(\ell^r) \cap \Gamma_1(N)$
whose eigenvalues are congruent to those of~$f$.


\proclaim{Step 3}
$\rho$ arises from $\Gammao(\ell) \cap \Gamma_1(N)$.
\endproclaim

\noindent
Assume that $\rho$ arises from an eigenform $f=\sum a_nq^n$ on
$\Gammao(\ell^r) \cap \Gamma_1(N)$, with $r>1$.  
Let $K$ be a finite Galois extension of~$\Q$ containing the $a_n$,
and let $\sigma\in\Gal(K/\Q)$ be a Frobenius element for~$v$.
Thus $\sigma a \equiv a^\ell$ mod~$v$ for all elements $a$ of the
ring of integers of~$K$.  The series $\sum \sigma^{-1}a_n q^n$
is the Fourier expansion of a normalized eigenform $\sigma^{-1}f$
of the same weight as~$f$.
We wish to show
that $f$ is congruent mod~$v$ to a cusp form of some weight  on
$\Gammao(\ell^{r-1}) \cap \Gamma_1(N)$.  Consider 
$g\seteq (\sigma^{-1}f)^\ell|U$, where
$U$ is the $\ell^{\roman{th}}$ Hecke operator $T_\ell$.  On the one hand, $g$ is
a form on  $\Gammao(\ell^{r-1}) \cap \Gamma_1(N)$ (see, e.g.,
\cite{\LI, Lemma~1}).
On the other hand, the Fourier expansion of~$g$ is
congruent mod~$v$ to $\sum (\sigma^{-1}a_n)^\ell q^n$, which in turn
is congruent to the Fourier expansion of~$f$.

\proclaim{Step 4}
$\rho$ arises from $\Gamma_1(N)$.
\endproclaim

\noindent
  To prove this, we use the argument given by Serre in 
\cite{\ANTWERP, \S3.2},
replacing the operators $W$, $U$, $V$, $\ldots$ by their analogues
for forms on~$\Gamma_1(N)\cap\Gammao(\ell)$.  More precisely, we let
$U$ be the $\ell^{\roman{th}}$ Hecke operator, as above, and let $V$
be the operator $\sum a_n q^n \mapsto \sum a_n q^{\ell n}$, which takes
forms on~$\Gamma_1(N)$ to forms of the same weight 
on~$\Gamma_1(N)\cap\Gammao(\ell)$.  Further, we let $W$ be the
 operator $V_\ell^{N\ell}$   as defined in~\S1 of~\cite\LI.  Thus $W$
is given by the matrix
$\pmatrix \ell x&y\cr N\ell z &\ell\crcr\endpmatrix $,
where $x$, $y$, and~$z$ are integers and we have $\ell x -Nyz=1$.
This operator is the inverse of the operator denoted $W_Q$ in~\cite\AL.
Indeed, by \cite{\AL, Proposition~1.1}, 
one has $W=\epsilon(\ell)W_Q$ on the space
of forms with character $\epsilon$ on
$\Gamma_1(N)\cap\Gammao(\ell)$.  (We consider $\epsilon$ as a 
Dirichlet character mod~$N$, which permits us to evaluate it at~$\ell$.)
At the same time, $W^2$ is multiplication by~$\epsilon(\ell)$,
according to Lemma~2 of~\cite\LI.

Next, suppose that
  $F$ is a form of weight~$w$ and character $\epsilon$ on
$\Gamma_1(N)\cap\Gammao(\ell)$.  Then \cite{\LI, Lemma~3}
implies that the form
$$ \Tr(F)\seteq F + \epsilon^{-1}(\ell)\ell^{1-w/2} F|W|U $$
is a form of weight~$w$ on~$\Gamma_1(N)$.
(Apply \cite{\LI, Lemma~3} to $F|W$.)
Further, if $G$ is a form of weight $w$ and character $\epsilon$
on $\Gamma_1(N)$, one has
$G|W = \ell^{w/2}\epsilon(\ell)G|V$ 
\cite{\AL, Prop.~1.5}.

If $\ell>3$, let $E$ be the normalized Eisenstein series %$E_{\ell-1}$
of
weight $\ell{-}1$ on~$\SLTWO\Z$, 
so that we have $E\equiv1$ mod~$\ell$.  Similarly, 
if $\ell=3$ let $E$ be the normalized 
Eisenstein series $E_4$ of weight four.  Let $a$ denote the weight of~$E$.
As in \cite\ANTWERP, we introduce the form
$$ g\seteq E - \ell^{a/2}E|W = E - \ell^a E|V.$$
One has $g\equiv1$ mod~$\ell$; furthermore, $g|W$ is divisible by
a positive power of~$\ell$ (in fact, by~$\ell^{1+a/2}$).

Let $f$ be an eigenform giving rise to~$\rho$, as above, and consider
$\Tr(fg^i)$, where $i$ is a positive integer.
The modular form $\Tr(fg^i)$ is a form on~$\Gamma_1(N)$.
On the other hand,
a calculation similar to 
that of \S3.2 of~\cite\ANTWERP\
shows that
$\Tr(fg^i)\equiv f$ mod~$\ell$ for sufficiently large~$i$.  
\endproof

A partial converse to (2.1) is the following result:

\proclaim{(2.2) Theorem}  Suppose that $\rho$ arises from $S_k(\Gamma_1(N))$,
with $N$ prime to~$\ell$ and $2\le k\le \ell{+}1$.  Assume that $\ell>3$
or that~$N>3$.
Then $\rho$ arises from $S_2(\Gamma_1(N\ell))$.
\endproclaim

\noindent
For $\ell\ge5$, the theorem is proved by Ash and Stevens
\cite{\AS, Theorem 3.5a} 
by methods involving parabolic
cohomology.  Another approach, which originated with Serre, is
worked out in Gross's article \cite{\BHG, Proposition~9.3}.  Although
Gross's prime~$p$ (which corresponds to our~$\ell$) is arbitrary,
he assumes that the level~$N$ satisfies $N\ge4$.  According to comments
in~\S10 of~\cite\BHG, Proposition~9.3 of~\cite\BHG\ extends to the case where
$N\le2$ and~$k$ is even.


\head 3. Adjustment of the weight.\endhead\noindent
Let $\rho\:\GalQ\to\GLTWO\F$ be an irreducible representation,
and let $k(\rho)$ be the ``weight" attached to~$\rho$ by
Serre \cite{\DUKE, \S2}.  
Edixhoven \cite{\EDIX, \S4} proved that
if $\rho$ arises from a cusp form of some weight, then $\rho$
arises from a cusp form of weight~$k(\rho)$.  More precisely,
\cite{\EDIX, Th.~4.3} gives the following result:

\proclaim{(3.1) Theorem}
Let $N$ be prime to~$\ell$, and
assume that $\rho$ arises from
an eigenform $f\in S_k(\Gamma_1(N))$ with character
$\epsilon$. 
Then $\rho$
arises from an eigenform $f'\in S_{k(\rho)}(\Gamma_1(N))$ with
character~$\epsilon$.  The integers $k$ and~$k(\rho)$
are congruent mod~$\ell{-}1$; moreover, we have $k\ge k(\rho)$
provided that $\ell$ is~odd.
\endproclaim

\noindent
The proof of~(3.1) relies on two points which
merit independent discussion.  First, let $D_\ell$ be a decomposition
group for~$\ell$ in~$\GalQ$ and
let $I_\ell$ be the inertia subgroup of $D_\ell$.  
Suppose that $$\rho\:\GalQ\to\Aut V\approx \GLTWO\F$$
arises from
an eigenform $f=\sum a_n q^n$ of weight~$k$ and character~$\epsilon$
on~$\Gamma_1(N)$, with $N$ prime to~$\ell$.  
The proof of~(3.1) requires information about
the restriction of~$\rho$ to~$D_\ell$, in the case where
$k$ satisfies $2\le k\le \ell$.

Specifically, suppose that this inequality is satisfied, and that
$f$ is ordinary in the sense that $a_\ell \not\equiv 0$ mod~$v$.
A theorem of Deligne (proved in \cite\BHG) states that $V$
has an unramified quotient $V/V_0$
on which $\Frob_\ell$ acts by multiplication
by~$a_\ell$ (or, more precisely, by the image of~$a_\ell$ in~$\F$).
The action of~$D_\ell$ on~$V_0$ is determined by this information,
since the determinant of~$\rho$ may be computed from~$k$ and~$\epsilon$.
On the other
hand, suppose that $f$ is supersingular, i.e., not ordinary.
Let $\psi$ and $\psi'$ be the two fundamental characters 
$I_\ell\to\F$ of level~2, as defined in~\cite{\SERREINV, \S1.7}.
A theorem of Fontaine states that the restriction of~$\rho$ to~$I_\ell$
is the direct sum of two one-dimensional representations, on which
$I_\ell$ operates by the $k{-}1^{\roman{st}}$ powers of 
$\psi$ and~$\psi'$.  This theorem was proved by Fontaine in
an exchange of letters with Serre in~1979; a new proof was given
by Edixhoven in~\cite\EDIX. 

Secondly, Edixhoven's proof requires B.\hskip 0.1667em
\relax H.~Gross's result about
companion forms \cite\BHG, which conversely may be viewed as a special case
of~(3.1).  Before stating Gross's result, we recall that there is
an operator
$\theta=q{ d\over dq}$ on mod~$\ell$ modular forms for~$\Gamma_1(N)$
when $N$ is prime to~$\ell$.  
This operator
maps forms of weight~$k$ to forms of weight $k{+}\ell{+}1$.  It
was considered initially  by Serre and Swinnerton-Dyer 
\cite\JPS, \cite\HPF\
for forms on~$\SLTWO\Z$, and then constructed by Katz
  \cite\KATZBONN\ for forms of full level~$N$, where $N\ge3$.  It is
introduced
in \cite{\BHG,\S4}
 in the case of forms on~$\Gamma_1(N)$, with $N\ge4$.
According to 
Edixhoven
\cite{\EDIX, \S2.1}, 
mod~$\ell$ modular forms of weight~$k$ on~$\Gamma_1(N)$ 
are $\GLTWO{\Z/n\Z}$-invariant sections 
of the line bundle $\omega^k$ on the curve $X$. Here $n\ge3$
is prime to~$\ell$, and $X$ is the modular curve parametrizing
elliptic curves with a $(\Gamma_1(N), \Gamma(n))$-structure.  
The integer~$N$ is assumed only to be prime to~$\ell$.
Using this fact, Edixhoven constructs $\theta$ for 
modular forms of weight~$k$ on~$\Gamma_1(N)$.

Consider now an
ordinary form $f$ as above, whose weight $k$ satisfies
$2\le k \le \ell{-}1$.  Suppose
in addition that the
exact sequence 
of $\F[D_\ell]$-modules
$$0\to V_0\to V\to V/V_0\to0$$
is {\it split}.  The inertia group $I_\ell$
acts on~$V_0$ via the character~$\chi^{k-1}$, since $\GalQ$
acts on $\det V$ via the product $\epsilon\chi^{k-1}$ and since
$I_\ell$ acts trivially on~$V/V_0$.
Consider the representation $\rho'\seteq\rho\otimes\chi^{1-k}$ of~$\GalQ$.  
Locally at~$\ell$, it is the direct sum of $V_0\otimes\chi^{1-k}$
and $(V/V_0)\otimes\chi^{1-k}$.  In particular, it is an extension
of the unramified line $V_0\otimes\chi^{1-k}$ by a line on which
$I_\ell$ acts via the character $\chi^{1-k} = \chi^{\ell{+}1{-k}-1}$.
This suggests that $\rho'$ arises from an eigenform of weight
$k'\seteq \ell{+}1{-}k$ and
character~$\epsilon$ on~$\Gamma_1(N)$;
and indeed Serre's recipe for weights sets
$k(\rho')=k'$.
It is clear
that
$\rho'$ is modular of some weight, because twisting representations
by~$\chi$ corresponds to applying the 
operator $\theta$ on modular forms.  
Hence (3.1)
asserts, in particular, that $\rho'$ arises from an eigenform $f'$
of weight~$k'$.  The construction of the ``companion" $f'$ to~$f$
is the main result of~\cite\BHG.
(If $k=\ell$ and $\rho$ is split locally at~$\ell$, then Gross
still produces a companion form, of weight~1, in certain cases.
For $\ell$ odd, the remaining cases of weight~1 are treated in a
recent article by Coleman and Voloch \cite{\CoV}.)
This completes the discussion of Theorem~3.1.

As above, we shall use the phrase
``$\rho$ arises from~$\Gamma_1(N)$" to
indicate that $\rho$ arises from an eigenform in $S_k(\Gamma_1(N))$
for some~$k\ge2$.

\proclaim{(3.2) Corollary}
Suppose that $\rho$ arises from $\Gamma_1(N)$, 
with $N$ prime to~$\ell$.  
Then there exists a power $\chi^i$ of the mod~$\ell$
cyclotomic character~$\chi$ such that $\rho\otimes\chi^i$ arises from
$S_k(\Gamma_1(N))$ for some $k\le\ell{+}1$.
\endproclaim

\noindent
{\sl Proof}.\enspace
Replacement of $\rho$ by any of its twists $\rho\otimes\chi^i$ 
($i=0,1,2,\ldots$) 
corresponds to applying the 
operator $\theta=q{d\over dq}$ on modular forms
$i$ times.  Hence, $\rho$ arises from a given~$\Gamma_1(N)$
 if and only if 
  $\rho_i\seteq\rho\otimes\chi^i$ arises from the same group. 
The definition of~$k(\rho)$ is such
that a suitable twist $\rho_i$ satisfies the inequality
$$2\le k(\rho_i)\le\ell{+}1.$$
(See \cite{\EDIX, Theorem~3.4},
and the discussion following the statement of that theorem.)
By Theorem~3.1, $\rho_i$ arises from $S_{k(\rho_i)}(\Gamma_1(N))$.
\endproof 

Let $\rho\:\GalQ\to\GLTWO\F$ be an irreducible
representation.  Consider the following four sets of 
positive integers:
$$ \eqalign{
\N_1&\seteq \{\, N \hbox{ prime to~$\ell$} \; | \; 
    \rho \hbox{ arises from } S_{k(\rho)}(\Gamma_1(N)) \,\}
;\cr \N_2&\seteq \{\, N \hbox{ prime to~$\ell$} \; | \; 
    \rho \hbox{ arises from } \Gamma_1(N) \,\}
;\cr \N_3&\seteq \{\, N \hbox{ prime to~$\ell$} \; | \; 
    \rho \hbox{ arises from } \Gamma_1(N\ell^\alpha) \hbox{ for
    some }\alpha\ge0\,\}
;\cr \N_4&\seteq \{\, N \hbox{ prime to~$\ell$} \; | \; 
    \rho \hbox{ arises from } S_2(\Gamma_1(N\ell^2))\,\}
.\cr}$$



\proclaim{(3.3) Theorem}
If $\ell\ge5$, then the four sets of
integers $\N_i(\rho)$ are equal.
\endproclaim

\noindent
{\sl Proof}.\enspace
The equality of $\N_1$ and~$\N_2$ is guaranteed by Theorem~3.1.
That $\N_2$ and~$\N_3$ are equal follows from Theorem~2.1.  Since
it is clear that $\N_4\subseteq\N_3$, it remains only to show that
$\N_3\subseteq\N_4$.

We will show, equivalently, that $\N_2\subseteq\N_4$.
Assume that $\rho$ arises from~$\Gamma_1(N)$, and choose $i\ge0$
such that $\rho\otimes\chi^i$ arises from $S_k(\Gamma_1(N))$,
with $2\le k \le\ell{+}1$.  By~(2.2), $\rho\otimes\chi^i$
arises from $S_2(\Gamma_1(N\ell))$.
  To see that $N\in\N_4$, we view
the twisting operator $\otimes\chi^{-i}$ as a
a Dirichlet twist on modular forms.  Such a twist changes the level
of a modular form, but not its weight.  From
\cite{\AL, Proposition~3.1},
we find that $\rho=(\rho{\otimes}\chi^i)\otimes
\chi^{-i}$ arises from $S_2(\Gamma_1(N\ell^2))$.
\endproof

In case $\ell\ge5$, we will write simply $\N(\rho)$ for the common
value of the four sets~$\N_i(\rho)$.

\head 4. The levels of $\rho$ and of $f$.\endhead\noindent
Suppose that an irreducible
representation $\rho\:\GalQ\to\GLTWO\F$ is ``modular" in the sense
that it
arises from an eigenform of some weight and some
level.
Problem~(1.4) requires that we find an eigenform of weight~$k(\rho)$
and level~$N(\rho)$ which gives rise to~$\rho$.  
Because of Edixhoven's theorem (Theorem~3.1 above),
the weight aspect of~(1.4) disappears.
In fact, suppose that
$\ell\ge5$.  Then
Theorem~3.3
translates (1.4) into the folllowing question:
$$ N(\rho) \buildrel ?\over\in \N(\rho). \leqno(4.1)$$
In other words, our goal is to show that $\rho$ is modular of level
$N\ell^\alpha$, for some $\alpha\ge0$.
With (4.1) in mind, we are led to examine the set 
$\N(\rho)$ more closely.
The
following theorem was proved by Carayol  \cite\CAR\ and, independently,
by Livn\'e \cite{\LIVNE, Proposition~0.1}:

\proclaim{(4.2) Theorem}  Suppose that $\rho$ arises from 
$\Gamma_1(N)$.  Then
Serre's invariant $N(\rho)$ divides $N$.
\endproclaim

\noindent
In attacking problem~(4.1), one
begins with an eigenform $f$ giving rise to~$\rho$.
If $N$ is the level of~$f$, we have $N(\rho) | N$.
Since $N(\rho)$ is, by definition, prime to~$\ell$, we have
$N(\rho) | N'$, where $N'$ is the prime-to-$\ell$ part of~$N$.
In case this divisibility is strict,
the aim is to ``lower"~$N$ by replacing it by a divisor.
For the most part,
the strategy is to do this ``locally": we consider a prime
$p\ne\ell$ which divides $N/N(\rho)$ and seek to replace $N$
by $N/p^i$ for some positive integer~$i$.


As a
preliminary step, we 
should certainly replace $f$ by the {\it newform\/}
(or primitive form)
associated to~$f$.  
This is an eigenform whose
system of $\lambda$-adic representations coincides with the system
$(\rho_\lambda)$ attached to~$f$, and whose level is minimal among
all such eigenforms.  
The level of the newform attached to~$f$ is a divisor of~$N$.
(For the theory of newforms,
see for instance the account in \cite\LI, or \cite{\MI, \S4.6}.)  
According to a theorem of Deligne, Langlands \cite\LANGLANDS, 
and~Carayol \cite\CAREARLY,
the conductor of the system~$(\rho_\lambda)$ coincides with~$N$ once
this replacement is made.

Assume, then, that $f$ is a newform.  Theorem~4.2 is actually a
consequence of a much more refined comparison of $N(\rho)$
with (the prime-to-$\ell$ part of)~$N$.
This comparison is again due to Carayol \cite\CAR\ and Livn\'e \cite\LIVNE.
We will discuss the analysis of Carayol and Livn\'e,
under the simplifying assumption $\ell\ge5$.
In view of (1.3), this assumption permits us to assume
that the character $\epsilon$
which is associated to~$f$ has prime-to-$\ell$ order, and thereby
facilitates the discussion.

Consider that representation $\rho_\lambda$ which is
attached to~$f$ and the prime $\lambda$ induced by the chosen
place $v$ of~$\Qbar$.  Then $\rho$ is the reduction 
of~$\rho_\lambda$ in the sense which is outlined above.  Since
$\rho$ is irreducible, $\rho$ is 
in fact the ``naive"
reduction of any model of~$\rho_\lambda$ over~${\Cal O}_\lambda$:
no semisimplification is required.  

Fix a prime $p\ne\ell$, and let
$D_p\subset\GalQ$ again be a decomposition group for~$p$.
Let $I_p$ be the inertia subgroup of~$D_p$,
and let $e=\ord_p(N/N(\rho))$.  Thus, $e$ is a non-negative integer,
according to~(4.2).

The inequality $e\ge0$ can be interpreted conceptually.
Consider $\rho_\lambda$ and~$\rho$ as homomorphisms
$$\GalQ\to\Aut(V_\lambda),\qquad\GalQ\to\Aut V,$$
where $V_\lambda$ and $V$ are two dimensional vector spaces over
$E_\lambda$ and~$\F$ respectively.  Then one has
$$ e = \dim_\F V^{I_p} - \dim_{E_\lambda} V_\lambda^{I_p},\leqno(4.3)$$
where $V^{I_p}$ is the space of $I_p$-invariants in~$V$ and
$V_\lambda^{I_p}$ is space of $I_p$-invariants in~$V_\lambda$.
In particular, we have $e>0$ if and only if 
$$\dim_\F V^{I_p} > \dim_{E_\lambda} V_\lambda^{I_p}.$$

Carayol and Livn\'e have classified
the situations in which we have $e>0$.  We will now describe
their classification.  For each situation which arises in the
classification, we will summarize what is known about the question
of showing that $\rho$ is modular of
level (dividing) $N/p$, i.e., that $\rho$ arises from~$\Gamma_1(N/p)$.
Our summary appeals to a number of results which are proved
only later in this article (for example, Theorems~5.1, 1.5, and~8.1).
The material in this~\S\ may thus be viewed as a motivation for
the theorems in \S\S5--8.

The first step in the classification is to
distinguish cases according to
the behavior of the component at~$p$ of the automorphic representation
attached to~$f$.
Recall that the newform~$f$ is associated to an automorphic
representation $\pi_{\bold A}$
of~$\GLTWO{\bold A}$, where $\bold A$ is the ad\`ele ring
of~$\Q$.  Let $\pi$ be the component of~$\pi_{\bold A}$ at~$p$,
so that $\pi$ is an admissible representation of~$\GLTWO\Qp$.
This representation may be classified as a principal series
representation, a special representation, or a cuspidal
representation of~$\GLTWO\Qp$.

The Langlands correspondence attaches to~$\pi$ a
$\lambda$-adic
representation $\rho_{\lambda,\pi}$
of the Weil group of~$\Qp$.  
(Strictly speaking, $\rho_{\lambda,\pi}$ is defined only up to
isomorphism.  See~\cite\TATECORVALIS, \S4 for a discussion of
the relation between $\lambda$-adic representations of the
Weil group of~$\Qp$ and complex representations of the
corresponding Weil-Deligne group.)
By the main result of~\cite\CAREARLY,
$\rho_{\lambda,\pi}$
is somorphic to the restriction of~$\rho_\lambda$ to
the Weil group of~$\Qp$.
Therefore, properties of~$\pi$ are mirrored by the
local behavior of~$\rho_\lambda$ at~$p$.
The restriction of~$\rho_\lambda$ to~$D_p$ is: reducible and
semisimple when $\pi$ is a principal series representation;
reducible, but not semisimple when $\pi$ is special;
and irreducible when $\pi$ is cuspidal.

We now discuss the inequality $e>0$ in each of these three
cases.


\subhead The principal series case \endsubhead


In this case, which was treated by Carayol,
the representation $\pi$ arises from an
(unordered) pair of Grossencharacters of $\Q_p^*$, say $\alpha$
and~$\beta$.  Identify these characters with characters
of~$D_p$, using the reciprocity map of local classfield theory.
Then
$\rho_\lambda|_{D_p}$ 
is the direct sum
$\alpha\oplus\beta$.  

\proclaim{(4.4) Proposition \cite\CAR}
Assume that~$e>0$. Then there is a Dirichlet character $\phi$ of
conductor~$p$ and $\ell$-power order such
that the newform associated to
$f\otimes\phi$ has level dividing $N/p$.  In particular, $\rho$
is modular of level~$N/p$.
\endproclaim

\noindent{\sl Proof}.\enspace
One of   
$\alpha$ and~$\beta$ is ramified, but has unramified reduction mod~$\lambda$.
After possibly permuting $\alpha$ and~$\beta$, we may assume that
the restriction of~$\alpha$
          to~$I_p$ is non-trivial, but has $\ell$-power order.
On the other hand,
because of formula~(1.1), and the assumption that the order
of~$\epsilon$ is prime to~$\ell$,
the restriction of~$\det\rho_\lambda$ to~$I_p$
has prime-to-$\ell$ order.
(Note that $\tilde\chi$ is ramified only at~$\ell$.)
Thus $\alpha\beta|_{I_p}$ has prime-to-$\ell$ order.
It follows from this that, on~$I_p$, $\alpha$ is a power of~$\beta$,
since $\beta$ is the product $\alpha^{-1}\cdot \alpha\beta$ of
characters of relatively prime orders.  Thus the kernel of
$\alpha^{-1}|_{I_p}$ contains that of~$\beta|_{I_p}$, and it follows
that the conductor of~$\alpha^{-1}\beta$ is at most that of~$\beta$.
(Incidentally, we will write the conductor multiplicatively: the
conductor of a character of~$D_p$ is a power of~$p$, and the
conductor of the trivial character of~$D_p$ is $1=p^0$.)

Since $\alpha|_{I_p}$ has
prime-to-$p$ order,  
$\alpha$ coincides on~$I_p$
with some power $\omega^i$ of the Teichm\"uller character~$\omega$
which appeared in the proof of~(2.1).  Note that $\omega$ is a
character which is defined naturally on~$\GalQ$.
Let $\phi=\omega^{-i}$, and
consider the twist $f\otimes\phi$, which corresponds to the
$\lambda$-adic representation $\rho_\lambda\otimes\phi$.
Since $\phi$ has $\ell$-power order, the reductions of
$\rho_\lambda\otimes\phi$ and $\rho_\lambda$ coincide.
On the other hand, $(\rho_\lambda\otimes\phi)|_{D_p}$
is a direct sum of the {\it unramified\/}
$\alpha \phi$ and the character $\beta \phi$.  This latter character
coincides with $\beta\alpha^{-1}$ on~$I_p$.  Therefore, its
conductor is a divisor of the conductor of~$\beta$.  At the same time,
the conductor of $\alpha \phi$, which is~1, is a strict divisor
of the conductor of~$\alpha$.
Therefore, the power of~$p$ dividing
the conductor of~$\rho_\lambda\phi$ is smaller than
the power of~$p$ dividing the conductor of~$\rho_\lambda$. 

In the language of cusp forms, this means that the level of the newform
associated to~$f\otimes\phi$ is divisible by a smaller power of~$p$
than the level $N$ of~$f$.  The two levels differ only at~$p$, since
the character $\phi$ is ramified only at~$p$. \endproof



\subhead The special case\endsubhead

Here, we have $\pi= \alpha\otimes{\roman{sp}}$,
where $\roman{sp}$ is a ``standard" special
representation, and $\alpha$ is a Grossencharacter of $\Q_p^*$, which we
identify as above with a character of~$D_p$.
We consider that $\roman{sp}$ has been normalized so that
$\rho_\lambda|_{D_p}$ is an extension of the one-dimensional representation
with character~$\alpha$
 by the one-dimensional representation 
with character~$\alpha\tilde\chi$.  (The fact that $\rho_\lambda|_{D_p}$
has this form, when $\pi$ is special, was proved by Langlands 
\cite\LANGLANDS.)
The semisimplification of~$\rho|_{D_p}$ then has the form 
$\bar\alpha\oplus \bar\alpha\chi$, 
where $\bar\alpha$ is the reduction of~$\alpha$ mod~$\lambda$.
Also, $\epsilon=\alpha^2\tilde\chi^{2-k}$.
Finally,
the conductor of~$\alpha\otimes{\roman{sp}}$ is the square of the conductor
of~$\alpha$ if $\alpha$ is ramified, and $p^1$ if not.

\proclaim{(4.5) Proposition \cite\CAR}
Assume that~$e>0$ and that $\alpha$ is ramified.
Then there is a Dirichlet character $\phi$ of
conductor~$p$ and $\ell$-power order such that the newform associated
 to $f\otimes\phi$ has level dividing $N/p$.  In particular, $\rho$
is modular of level~$N/p$.
\endproclaim

\noindent{\sl Proof}.\enspace
Since $e>0$, we have $\dim_\F V^{I_p}>0$, so that $\bar\alpha$ is
unramified.  Thus $\alpha|_{I_p}$ has $\ell$-power order, and
coincides with some power $\omega^i$ of~$\omega$.  
It follows that the conductor of~$\alpha$ is~$p$.
As above, we
set $\phi=\omega^{-i}$.  The representation $\pi\otimes\phi$ then
has conductor~$p$, whereas $\pi$ itself has conductor $p^2$.
Hence replacing
$f$ by the newform associated to $f\otimes\phi$ 
replaces $N$ by~$N/p$.\endproof

In the situation of Proposition~4.5, $\bar\epsilon$ is
unramified at~$p$, since the determinant of~$\rho$ 
is unramified at~$p$.  Since we have assumed that the order
of~$\epsilon$ is prime to~$\ell$, this implies that $\epsilon$
is unramified at~$p$.  Hence $f$ is in fact a newform on the
group $\Gamma_1(M)\cap\Gammao(p^2)$, where $M=N/p^2$ is prime
 to~$p$, and the character~$\phi$ is quadratic.
In the case where $\pi$ is special
but Proposition~4.5 does {\it not\/} apply,
  $\alpha$ is an unramified character,  so
that $\epsilon$, in particular,
is unramified.  The newform $f$ then has level $Mp$, with $M$  
prime to~$p$, and $f$
is a cusp form on the group $\Gamma_1(M)\cap\Gammao(p)$.
We have $e>0$ if and only if the representation $\rho$ is
unramified.  When this is the case, $N(\rho)$ is prime to~$p$,
and one must show that $\rho$ is represented by an eigenform
of level~$M$.  Here is a statement of the problem to be solved
(cf.~\cite{\CAR, Conjecture~B}):

\proclaim{(4.6) Problem}  Suppose that $\ell\ge5$ and that $p$ is a prime
distinct from~$\ell$. 
Assume that $\rho$ arises from an eigenform $f$ of weight~$k$
on~$\Gamma_1(M)\cap\Gammao(p)$,
where $M$ is prime to~$p$ and that $\rho$ is unramified
at~$p$.
Show that $\rho$ arises from an eigenform on~$\Gamma_1(M)$.
\endproclaim

\noindent
The representation $\pi$
is automatically special in all non-trivial cases of~(4.6).  Indeed,
let $f$ be an eigenform on~$\Gamma_1(M)\cap\Gammao(p)$.  If
the newform associated to~$f$ has level dividing~$M$, then in particular 
$\rho$ arises from an eigenform on~$\Gamma_1(M)$.  If, to the contrary,
the level of the newform associated to~$f$ is divisible by~$p$, then
$\pi$ is special.

We now describe the extent to which (4.6) has been treated.
A first remark, which is
a consequence of
the discussion concerning the four sets $\N_i(\rho)$, 
is that it suffices
 to solve~(4.6) in {\it either\/} of the following situations:
%
\item{{\bf a.}} $k=2$ and $M$ is of the form $N\ell^\alpha$ with $N$ 
prime to~$\ell$; %  and~$\alpha\le2$
%
\item{{\bf b.}} $M$ is prime to~$\ell$ and $2\le k\le \ell{+}1$.

\noindent
One can assume $\alpha\le2$ in case~(a),
but this does not seem to be helpful.


\proclaim{(4.7) Mazur's Principle}
Suppose that $\ell\ge5$ and that $p\ne\ell$ is a prime
such that $p\not\equiv1$ {\rm mod}~$\ell$.
Assume that $\rho$ arises from an eigenform $f$ of weight~$k$
on~$\Gamma_1(M)\cap\Gammao(p)$,
with $M$ prime to~$p$, and that $\rho$ is unramified at~$p$.
Then $\rho$ arises from an eigenform on~$\Gamma_1(M)$.
\endproclaim

\noindent
This theorem was proved by Mazur in a letter to J-F.~Mestre 
\cite\MAZURLETTER\ in the special case where $k=2$ and $f$ has
trivial character, i.e., is a cusp form on~$\Gammao(Mp)$.
Mazur's proof was presented by the author in \cite{\FERMAT,~\S6}.
In proving the theorem more generally,
we may apply the analysis above to work either in case~(a)
or in case~(b).  We choose to work in situation~(a), i.e.,
in the case of forms of weight $k=2$.  This is essentially
the case we have already treated in~\cite{\FERMAT,~\S6};
the main difference is that we now allow the Nebentypus
character of~$f$ to be non-trivial.
(This character
 is naturally defined mod~$M$, because $f$
is a form on~$\Gamma_1(M)\cap\Gammao(p)$.)
For the convenience of the reader, we establish
Mazur's principle, in case~(a),
in~\S8 below.  (In \S8, 
we require only that $\ell$ be odd:
we do not exclude the case $\ell=3$.  Also, we use
slightly different notation: the prime~$q$ of~\S8
plays the role of~$p$ in~4.7.)

After Mazur formulated his principle in 1985,
problem~4.6 was studied extensively when $k=2$
and $f$ is a cusp form on~$\Gammao(Mp)$.
Here, one wishes to show
that $\rho$ arises from a weight-two eigenform on~$\Gammao(M)$.
In \cite\FERMAT, the author proved this result when $M$ is prime
 to~$\ell$.
Next, in a joint article \cite\MAZR, Mazur and the
author established a ``multiplicity one" theorem which implies, by
the techniques of \cite\FERMAT, that Theorem~1.5 holds when $\ell$, but
not $\ell^2$, divides~$M$.  Then, in a recent note \cite\NOMULT,
the author
proved Theorem~1.5
when $M$ is ``exactly divisible" by a prime~$q$, different from $p$
and~$\ell$, at which $\rho$ is ramified. 
The argument given in \cite\NOMULT\
relies heavily on a recent theorem of N.~Boston, H.W.~Lenstra, Jr.,
and the author \cite\BLR, but avoids appeal to the ``multiplicity one"
principle which appears in~\cite\FERMAT.
Theorem~1.5, which will be proved in~\S7, may be viewed as the
most recent link in this chain.


Theorem~1.5 falls short of solving~(4.6) because of the requirement
that the character of~$f$ be trivial.  However, 
as already mentioned in~\S1, the author
expects that the proof of~(1.5) will extend without difficulty to
the case where the character associated to~$f$ is an arbitrary
character on $(\Z/M\Z)^*$.

Problem~4.6 has been studied from perspective~(b) by Jordan
and~Livn\'e \cite\JL, who expect to treat this case (at least) in the
case where $f$ has trivial character and the weight $k$ satisfies
the inequality $2\le k<\ell$.  Their arguments are an
adaptation to weight~$k$ of arguments given in~\cite\FERMAT.
In particular, the arguments of Jordan and~Livn\'e
rely
on a weight-$k$
``multiplicity one" principle, which will be treated
in a forthcoming article of Faltings and Jordan \cite\FJ. 
It is quite possible that the arguments of~\S7 can be adapted to
their situation, thereby obviating the necessity of using a 
weight-$k$ multiplicity-one principle.


\subhead The cuspidal case\endsubhead


In this case, the restriction of~$\rho_\lambda$ to~$D_p$ is irreducible.
Assume that $e>0$.  Then,
according to Carayol \cite{\CAR, \S\S1.1--1.2}, $\rho_\lambda|_{D_p}$
is the two-dimensional representation induced from 
an abelian character $\xi\: H\to\Qbar^*$, where $H$ is the subgroup
of index two in~$D_p$ which corresponds to the unramified quadratic
extension of~$\Q_p$.  Moreover, the reduction of~$\xi$ mod~$v$ is
unramified.  The level~$N$ is divisible exactly by~$p^2$; i.e., it is
divisible by~$p^2$, but not by~$p^3$.  The Serre invariant $N(\rho)$
is divisible at most by~$p$. One wishes to show that $\rho$
arises from an eigenform on~$\Gamma_1(N/p)$ of some weight, i.e., that
$\rho$ arises from~$\Gamma_1(N/p)$.

For this, we can (and will) assume that the weight $k$ of~$f$ 
satisfies $2\le k \le \ell{+}1$.

By Theorem~5.1 below, there are infinitely many prime numbers $q$,
prime to~$N$ and congruent to~$-1$ mod~$\ell$, such that:
$\rho$ arises from a weight-$k$ eigenform
on the group~$\Gamma_1(N)\cap\Gammao(q)$ for which the associated
newform has level divisible by~$q$.  (One takes $q$ to be
an ``auxiliary prime" as defined in \S5.)
Fix such a prime, together with an eigenform $F$
satisfying the given condition.  Let $\pi_{\bold A}$ be
the representation of~$\GLTWO{\bold A}$ associated with~$F$.
In the language of automorphic
representations, the condition on~$F$
ensures that the component at~$q$
of~$\pi_{\bold A}$ is a special representation.  

Consider the twisted form of~${\bold{GL}}(2)$ which arises from the
quaternion algebra over~$\Q$ which is ramified exactly at~$p$ and
at~$q$.
Since the components
at both $p$ and~$q$ of~$\pi_{\bold A}$ are discrete series representations,
we can conclude that $\pi_{\bold A}$
arises by the Jacquet-Langlands correspondence from an automorphic
representation on this twisted form.  Using this correspondence, and a
lemma which generalizes~(1.3), Carayol proves that 
$\rho$ arises from an eigenform on~$\Gamma_1(N/p)\cap\Gammao(q)$. 
Note that, since $\ell$ is odd, the congruence $q\equiv-1$ mod~$\ell$
implies that one has $q\not\equiv +1$ mod~$\ell$.  Using this fact,
together with
Mazur's Principle (4.7), we conclude, as desired, that
$\rho$ is modular of level~$N/p$.



\head 5.  Diamond's theorem.\endhead\noindent
This \S\ concerns the result of F.~Diamond \cite\DIA\ which was alluded to
above.  We establish an analogue (Theorem~5.1)
of a result proved by the author \cite\ICM,
\cite\DPP\ in the case of weight-two forms on~$\Gammao(N)$.  
This analogue is a simple variant of
Theorem~1 of~\cite\DIA.  More precisely,
we shall focus attention on some
intermediate results obtained by Diamond during the
course of the proof of \cite{\DIA, Theorem~1} and then derive
Theorem~5.1 from these.

Results like Theorem~5.1 have been useful both in ``level lowering"
(as in\S4 above) and in other contexts.  For example, \cite\BOS\
applies the theorem of~\cite\DPP\ to the study of deformations
of mod~$\ell$ representations arising from weight-two modular forms.

Let $\rho\:\GalQ\to\GLTWO\F$ be an irreducible mod~$\ell$
representation arising from~$S_k(\Gamma_1(N))$, with $N$ prime
 to~$\ell$.    Consider the following two conditions on a prime
number $q$, assumed to be prime to~$N\ell$:
\item{I.} The representation $\rho$ arises from a weight-$k$ eigenform
on~$\Gamma_1(N)\cap\Gammao(q)$ for which the associated
newform has level divisible by~$q$.
\item{II.} The characteristic polynomial of  $\rho(\Frob_q)$
is of the form $(T-a)(T-qa)$, with $a\in\F^*$.

\proclaim{(5.1) Theorem {\rm (Diamond)}}
Assume that $k$ satisfies $2\le k\le\ell{+}1$.  Then
conditions\/ {\rm I} and\/
{\rm II} are equivalent.
\endproclaim

\noindent We first make some comments 
concerning the
two conditions. 
Let $\Cal S$ be the space $S_k(\Gamma_1(N)\cap\Gammao(q))$,
and let ${\Cal S}^{q\roman{-new}}$
be the ``$q$-new" subspace
of~$\Cal S$, defined,
for example, as the kernel of the natural
trace map from~$\Cal S$ 
 to the direct sum of two copies
of~$S_k(\Gamma_1(N))$.  Condition~I means simply that $\rho$ arises from
an eigenform in the space~${\Cal S}^{q\roman{-new}}$.

Next, we should point out that Condition~II is satisfied in the
case where $q$ is an {\it auxiliary prime}.
Let $\sigma$ be the three-dimensional
representation $\rho\times\chi$, where
 the
mod~$\ell$ cyclotomic character $\chi$ is  regarded as a one-dimensional
representation of~$\GalQ$ over~$\F$.
An auxiliary prime is a
prime number~$q$, prime to~$\ell N$, such that
$\sigma(\Frob_q)$ is conjugate to the matrices~$\sigma(c)$, with
$c\in\GalQ$ a complex
conjugation.  If $q$ is such a prime,
the characteristic polynomial of~$\rho(\Frob_q)$ coincides
with that of the~$\rho(c)$.  Since $\rho$ is an odd representation,
$\rho(c)$ is conjugate to $\pmatrix -1&0\cr\hfill0&1\cr\endpmatrix$,
and thus has
characteristic polynomial $(T-1)(T+1)$.
Because $\chi(\Frob_q)=\chi(c)=-1$, we have
$q\equiv -1$ (mod~$\ell$).
Thus the characteristic polynomial of~$\rho(\Frob_q)$ may be written
$(T-a)(T-qa)$, where $a$ can be taken to be either
$+1$ or~$-1$, so that condition~II of~\S5 is satisfied.


By the \v Cebotarev
Density Theorem, the set of prime numbers for which
$c$ and $\Frob_q$ map to conjugate elements in 
the image of~$\rho\times\chi$ has positive density.  In particular,
there are an infinite number of auxiliary primes.


Also, we should
point out that the implication
 ``Condition~I  $\Rightarrow$ Condition~II" is a direct
consequence of Langlands's theorem (\cite\LANGLANDS, see also \cite\CAREARLY).
Indeed, suppose that I  is satisfied, and let
$f$ be a newform in~${\Cal S}^{q\roman{-new}}$ which gives rise to~$\rho$.
Let $\pi$ be the admissible representation of~$\GLTWO\Qq$ which
is associated to~$f$.  The fact that $f$ is associated with
$\Gamma_1(D)\cap\Gammao(q)$, for some $D|N$, implies that $\pi$ is a
special representation $\alpha\otimes{\roman{sp}}$, with $\alpha$ unramified.
 As discussed above, a theorem of Langlands states
that $\rho_\lambda|_{D_q}$
is an extension of the one-dimensional representation
with character~$\alpha$
 by the one-dimensional representation 
with character~$\alpha\tilde\chi$.  The semisimplification 
of~$\rho|_{D_q}$   has the form 
$\bar\alpha\oplus \bar\alpha\chi$, 
where $\bar\alpha$ is the reduction of~$\alpha$ mod~$\lambda$.
Applying $\bar\alpha\oplus \bar\alpha\chi$ to~$\Frob_q\in D_q$,
we obtain the matrix $\pmatrix a&0\cr 0&qa\endpmatrix$,
with $a=\alpha(\Frob_q)$.
The characteristic polynomial of this matrix is $(T-a)(T-qa)$.
Hence condition~I implies condition~II.

We now turn to the implication ``Condition~II  $\Rightarrow$ Condition~I,"
which is the essential content of Theorem~5.1.
We  first
compare eigenforms in $S_k(\Gamma_1(N))$ and in~$\Cal S$.
Suppose that $\rho$ arises from a weight-$k$ eigenform 
$f=\sum c_ne^{2\pi i n\tau}$
on~$\Gamma_1(N)$, and let
$\epsilon$ be the character of $(\Z/N\Z)^*$ associated with~$f$.
(For obvious reasons, we must refrain from writing $q=e^{2\pi i\tau}$.)
The forms $f$ and $f'\seteq \sum c_n(e^{2\pi i n\tau})^q$
may be considered as elements of~$\Cal S$; indeed, this construction 
defines the standard inclusion of $S_k(\Gamma_1(N))\oplus S_k(\Gamma_1(N))$
into~$\Cal S$ as a space of oldforms.
Both forms $f$ and~$f'$ in~$\Cal S$
are eigenvectors for the
diamond operators $\diamond d$ with $d\in(\Z/N\Z)^*$
and the Hecke operators $T_n$ with
$(n,q)=1$; the eigenvalues are the same as those for $f$ when 
considered as an element of~$S_k(\Gamma_1(N))$.  
(Some authors would write $\diamond d_N$ for the operator~$\diamond d$,
 to emphasize that only the group $(\Z/N\Z)^*$, rather than~$(\Z/Nq\Z)^*$,
is operating.  The group $(\Z/Nq\Z)^*$ operates through its
quotient $(\Z/N\Z)^*$ because we are considering cusp forms
on~$\Gammao(q)\cap\Gamma_1(N)$.)  However, neither of the forms
$f$, $f'$ is (in general) an eigenvector for the $q^{\roman{th}}$
Hecke operator $U\seteq T_q$ on~$\Cal S$.

In order to obtain a true eigenform in~$\Cal S$, we consider the two
roots
$\alpha$ and~$\beta$
of the polynomial $T^2 - c_qT + q^{k-1}\epsilon(q)$.
A computation shows that
the two forms $f-\alpha f'$ and $f-\beta f'$ are eigenvectors for~$U$
on~$\Cal S$, with eigenvalues $\beta$ and
$\alpha$, respectively.  

Consider the Hecke algebra $\T=\T_{\Cal S}$
associated with~$\Cal S$, i.e., the subring of~$\End\Cal S$ generated
by the $T_n$ with $n\ge1$.  We have $\diamond d\in\T$ for all
$d\in(\Z/N\Z)^*$.  Indeed, if $r$ is a prime number 
which is prime to~$qN$, then
$T_r^2 -  T^{\vphantom 2}_{r^2}
=\diamond r r^{k-1}$, so that $r^{k-1}\diamond r\in\T$.
By taking two distinct $r$'s which map to~$d$ mod~$N$, 
we find~$\diamond r\in\T$.  

 
The action of~$\T$ on $g\seteq f-\beta f'$
is given by a homomorphism $\varphi\:\T\to\C$, which is characterized
by the formulas $\varphi(\diamond d)=\epsilon(d)$, $\varphi(U)=\alpha$,
and $\varphi(T_n)=c_n$, the latter valid for $n$ prime to~$q$.  We may view
$\varphi$ as a homomorphism $\T\to{\Cal O}$, where $\Cal O$ is the integer
ring of a suitable number field.  Our fixed place $v|\ell$ of~$\Qbar$
induces a prime ~$\lambda|\ell$ of~$\Cal O$, together with an inclusion
${\Cal O}/\lambda\hookrightarrow\F$.  On composing $\varphi$ with the
map ${\Cal O}\to\F$, we obtain a ring homomorphism
$$\bar\varphi\:\T\to\F.$$
This map depends on~$f$, and also on $(\alpha,\beta)$ as an ordered pair.


Assume now that condition II is satisfied and
fix $a\in\F^*$ as in the condition.
Then $T^2 - c_qT + q^{k-1}\epsilon(q) \equiv (T-a)(T-qa)$
mod~$\lambda$.  After permuting the roots $\alpha$ and~$\beta$ if necessary,
we have $\alpha\equiv a$ mod~$\lambda$.  Setting
$$ \eta = U^2 - q^{k-2}\diamond q,$$
we find from condition~II that $\bar\varphi(\eta)=0$.  Indeed, one has
$\bar\varphi(q\eta) = qa^2 - \bar\epsilon(q)q^{k-1}$, but 
$\epsilon(q)q^{k-1}$ is congruent to~$qa^2$ in view of condition~II.
In other words, the kernel $\m$ of~$\bar\varphi$ is a maximal ideal
of~$\T$ which contains~$\eta$.

The subspace ${\Cal S}^{q\roman{-new}}$ of~$\Cal S$
is stable under all Hecke operators $T_n$ and~$\diamond d$.  Thus
${\Cal S}^{q\roman{-new}}$ cuts out a quotient $\T_{q\roman{-new}}$ of~$\T$:
the image of~$\T$ in~$\End{\Cal S}^{q\roman{-new}}$.  One verifies
condition~I by proving that~$\m$ arises by pullback from a maximal
ideal $\overline\m$
of~$\T_{q\roman{-new}}$, i.e., that $\m$ lies in the support of
the $\T$-module ${\Cal S}^{q\roman{-new}}$.  
It is well known that this property of~$\m$ implies that $\rho$
arises from an eigenform in~$\Cal S$; see \cite{\BHG, Proposition 9.3} and 
\cite{\DS,  Lemma 6.11}, and the proofs of these results.   
Therefore, to prove that
condition~II implies condition~I, it suffices to
verify 

\proclaim{(5.2) Proposition}  Suppose that $\ell$ is prime to~$N$
and that $k$ satisfies $2\le k \le\ell{+}1$.  Let $\m$ be a maximal
ideal of~$\T$ associated with 
${\Cal S} = S_k(\Gamma_1(N)\cap\Gammao(q))$, where $q$ is a prime
number not dividing~$N\ell$.  Assume that $\m$ divides~$\ell$ (i.e.,
$\m$ contains~$\ell$) and that $\m$ contains 
$\eta = U^2 - q^{k-2}\diamond q$.  Then $\m$ arises by pullback from
the $q$-new quotient of~$\T$.
\endproclaim


\noindent
This proposition is implicit in Diamond's work, although it is not
explicitly stated in~\cite\DIA.

  To prove the proposition, we recall some results from~\cite\DIA,
using somewhat different notation.
Let $L$  
be the 
parabolic cohomology group with $\Z_\ell$-coefficients which
is associated with~$\Cal S$, so that $L$ is a free $\Z_\ell$-module
of rank $2\dim_\C {\Cal S}$.  To construct~$L$, we let $\Gamma$
be the image of $\Gamma_1(N)\cap\Gammao(q)$ in $\PSLTWO\Z$,
and consider the parabolic cohomology group
$H^1_{\roman{par}}(\Gamma,\Sym^{k-2}(\Z_\ell\oplus\Z_\ell))$.  
This group is a finitely-generated $\Z_\ell$-module $M$
which is not
necessarily torsion free; $L$ is defined as the image of~$M$ in
$M\otimes\Q_\ell$, i.e., the largest torsion free quotient of~$M$.
We let $X=X_1\oplus X_2$ be the
direct sum of two copies of 
the corresponding parabolic cohomology group made with $\Gamma_1(N)$
in place of~$\Gamma_1(N)\cap\Gammao(q)$.  
Diamond considers the degeneracy
map $X\hookrightarrow L$ 
which corresponds to the inclusion of 
$S_k(\Gamma_1(N))\oplus S_k(\Gamma_1(N))$ in~$\Cal S$ on the level of
modular forms.  We will denote this map by $\alpha$.  (There is no
further
need to refer to the roots $\alpha$ and~$\beta$ which occurred in
the discussion above.)
The map $\alpha$ becomes $\T$-equivariant when one endows
$X$ with its natural action of~$\T$.  
(In this action, the $T_n$ with $n$ prime to~$q$ act ``diagonally"
on~$X$, while $U$ acts as the two-by-two matrix
$\pmatrix T_q & q^{k-1}\cr -\diamond q&0 \endpmatrix$.)
In particular, the image of~$\alpha$
is stable under~$\T$.

Using results of Ihara, and the hypotheses on~$\ell$, Diamond proves
that the cokernel~$Y$ of~$\alpha$ is {\it torsion free\/}.
The tautological exact sequence
$$ 0\to X\to L\to Y\to 0\leqno(5.3)$$
is thus an exact sequence
of free $\Z_\ell$-modules.  
The quotient of~$\T\otimes\Zell$
cut out by~$Y$ is the $\ell$-adic completion of the
$q$-new quotient of~$\T$, while the quotient
of~$\T\otimes\Zell$ 
cut out by~$X$ is the completion of
the analogous $q$-old quotient of~$\T$;
this is the same quotient which is defined by the action of~$\T$
on $S_k(\Gamma_1(N))\oplus S_k(\Gamma_1(N))$.  

In particular, let $\m$ be a maximal ideal of~$\T$ with residue
characteristic~$\ell$.  Because $\T$ acts faithfully
on~$L$, $\m$ lies in the support of~$L$.
In view of~(5.3), it follows that $\m$ lies either in~$\Supp X$
or in~$\Supp Y$, and perhaps in both.  
The statement to be proved may be rephrased as follows: if
$\m\in\Supp X$ and if $\eta\in\m$, then $\m\in\Supp Y$.
Note that the two
hypotheses on~$\m$ may be combined into the statement
that $\m$ belongs to the
support of~$X/\eta X$.  Indeed, if $\m\in\Supp X$, then $X/\m X$
is non-zero, by Nakayama's lemma.  If furthermore we have
$\eta\in\m$, then $X/\m X$ is a quotient of~$X/\eta X$.  Hence
$\m$ belongs to the support of~$X/\eta X$, since $\m$ belongs to
the support of~$X/\m X$.  Conversely, if $\m$ lies in the support
of~$X/\eta X$, it is clear that $\eta$ is an element of~$\m$ and
that $\m\in\Supp X$.  Thus one must prove
\proclaim{(5.4)}  If $\m\in\Supp(X/\eta X)$, then $\m\in\Supp Y$.
\endproclaim

\noindent
Diamond proves (5.4) by constructing a $\T$-equivariant
surjection $Y\to X/\eta X$.
  To construct this map, we first dualize (5.3), thereby
obtaining
an exact sequence
$$ 0\to\Hom(Y,\Zell)\to\Hom(L,\Zell)\to\Hom(X,\Zell)\to0.$$
As Diamond recalls, perfect pairings constructed by~Hida \cite{\HIDAZERO,
Theorem~3.2} identify $\Hom(L,\Zell)$ with~$L$ and $\Hom(X,\Zell)$
with~$X$.  Using these pairings, we obtain an exact sequence
$$ 0\to\Hom(Y,\Zell)\to L \buildrel \beta\over\to X \to0.\leqno(5.5)$$
The map $\beta$ is ``almost" $\T$-equivariant: it intertwines
the maps labeled $T_n$ on $L$ and on~$X$ whenever $(n,q)=1$,
but intertwines the $q^{\roman{th}}$ Hecke operator $U$ of~$L$
with the endomorphism
$\pmatrix 0&q\cr -\diamond q q^{k-2} & T_q\cr\endpmatrix$
of~$X$ (cf.~\cite{\HIDAZERO, Prop.~3.3}).

Consider the automorphism 
    $\omega=\pmatrix -q^{k-2}\diamond q&T_q\cr0& -\diamond q\endpmatrix$
of~$X$.  A computation shows that the composite $\omega\circ\beta\:
L\to X$ is $\T$-equivariant.  Moreover, $(\omega\circ\beta)\circ\alpha$
is the endomorphism~$\eta$ of~$X$.
It follows that $\omega\circ\beta$ induces a surjection
$L/\alpha X  \to X/\eta X$.  Since $Y$ is, by definition, $L/\alpha X$,
we obtain the desired map.

\head 6. Character groups and component groups.\endhead\noindent
In this \S, we recall some material from \cite\FERMAT\ concerning the
component groups and character groups associated with
bad reductions of ordinary modular curves and Shimura curves.  
In addition, we explore in further detail a relation that was
first found by Jordan and Livn\'e \cite\JLEARLY\ and then deepened by
\cite{\FERMAT, Theorem~4.3}.

For each $N\ge1$, we let $\Xo N $ be the classical modular curve
(over~$\Q$) which classifies elliptic curves which are endowed with
cyclic subgroups of order~$N$.  The space of holomorphic differentials
on ${\Xo N }_{/\C}$ may be identified with~$S_2(\Gammao(N))$
in a canonical fashion.
The curve $\Xo N $ comes equipped with a family of correspondences $T_n$
($n\ge1$); the correspondence $T_n$ induces on~$S_2(\Gammao(N))$
the endomorphism~$T_n$.
The Jacobian $\Pico \Xo N $ will be denoted~$\Jo N $, as usual; we
write again simply $T_n$ for the endomorphism of~$\Jo N $ which is
induced by the correspondence~$T_n$ of~$\Xo N $.

Consider first a prime number $q$ and a positive integer $N$ prime
 to~$q$.  We are interested in the modular curves $\Xo{qN}$ and~$\Xo N$,
and in their Jacobians $\Jo{qN}$ and~$\Jo N$.
The two standard degeneracy coverings $\Xo{qN}\doublemap \Xo{N}$ 
induce, by pullback, a map 
$\delta\: \Jo N \times \Jo N \to \Jo{qN}$.
The kernel of~$\delta$ is finite; it is the ``antidiagonal" image in
$ \Jo N \times \Jo N $ of the kernel of the pullback of the
natural covering $X_1(N)\to \Xo{N}$ \cite\ICM.  (The curve
$X_1(N)$ classifies elliptic curves which are furnished with a point
of order~$N$.)  

For each $n\ge1$, a Hecke operator labeled $T_n$ acts diagonally on
the product $\Jo N \times \Jo N $, and a similarly-named operator
acts on the target $\Jo{qN}$ of~$\delta$.  When $n$ is prime to~$q$,
$\delta$ is compatible with the two operators $T_n$.  However,
when $n=q$, the situation is more subtle.
Reserve the symbol $T_q$ for the $q^{\roman{th}}$ Hecke operator on~$\Jo {N}$
and let $U$ denote the $q^{\roman{th}}$ Hecke operator on~$\Jo{qN}$.
Then $U\circ \delta = \delta\circ U$,
where the latter operator $U$ denotes the matrix 
$\pmatrix T_q & q\cr -1& 0 \endpmatrix$ of endomorphisms
of~$\Jo N $.  (We view this matrix as an endomorphism
of~$\Jo N {\times} \Jo N $.)
Let $\T=\T_{qN}$ be the subring of~$\End \Jo{qN}$ generated by
the $T_n$ for $n\ge1$, i.e., by the $T_n$ for $n$ prime to~$q$
and the operator $U=T_q$.  
We endow $\Jo N\times \Jo N$ with the operation of~$\T$ such that
the elements $U$ and $T_n$ of~$\T$ (with $n$ prime to~$q$) 
act as the operators of $\Jo N\times \Jo N$ with the same names.
This is the unique operation for which~$\delta$ is $\T$-equivariant.

We note in passing that the endomorphisms $T_n$ and~$U$ and
the homomorphism~$\delta$ are
defined over~$\Q$.
Also,
in analogy with the situation of~\S5,
we let $\eta=U^2-1$ in~$\T$.


\proclaim{(6.1) Theorem} There is a unique homomorphism of
abelian varieties $$\sigma\:\Jo{qN}\to\Jo N\times\Jo N$$
such that $\sigma\circ\delta =\eta$.  This homomorphism
is $\T$-equivariant.
\endproclaim


\noindent
  To construct $\sigma$,
we consider the map $\delta'\:\Jo{qN}\to\Jo N\times\Jo N$
which is induced by the
degeneracy coverings $\Xo{qN}\doublemap \Xo{N}$ using
Albanese (i.e., covariant) functoriality of the Jacobian.
The definition of~$T_q$ as a correspondence on~$\Xo N$
shows that $\delta'\circ\delta=
\pmatrix q+1 & T_q\cr T_q & q+1\cr \endpmatrix$.  Define 
$ \sigma = \pmatrix -1 & T_q\cr \hfill0 & -1 \endpmatrix\circ\delta'$.
A computation shows that $\sigma\circ\delta$ coincides
with $\eta$ as a matrix of endomorphisms of~$\Jo N$.

For the unicity and the $\T$-equivariance, we observe that
$$\Hom(Q,\Jo N{\times}\Jo N)=0,$$
where $Q=\Jo{qN}/\delta(\Jo N{\times}\Jo N)$.  This fact may
be seen arithmetically, by noting that $Q$ has purely toric
reduction at the prime~$q$ \cite\DERA, whereas $\Jo N$ has good
reduction at~$q$.  It proves the unicity, since the difference
between two operators $\sigma$ which satisfy $\sigma\circ\delta=\eta$
is an operator which vanishes on the image of~$\delta$.
Similarly, let $T\in\T$, and consider $\sigma\circ T- T\circ\sigma$,
which is a~priori a homomorphism $h\:\Jo{qN}\to\Jo N{\times}\Jo N$.
We have $h\circ\delta= \eta T - T\eta =0$, so that $h$ factors
through~$Q$; as a consequence, $h=0$.\endproof

Now specialize to the case where $N=pM$, where 
$p$ is a prime number which does not divide~$M$.  The level
$qN$ becomes the product~$pqM$ in which the two prime numbers
$p$ and $q$ play symmetrical roles.
The space $\Cal S$ of weight-two cusp forms on~$\Gammao(pqM)$ is then
analogous to the space denoted $\Cal S$ in~\S5, in
the special case $N=pM$, $k=2$.  The main difference is that
we have replaced $\Gamma_1(N)\cap\Gammao(q)$ by~$\Gammao(qN)$.






Let $\Jo{pqM}_{/\Fp}$ be the fiber at~$p$ of the
N\'eron model of~$\Jo{pqM}$.  This group variety is an extension of
a finite ``component group" $\Theta_p$ by a connected group
$\Jo{pqM}^o_{/\Fp}$, which is in turn an 
extension of the product of two copies of~$\Jo{qM}$
by a torus $\Cal T$ over~$\Fp$.
Let $L_p$ be the character group of this torus, and let
$L_q$ be the character group of the analogous torus for~$\Jo{pqM}_{/\Fq}$.
Similarly, let $X_p$ be the analogue of~$L_p$ for
$(\Jo{pM}{\times} \Jo{pM})_{/\Fp}$ and let $X_q$ be the analogue of~$X_p$
with $p$ replaced by~$q$.
The groups $X_p$ and $X_q$ are naturally direct sums of two copies of
the character groups coming from $\Jo{pM}_{/\Fp}$ and $\Jo{qM}_{/\Fq}$,
respectively; thus, we may represent endomorphisms of $X_p$, say,
as two-by-two matrices of endomorphisms of the character group
associated to~$\Jo{pM}_{/\Fp}$.
The endomorphisms $T_n$
(and~$U$) of   $\Jo{pqM}$, $\Jo{pM}{\times} \Jo{pM}$ and
$\Jo{qM}{\times} \Jo{qM}$ induce maps on   the four character groups
$L_p$, $X_p$, $L_q$, and $L_p$.
In what the author hopes is an acceptable abuse of notation,
these maps will be denoted simply $T_n$ and~$U$.
In particular, the endomorphism $U$ of~$X_p$ is the matrix of endomorphisms
$\pmatrix T_q& -1\cr q&\hfill0 \endpmatrix$.


Continuing the abuse of notation, we will write simply $\delta$
for the map $L_p\to X_p$ induced by the degeneracy map
$\delta\: \Jo{Mp}{\times}\Jo{Mp} \to \Jo{Mpq}$.
The surjectivity of $\delta\: L_p\to X_p$ was
established in \cite{\FERMAT, Th.~3.15}.  
(In \cite\FERMAT, the author
wrote ``$X$" for the character group associated to a {\it single\/}
copy of~$\Jo{pM}$ or~$\Jo{qM}$.)
Theorem~4.1 of~\cite\FERMAT\ identifies the kernel
of~$\delta$ with an analogue of~$L_q$ in which $\Xo{pqM}$
is replaced by an appropriate Shimura curve.
Namely, let $C$ be the Shimura curve made with 
a quaternion algebra of discriminant~$pq$ and
$\Gammao(M)$-type level structure (cf.~\cite{\FERMAT, \S4}).  Let $J$ 
be the Jacobian of~$C$, and let $Y_p$ and $Y_q$ be the analogues
of $L_p$ and $L_q$ for~$J$.
The groups $J_{/\Fp}$ and $J_{/\Fq}$ are extensions of their
component groups $\Psi_p$ and~$\Psi_q$
by the tori whose character groups are $Y_p$ and~$Y_q$.
(As was proved by Cerednik \cite\CERED\ and Drinfeld \cite\DRIN,
all components of $C_{/\Fp}$ and~$C_{/\Fq}$ have genus zero.)
In other words, the reductions of~$J$ at $p$
and~$q$ are ``semiabelian," with trivial 
abelian variety parts.  
Then theorem~4.1 of~\cite\FERMAT\ provides an exact sequence
$$ 0 \to Y_q \buildrel \iota\over\to L_p \buildrel\delta\over\to X_p\to0
\leqno(6.2)$$ and an analogue
$$ 0 \to Y_p \to L_q \to X_q \to 0 \leqno(6.3)$$
in which the roles of $p$ and~$q$ have been permuted.
These sequences are compatible with the Hecke operators labeled $T_n$
on~$J$ and on~$\Jo{pqM}$; note, incidentally, that the operators
$T_p$ and $T_q$ on~$J$ are in fact {\it involutions}.
(See \cite{\FERMAT, Theorem~4.1}
and also the results proved in
\cite\BIMODULES.)
The sequences (6.2) and (6.3) seem to play a role analogous to that
played by (5.3) and its $\Zell$-dual in the discussion of~\S5.

One deduces from (6.2) or~(6.3) that the Hecke ring
$\Z[\ldots, T_n, \ldots]\subseteq\End J$ associated to~$J$ is
naturally a quotient of the ring~$\T$ associated to~$\Jo{pqM}$.
Thus, we may speak of the action of~$\T$ on~$J$.

Let us introduce the notation~$'$ to denote a dual abelian variety;
thus $\Jo{pqM}^\prime$, for example, will be the abelian
variety dual to~$\Jo{pqM}$.  We use $T^\prime_n$ and~$U'$ to
denote the endomorphisms of the dual objects which are induced
functorially by the operators $T_n$ and~$U$.
Thus, for instance, there is a natural action of the ring~$\T$
on~$J'$, in which the element of~$\T$ labeled~$T_n$ acts on~$J'$
as the endomorphism~$T^\prime_n$.
We let $Y^\prime_q$, $L^\prime_p$, and so on, represent
the character groups of the tori arising from the dual abelian
varieties.  The operator $T_n$ (say) in~$\T$ then induces an
endomorphism of~$L_q^\prime$ which will be denoted $T_n^\prime$;
this endomorphism is induced by the endomorphism $T^\prime_n$ 
of~$\Jo{pqM}^\prime$.  Similar remarks apply to the component
groups which we have introduced: $\Theta_p$ and $\Theta_q$ in the
case of~$\Jo{pqM}$, and $\Psi_p$ and $\Psi_q$ in the case of~$J$.

Next, we discuss monodromy pairings. To fix ideas we will first
consider $L_p$.  The pairing associated with $L_p$ is
a bilinear map $\langle \; ,\; \rangle: L_p \times L^\prime_p \to \Z$,
which induces an injection $L^\prime_p \hookrightarrow \Hom(L_p,\Z)$
with finite cokernel.  According to \cite{\SGA, \S11}, the cokernel
of this injection is canonically isomorphic to $\Theta_p$.  The exact
sequence
$$ 0 \to L^\prime_p \to \Hom(L_p,\Z) \to \Theta_p \to 0$$
is $\T$-equivariant when the elements of~$\T$
act in the natural way, 
i.e., when $T_n\in\T$ acts as $T_n$ on~$\Theta_p$,
as $T^\prime_n$ on~$L^\prime_p$, and as $\Hom(T_n,\Z)$ on $\Hom(L_p,\Z)$.
In a similar way, we have monodromy pairings
$$ Y_q \times Y^\prime_q \to \Z, \qquad X_p \times X^\prime_p \to \Z.$$

One potential source of confusion is that
the abelian varieties under discussion
are Jacobians of modular curves, or else products of two Jacobians.
It follows that they are naturally self-dual.
After using the autoduality of the Jacobian to identify
$J'$ and~$J$, for example, we find that the lattices
$Y_q$ and~$Y_q^\prime$ are {\it equal}, or at least canonically
isomorphic.
In point of fact,
it is often fruitful to identify both $Y_q$ and~$Y_q^\prime$
with $H_1(\Gamma,\Z)$, where $\Gamma$ is the ``dual graph"
attached to the mod~$q$ reduction of~$C$.
The group $H_1(\Gamma,\Z)$
thereby acquires two actions of~$\End J$; these differ by the Rosati
involution of~$\End J$ which arises from the theta divisor on~$C$.
Similarly, the group $X_p$ has two natural actions of~$\T$, which
differ by the Rosati involution
$\pmatrix a & b\cr c &d \endpmatrix\mapsto
\pmatrix a'&c'\cr b' & d'\endpmatrix $
on $\End \Jo{pM} {\times} \Jo{pM}$,
where $'$ is the Rosati involution on $\End \Jo{pM}$.
For example, the operator $U$, which acts on~$X_p$ as
$\pmatrix T_q& -1\cr q&\hfill0 \endpmatrix$ in its standard action, becomes
$\pmatrix T_q^\prime & q \cr -1 & \hfill0\endpmatrix $ in the other action.
(It might be worth pointing out that the endomorphisms $T_q$ and $T_q^\prime$
of $\Jo N$ are equal.)
As a mnemonic device,
we will attempt whenever possible to use the $'$~notation when 
the Hecke
operators in~$\T$ act as $T_n^\prime$ and~$U'$.

These examples show that the natural autoduality
$\Jo N \approx \Jo{N}'$ does not necessarily intertwine the actions
of~$\T$ on these two abelian varieties.
Nonetheless, it is a fact that are natural
$\T$-equivariant isomorphisms $J\approx J'$  and $\Jo{N}\approx
\Jo{N}'$ for each~$N$.  
In the case of~$\Jo{N}$, the isomorphism
is provided by the involution $w=w_N$ on~$\Jo{N}$ which is
induced by the Atkin-Lehner
involution $(E,C)\mapsto (E/C,E[N]/C)$ on the set
of elliptic curves with cyclic subgroups of order~$N$ 
(see~\cite{\SHIM, Proposition~3.54}).  In the case of~$J$, there is
an analogous involution $w$
on the set of ``fake elliptic curves" which
are classified by~$C$.  
These $\T$-equivariant isomorphisms are
occasionally useful.


Once we identify $L_p$ with~$L_p^\prime$,
the monodromy pairings associated to $L_p$ and~$Y_q$ become maps
$L_p\times L_p\to\Z$ and 
$Y_q\times Y_q\to\Z$, respectively.
Both pairings are symmetric; they are given by explicit formulas
involving the dual graphs of the mod~$p$ reduction of~$\Xo{pqM}$
and the mod~$q$ reduction of~$C$.
According to \cite{\FERMAT, Th.~4.1}, the restriction
      to~$Y_q\times Y_q$ of the
monodromy pairing $L_p\times L_p\to\Z$ agrees with the
monodromy pairing on~$Y_q$.  Here, one uses $\iota$ to
embed $Y_q$ in~$L_p$.

\proclaim{(6.4) Proposition}
The embedding $\iota\:Y_q\hookrightarrow L_p$ remains $\T$-equivariant
when it is regarded as a map $Y_q^\prime\to L^\prime_p$.
\endproclaim

\noindent
We regard $L_p^\prime$ and~$Y^\prime_q$ as the same physical
groups as $L_p$ and~$Y_q$, only with different actions of~$\T$.
Fix $T\in\T$, and use the symbols $T_Y$ and~$T_L$
to
denote the 
endomorphisms of~$Y_q$ and of~$L_p$ induced by~$T$.
Write $T^\prime_Y$ and $T^\prime_L$ for the endomorphisms
of these
groups which $T$ induces in its ``dual" actions.
The endomorphism $T_L^\prime$ (say) is the adjoint of 
$T_L$ with respect to the monodromy
pairing $\langle \, ,\,\rangle_L$ on~$L_p$.
Proposition~(6.4) states the formula
$T^\prime_L\circ\iota=\iota\circ T^\prime_Y$.

Consider the homomorphism
$\kappa\:\Jo{Mp}'{\times}\Jo{Mp}' \to \Jo{Mpq}'$
which is dual to the map $\sigma$ of~(6.1).
It induces a $\T$-equivariant map $\kappa\:L_p^\prime\to X_p^\prime$;
this map is adjoint to the map $X_p\to L_p$ induced by~$\sigma$.
The construction of~$\sigma$ shows that $\kappa$
differs from~$\delta$ by an automorphism of~$X_p$.
Therefore, the kernel of~$\kappa$ coincides with the
kernel of~$\delta$, which is the image of~$\iota$.
It follows that $T_L^\prime$ preserves this image, since
we may write $\kappa\circ T^\prime_L\circ \iota = T^\prime_X \circ
\kappa \circ \iota =0$.  Hence the map $\iota\circ T^\prime_Y -
T^\prime_L\circ\iota$ maps $Y_q$ to~$\iota(Y_q)$.
Consequently, to show that $\iota\circ T^\prime_Y=
T^\prime_L\circ\iota$, it suffices to verify the equality of
$ \langle T^\prime_Y y, z \rangle_Y$ and
$\langle T^\prime_L\circ\iota y, \iota z \rangle_L$ for all $y,z\in Y_q$.
Here, we 
make use of the fact that the monodromy pairing
$\langle \, , \, \rangle_Y$ on $Y_q$
is the
restriction to~$Y_q$ of the pairing $\langle \, , \, \rangle_L$
on~$L_p$.

To verify the desired equation, we note the series of equalities
$$ \langle T^\prime_L\iota y, \iota z \rangle_L
     =
  \langle \iota y, T_L\iota z\rangle_L 
     =
  \langle \iota y, \iota T_Y z\rangle_L
     =
   \langle y, T_Y z\rangle_Y
     =
   \langle T^\prime_Y y, z\rangle_Y.$$
These follow from the adjuction relations and the equivariance
of~$\iota$ with respect to the standard actions of~$\T$, plus
the compatibility between the two monodromy pairings.\endproof



\proclaim{(6.5) Corollary}  We have an exact sequence of
$\T$-modules
$$ 0 \to Y_q^\prime \buildrel\iota\over\to L_p^\prime
 \buildrel\kappa\over\to X_p^\prime \to 0.\hbox{\endproof}$$
\endproclaim


\noindent
Theorem~4.3 of~\cite\FERMAT\ asserts that there is an exact sequence
of~$\T$-modules
$$ 0\to{\Cal K}\to X_p/\eta X_p \to \Psi_q \to {\Cal C} \to 0.$$
Here, $\Cal K$ and~$\Cal C$ are each
closely related to $\Theta_p$ and to the component group of the
group variety $(\Jo{pM}{\times}\Jo{pM})_{/\Fp}$;
in particular, they are Eisenstein $\T$-modules in the sense that
they are annihilated by $T_r - (r{+}1)$ for almost all prime
numbers~$r$, cf.~\cite{\FERMAT, Th.~3.12}, or \cite\BX.
(This allows us to neglect $\Cal C$ and~$\Cal K$ in
practice.)  The author now believes that he was not sufficiently
attentive to the distinction between the two actions of~$\T$
and that the exact sequence of \cite{\FERMAT, Th.~4.3} should
read
$$0\to{\Cal K}'\to X_p/\eta X_p \to 
     \Psi^\prime_q \to {\Cal C}' \to 0, \leqno(6.6)$$
where ${\Cal C}'$ and ${\Cal K}'$ are variants of
$\Cal K$ and~$\Cal C$, which are again Eisenstein.
Note,
however, that $\Psi_q$ and~$\Psi^\prime_q$ are
{\it isomorphic\/} as $\T$-modules, in view of the Atkin-Lehner
automorphism $w\: J\approx J'$ which intertwines the two actions
of~$\T$.  Hence \cite{\FERMAT, Th.~4.3} can be used as stated.

To derive a version of~(6.6),
we consider the monodromy pairings on~$L_p$ and~$X_p$
as injections
$$ X_p\hookrightarrow \Hom(X^\prime_p,\Z)\quad\hbox{and}\quad
    L_p\hookrightarrow \Hom(L^\prime_p,\Z),$$
respectively.  Their cokernels are the component groups $\Phi^\prime_p$
and $\Theta^\prime_p$ associated with the mod~$p$ reductions of
$\Jo{pM}{\times}\Jo{pM}$ and~$\Jo{pqM}$.  Using $\iota$, we regard
$Y_q$ as a submodule of~$L_p$.  We consider the $\T$-equivariant
maps $\sigma\: X_p \to L_p$ and $\kappa\: L^\prime_p\to X^\prime_p$
induced by the homomorphism $\sigma\:\Jo{pqM}\to\Jo{pM}{\times}\Jo{pM}$,
and use the abbreviation $\kappa^*$ to refer to the map $\Hom(\kappa,\Z)$
which is the $\Z$-linear dual of~$\kappa$.  We have a commutative square
$$ \matrix L_p &\to& \Hom(L^\prime_p,\Z)\cr
           \phantom{a}\uparrow\sigma&&\uparrow\kappa^*\cr
  X_p&\to& \Hom(X^\prime_p,\Z)\endmatrix $$
in which the horizontal maps are the injections coming from the
two monodromy pairings.
On dividing $L_p$ by its submodule $Y_q$, we obtain a commutative
diagram
%%%
$$ \matrix  0 & \to & L_p/Y_q & \to & \Hom(L^\prime_p,\Z)/Y_q &\to&
   \Theta^\prime_p &\to& 0\strut \cr
  &  & \phantom{a}\uparrow\sigma &  & \uparrow \kappa^* &&
    \uparrow\phantom{i} &&  \cr
 0 & \to & X_p & \to & \Hom(X^\prime_p,\Z) &\to&
    \Phi^\prime_p &\to& 0\strut \endmatrix$$
in which the rows are exact.
%
%
The map $\delta\: L_p\to X_p$ identifies $L_p/Y_q$ with~$X_p$.  
Also, we have on~$X_p$ the identity $\delta\circ\sigma=\eta$,
in view of the equation $\sigma\circ\delta=\eta$ involving
homomorphisms of abelian varieties.
Hence
the diagram may be rewritten
$$ \matrix  0 & \to & X_p\hfill & \to & \Hom(L^\prime_p,\Z)/Y_q &\to&
   \Theta^\prime_p &\to& 0\phantom{.}\strut \cr
     &  &\phantom{|} \uparrow {\eta} &  & \uparrow \kappa^* &&
	 \uparrow\phantom{|} &&  \cr
	  0 & \to & X_p \hfill& \to & \Hom(X^\prime_p,\Z) &\to&
	      \Phi^\prime_p &\to& 0.\strut \endmatrix$$
%
In view of (6.5), the cokernel of~$\kappa^*$ is~$\Hom(Y^\prime_q,\Z)$.
Therefore, the cokernel of the middle vertical map
is the finite group
$\Hom(Y^\prime_q,\Z)/Y_q =\Psi^\prime_q$.  By a dimension count, this
shows, in particular, that the middle vertical map is injective.  From
the snake lemma, we deduce a version of~(6.6) in which
the groups ${\Cal K}'$ and ${\Cal C}'$ are the kernel and cokernel
of the right-hand vertical map.

\proclaim{(6.7) Proposition}
The $\T$-modules $\Psi_q$ and $\Psi^\prime_q$ are mutually
$\Q/\Z$-dual.
\endproclaim

\noindent{\sl Proof}.\enspace
The indicated duality between the two
component groups may be read directly from the recipe for making
$\Psi_q$ and~$\Psi^\prime_q$ from the monodromy pairing on~$Y_q$.  
For more details,
see \cite{\SGA,\S11.4}, which treats the more general case of
an abelian variety with semistable reduction, and also
\cite{\SGA,\S11.3}, which discusses the prime-to-$p$ parts of
component groups associated to abelian varieties over a
local field of residue characteristic~$p$ which do not 
necessarily have semistable reduction.
\endproof

As O.~Gabber and R.~Livn\'e explained to the author,
Grothendieck conjectured
in \cite{\SGA, 1.2.1} that there is a perfect $\Q/\Z$-duality
between the component groups associated to two dual abelian
varieties over a local field.
In \cite\SGA, Grothendieck did not treat the $p$-primary
parts of component groups associated to arbitrary abelian
varieties over local fields of residue characteristic~$p$.
This gap was filled,
in the case of a perfect residue field,
by L.~B\'egueri as Th.~8.3.3 of~\cite\BEG.
Another proof, for an abelian variety over the fraction field of a
discrete valuation ring with finite residue field, is given
in~\cite\WMC.


In the statement of the following corollary, we introduce
Mazur's notation ``$[\m]$" for a kernel: if $\m$ is an ideal of~$\T$,
and $M$ is a $\T$-module, then $M[\m]$ denotes the submodule of~$M$
consisting of those element of~$M$ which are annihilated by all
elements of~$\m$.

\proclaim{(6.8) Corollary}
Let $\m\subset\T$ be a maximal ideal.  Then $\m$ lies in the support
of~$\Psi_q$ if and only if $\m$ lies in the support of~$\Psi^\prime_q$.
Moreover, the $\T/\m$-dimensions of $\Psi_q/\m\Psi_q$ and~$\Psi^\prime[\m]$
are equal; similarly, the dimensions of
$\Psi^\prime_q/\m\Psi^\prime_q$ and $\Psi_q[\m]$ coincide.
\endproclaim

\noindent{\sl Proof}.\enspace
The two equalities of dimension result directly from~(6.7).
Suppose now that $\m$ lies in the support of~$\Psi_q$.  By
Nakayama's Lemma, $\Psi_q/\m\Psi_q$ is non-zero.  
According to~(6.7), however, $\Psi_q/\m\Psi_q$ and~$\Psi^\prime_q[\m]$
are duals of each other.  Thus ~$\Psi^\prime_q[\m]$ is non-zero.
This non-vanishing implies
immediately that $\m$ lies in the support of~$\Psi^\prime_q$.
Similarly, the support of $\Psi^\prime_q$ is contained
in the support of~$\Psi_q$.
\endproof

\proclaim{(6.9) Theorem}
Let $\m$ be a maximal ideal of $\T$ for which the associated
representation of~$\GalQ$ is irreducible.  
Then $\m$ lies in the support of~$\Psi_q$ if and only if $\m$
contains $\eta$ and lies in the support of~$X_p$.  Moreover, in
this case we have the 
equalities of $\T/\m$-dimensions
$$ \dim X_p/\m X_p = \dim \Psi^\prime_q/m\Psi^\prime_q
          = \dim \Psi_q[\m].$$
\endproclaim

\noindent{\sl Proof}.\enspace
Assume that the representation of~$\GalQ$ associated with~$\m$
is irreducible.
By \cite{\FERMAT, 5.2c}, this hypothesis implies that $\m$
does not lie in the support of either of the modules ${\Cal K}'$
or~${\Cal C}'$ which appear in~(6.6).  Accordingly, $X_p/\eta X_p$
and $\Psi_q^\prime$ are isomorphic locally at~$\m$.  Thus $\m$
lies in the support of $\Psi_q^\prime$ if and only if it lies in
the support of $X_p/\eta X_p$.  By~(6.8), the supports of
$\Psi_q$ and~$\Psi_q^\prime$ coincide. 
Also, the support of~$X_p/\eta X_p$
clearly consists of those~$\m$ in the support
of~$X_p$ which contain~$\eta$.
Finally, whenever $\m$ is prime to the flanking
modules in~(6.6) and~$\eta\in\m$, we have an isomorphism
between $X_p/\m X_p$ and~$\Psi^\prime_q/m\Psi^\prime_q$.
This observation, together with~(6.8), gives the assertion
concerning dimensions.
\endproof

\head 7. Proof of Theorem 1.5.\endhead\noindent
Let $\rho$, $\ell$, $M$, and~$p$ be as in the statement of
Theorem~1.5.  In other words, we assume that
$\ell\ge3$ is a prime, that $\rho\:\GalQ\to\GLTWO\F$
is an irreducible representation arising from
an eigenform $f$ of weight two 
on~$\Gammao(Mp)$, and that
$p$ is prime to~$\ell M$.  
Suppose that
$\rho$ is unramified at~$p$.  We shall prove
that $\rho$ arises from a weight-two
eigenform on~$\Gammao(M)$.

The first step in the proof is the introduction
of an ``auxiliary prime" into the level
(cf.~the ``cuspidal case" discussion in~\S4 above and
\cite{\FERMAT, \S7}).
Choose a prime number~$q$,
prime to~$p\ell M$, such that
$\rho(\Frob_q)$ is conjugate to each matrix~$\rho(c)$, with
$c\in\GalQ$ a complex
conjugation.  
The characteristic polynomial of~$\rho(\Frob_q)$ then coincides
with that of the~$\rho(c)$; since $\rho$ is an odd representation,
$\rho(c)$ is conjugate to $\pmatrix -1&0\cr\hfill0&1\cr\endpmatrix $,
and thus has
characteristic polynomial $(T-1)(T+1)$.
Since the determinant of~$\rho$ is the mod~$\ell$ cyclotomic
character~$\chi$, we have
$q\equiv -1$ (mod~$\ell$).  
(Because $\ell$ is odd, this congruence
gives $q\not\equiv +1$ mod~$\ell$.)
Thus $(T-1)(T+1)=(T-a)(T-qa)$, where $a$ can be taken to be either
$+1$ or~$-1$, so that condition~II of~\S5 is satisfied.

An antecedent of Theorem~5.1 implies that
$\rho$ arises from a weight-two newform on
$\Gammao(pqM)$ whose level is divisible by~$q$. 
(See \cite\ICM, \cite\DPP, and \cite{\FERMAT, \S7}.)
Curiously, we will not use this fact in what follows, although
it will appear implicitly.
Instead, we consider the Hecke algebra
$\T=\T_{pqM}$ which appeared in~\S6 and mimic
the construction of the maximal ideal~$\m$ of~\S5.  Namely, let
$U$ again be the $q^{\roman{th}}$ Hecke operator in~$\T$, and let
$\eta=U^2-1$.  The construction of~\S5 enables one to
find a maximal ideal $\m$ of~$\T$ such that
$\m$ contains~$\eta$, together with an embedding
$\varphi\:\T/\m\hookrightarrow\F$,
such that $T_r$ is mapped to $\trace(\rho(\Frob_r))$ for almost all
prime numbers~$r$.
Fix such a maximal ideal in what follows.
(For the purposes of the proof of Theorem~1.5,
  $\T$ could be defined, more sparsely,
as the subring of~$\End S_2(\Gammao(pqM))$ which 
is generated by the $T_n$ with $n$ prime to~$pqM$, along with
the operator $U=T_q$.  Since no ``multiplicity-one"
theorems appear in the following argument, it is possible,
and perhaps desirable, to work with this more economical Hecke
algebra.)

Our next step is to replace $\rho$ by a model
$\rho_\m\:\GalQ\to\GLTWO{\T/\m}$ of~$\rho$ over the
finite field $\T/\m$.  Note that $\rho$ is an
irreducible finite-dimensional representation of $\GalQ$
over~$\F$ and that the characteristic polynomials of~$\rho(g)$
lie in~$\T/\m$ for each $g\in\GalQ$.  It is well known
that this necessary condition for descending
$\rho$ to~$\T/\m$ is also sufficient.

To descend $\rho$ directly, we write $K$ for $\T/\m$, and observe that
$\rho$ descends, in any case, to some finite extension $L$
of~$K$.  (Only a finite number of matrices are required
to describe~$\rho$.)  Let $W$ be an $L[\GalQ]$-module
which affords a model of~$\rho$ over~$L$, and let $V$ be
the vector space $W$, viewed as a module over $K[\GalQ]$.
(Thus $V$ is the ``restriction of scalars" of~$W$ from $L$
to~$K$.)  Let $X$ be a minimal (non-zero) $K[\GalQ]$-submodule
of~$W$.  The commutant $E\seteq\End_{K[\GalQ]} X$ of~$X$ is a 
division algebra by Schur's lemma; since $E$ has
finite cardinality, it is a finite commutative field.
It follows that the $L[\GalQ]$-module $X\otimes_K L$ has
commutant $E\otimes_K L$.  Because this commutant is
commutative, $X\otimes_K L$ must be multiplicity free.

On the other hand, $X\otimes_K L$ is a submodule of
$V\otimes_K L$.  This latter module is a direct sum
$\oplus_{\gamma} {}^\gamma\mkern-\thinmuskip W$,
where $\gamma$ runs through $\Gal(L/K)$ and where 
${}^\gamma\mkern-\thinmuskip W$ represents the twist of~$W$
by~$\gamma$, i.e., the tensor product $W\otimes_L L$ where $L$ is
viewed as a base extension of itself via $\gamma\:L\to L$.
The hypothesis on characteristic polynomials ensures that
the various ${}^\gamma\mkern-\thinmuskip W$ are isomorphic
to~$W$ (Brauer-Nesbitt theorem).  Hence $V\otimes_K L$ is
isomorphic to a sum of copies of~$W$, and it follows in
particular that $X\otimes_K L$ is a sum of copies of~$W$.
Since $X\otimes_K L$ is multiplicity free, there can
be only one summand.  This means that $X\otimes_K L$ is
isomorphic to~$W$, so that $X$ affords a model of~$\rho$
over~$K$.


Now that $\rho_\m$ has been constructed, we will abuse notation
by writing simply $\rho$ for this representation.


We will prove Theorem~1.5 by an
indirect reasoning: we will assume that $\rho$
is {\it not\/} modular of level~$qM$, and obtain a contradiction from
this assumption.  This reasoning will show that $\rho$
is modular of level~$qM$, but then Mazur's Principle (4.7)
will enable us to deduce the
desired conclusion that $\rho$ is modular
of level~$M$.


We continue to work with
the series of abelian varieties which appeared in~\S6:
$\Jo{pqM}$, $\Jo{pM}^2=\Jo{pM}{\times} \Jo{pM}$, 
  $\Jo{qM}^2=\Jo{qM}{\times} \Jo{qM}$, and the  
Jacobian $J=\Pico C$ of the Shimura curve
$C$.  We attach
``multiplicities" $\lambda$ and $\mu$ to $\Jo{pqM}$ and~$J$
in the following manner.  Consider
the kernel $J[\m]$
of $\m$ on $J(\Qbar)$.  The main theorem of~\cite\BLR\ implies
that this $\T/\m[\GalQ]$-module is isomorphic to a direct sum
$V\oplus\cdots\oplus V$, where $V$ is a two-dimensional $\T/\m$-vector
space with a $\GalQ$-action equivalent to~$\rho$.  We define $\mu$ to
be the number of factors, and refer to it as the multiplicity of~$\rho$
in~$J$.  The multiplicity $\lambda$ is defined analogously,
by considering $\Jo{pqM}[\m]$ in place of~$J[\m]$.

{\it Assume from now on  that $\rho$ is not modular of level~$qM$.}
Using this assumption, we shall
prove that both multiplicities $\lambda$ and~$\mu$ are zero.  This will
give the desired contradiction, since $\lambda$ is positive.  Indeed,
$\T$, by definition, operates faithfully on~$\Jo{pqM}$, and it is well
known that this implies the inequality $\lambda>0$.

  To prove that $\lambda$ and $\mu$ are both zero, we will establish a
series of relations among the dimensions of the
$\T/\m$-vector spaces obtained as the
mod~$\m$ reductions of the character groups which appear in
(6.2) and~(6.3).  The hypothesis that $\rho$ is not modular of level~$qM$
implies that $\m$ generates the unit ideal in the ``$p$-old" quotient
of~$\T$.  This  implies, for example, that $\m$ does not lie in the
support of the $\T$-module~$X_q$, since $\T$ operates on~$X_q$
through its $p$-old quotient.


 

\proclaim{(7.1) Proposition}  There are exact sequences of $\T$-modules
$$ \leqalignno{
& Y_p/\m Y_p \approx L_q/\m L_q; &(7.2)\cr
 \cdots\to &Y_q/\m Y_q \to L_p/\m L_p \to X_p/\m X_p \to 0. &(7.3)}$$
\endproclaim

\noindent
{\it Proof.}
The isomorphism (7.2) and the
exact sequence~(7.3) are obtained from (6.2) and~(6.3) by first
localizing at~$\m$ and then reducing mod~$\m$.
The localization at~$\m$ of $X_q$ vanishes, as remarked
above. \endproof
  
Proposition~7.1 implies the following relations among 
$\T/\m$-dimensions:
$$\leqalignno{
 \dim Y_p/\m Y_p & =\dim  L_q/\m L_q; &{(7.4)}\cr
\dim L_p/\m L_p\,&\le\, \dim Y_q/\m Y_q+\dim X_p/\m X_p . &{(7.5)}}$$


\proclaim{(7.6) Proposition}
We have
$$\dim Y_p/\m Y_p = 2\mu,\quad \dim L_p/\m L_p = 2\lambda.$$
\endproclaim

\noindent
As in~\cite\NOMULT,
these equalities are obtained by considering actions of a
decomposition group $D_p$ for~$p$ in~$\GalQ$.
By the main theorem of \cite\BLR,
the kernels of $\m$ on $\Jo{pqM}$,
and~$J$ are direct sums of copies of~$\rho$, first as
representations of~$\GalQ$, and then (by restriction) as
representations of~$D_p$.
In particular, they are unramified at~$p$.

By a well known theorem of Serre-Tate \cite\ST, the kernel of~$\m$
in the mod~$p$ reduction of~$J$ may be identified with the group
of $I_p$-invariants in $J[\m]$, where $I_p$ is the inertia subgroup 
of~$D_p$.  Since
$J[\m]$ is unramified at~$p$, the group of $I_p$-invariants is
all of~$J[\m]$, and therefore
$J_{/\Fp}(\Fpbar)[\m]$ is of dimension~$2\mu$. 

Now, as we recalled above,
$J_{/\Fp}$ is an extension of its ``group of components"
$\Psi_p$ by its toric part.
Also, $\m$ does not lie in the support of~$X_q$.  By (6.9),
or more precisely the variant of (6.9) gotten by interchanging
$p$ and~$q$,
the $\T$-module $\Psi_p$ is prime to $\m$.
In view of this fact,
we may deduce that the toric
part of $J_{\Fpbar}(\Fpbar)[\m]$ has dimension~$2\mu$.
This gives $\dim Y_p/\m Y_p = 2\mu$, which was the first of the
two formulas to be proved.

The formula involving~$L_p$ is
proved by an analogous argument in which we consider the mod~$p$
reduction of~$\Jo{pqM}$.  Let $A$ be the fiber at~$p$ of the
N\'eron model of~$\Jo{pqM}$, and let $B$ be the analogue of~$A$
for $\Jo{qM}\times \Jo{qM}$.  Thus $B$ is an abelian variety
over~$\Fp$.  As in the discussion involving~$J$, we know
that $A(\Fpbar)[\m]$ is of dimension~$2\lambda$.  In contrast to
the situation for~$J$, however, the group of components of~$A$
is ``Eisenstein" (\cite{\FERMAT, Th.~3.12}, or \cite\BX),
and therefore prime to~$\m$
\cite{\FERMAT, Th.~5.2c}.  Thus, 
if $A^o$ is the connected component of~0 in~$A$, 
$A^o(\Fpbar)[\m]$ 
is of dimension~$2\lambda$.

By results of Deligne-Rapoport \cite\DERA\
and Raynaud, there is an exact sequence
$$ 0\to T \to A^o \to B\to 0,$$
where $T$ is the ``toric part" of~$A^o$, i.e., that torus whose
character group is~$L_p$ (cf.~\cite{\FERMAT, p.~446}).  
Since $\m$ is not modular of level $qM$,
we have $B(\Fpbar)[\m]=0$.  (Compare \cite{\FERMAT, Theorem~3.11},
and the discussion in~\S8 below.)
Therefore, $T(\Fpbar)[\m]$ has 
$\T/\m$-dimension $2\lambda$.  Consequently, $L_p/\m L_p$
is of dimension~$2\lambda$.
\endproof

\proclaim{(7.7) Proposition}
We have
$$\dim Y_q/\m Y_q \le \mu,\quad \dim L_q/\m L_q \le\lambda.$$
\endproclaim

\noindent
This proposition is proved, analogously, by considering the action
of a decomposition group $D_q$ in $\GalQ$ for the prime~$q$.  Pick such
a group, together with a Frobenius element $\Frob_q$ in it.
It will be useful to view $D_q$ as $\Gal(\Qqbar/\Qq)$.

By the choice of~$q$, $\rho$ is unramified
at~$q$, and $\rho(\Frob_q)$ has the distinct eigenvalues $+1$ and~$-1$.
Therefore, on $J[\m]$, say, $\Frob_q$ has the eigenvalues $+1$ and~$-1$,
each with multiplicity~$\mu$, in view of the fact that $J[\m]$ is
a direct sum of $\mu$ copies of~$\rho$.

By \cite{\SGA, \S5.1} (or by the results of~\cite\ST), the $\T[D_q]$-module
$J[\m]$ contains the submodule
$\Hom(Y_q/\m Y_q,\mu_\ell)$.
(Here, $\mu_\ell$ denotes the group of $\ell^{\roman{th}}$ roots of unity of
$\Qqbar$; the author expects that there will be no confusion
with the multiplicity~$\mu$.)

We claim that
the Frobenius element $\Frob_q$ acts on~$Y_q$ as the $q$th Hecke
operator~$U=T_q$.  This claim implies that $\Frob_q$ acts on
$\Hom(Y_q/\m Y_q,\mu_\ell)$ as the product $qT_q$, and
therefore in particular as an element of
the field~$\T/\m$.
This scalar will be $\pm 1$, since $U$ is an involution and
since $q$ is congruent to~$-1$ (mod~$\m$).
It follows from this
that the dimension of~$Y_q/\m Y_q$ is bounded from above by the
multiplicity~$\mu$, since each
eigenvalue $\pm 1$ occurs 
at most $\mu$ times on~$J[\m]$.  In other words, we obtain
the desired inequality $\dim Y_q/\m Y_q \le \mu$.

To prove the claim, we view $Y_q$ as the first integral homology
group of the dual graph associated with the reduced
Shimura curve $C_{/\Fq}$.
This graph may be described explicitly in terms of certain
abelian
varieties in characteristic~$q$
with ``quaternionic multiplication" by an order $\Cal O$
in a rational quaternion algebra
\cite{\BIMODULES, \S5}.  The abelian
varieties $A$ which occur in the description
of the graph are all ``exceptional"
in the language of \cite{\BIMODULES, \S4}.  
This means that the kernel of the Frobenius morphism $A\to A^{(q)}$
coincides with the kernel of multiplication by~$\goth q$
on~$A$, where $\goth q$ is the unique two-sided maximal ideal
of~$\Cal O$ whose residue field has $q^2$ elements.
Since operator~$T_q$ has the modular description
$A\mapsto A/A[\goth q]$, we may deduce that the actions of
$T_q$ and of the Frobenius automorphism coincide on the graph.


The inequality $\dim L_q/\m L_q \le \lambda$ is obtained similarly.
\endproof


\proclaim{(7.8) Proposition}  We have $\dim X_p/\m X_p \le \mu$.
\endproclaim

\noindent
Let $J'$ again denote the
abelian variety dual to~$J$ and let $Y^\prime_q$ again be the 
analogue of~$Y_q$
for~$J'$. 
Because $J$ has multiplicative reduction
at~$q$, one has an exact sequence of $\T[\Gal(\Qqbar/\Qq)]$-modules
$$
  0\to\Hom(Y_q/\ell Y_q,\mu_\ell) \to J[\ell] 
   \to Y^\prime_q/\ell Y^\prime_q \to 0.
  \leqno(7.9)
$$  
Indeed, Grothendieck has constructed a
``canonical filtration" \cite{\SGA,11.6.5} on the $\Gal(\Qqbar/\Qq)$-module
$A[\ell]$, whenever $A$ is an abelian variety with semistable reduction 
over~$\Qq$.  This filtration has the form
$$
0 \subset A[\ell]^{\roman t} \subset A[\ell]^{\roman f} \subset A[\ell],
$$
where $A[\ell]^{\roman t}$, the ``toric part" of~$A[\ell]$, may be 
written
$\Hom(Y/\ell Y,\mu_\ell)$; here, $Y$ is the character group of the
toric part of the special fiber of the N\'eron model of~$A$.
Analogously
(and dually), the quotient $A[\ell]/A[\ell]^{\roman f}$ of Galois modules
may be identified with $Y'/\ell Y'$, where $Y'$ is the analogue of~$Y$
for the abelian variety dual to~$A$ (\cite{\SGA,11.6.6},
\cite{\SGA,11.6.7}).
In case $A$ has purely multiplicative reduction, $Y$ and~$Y'$
are free of rank $\dim A$, and the middle terms
$A[\ell]^{\roman t}$ and~$A[\ell]^{\roman f}$ of the filtration
coincide.  One therefore obtains an exact sequence like (7.9) 
whenever the reduction of~$A$ is multiplicative.
(For a discussion of abelian varieties with {\it split\/}
multiplicative reduction, the reader may consult the expository
account in Chapter~III of~\cite\MYTHESIS.  The exact sequence~(7.9)
appears as Lemma~3.3.1 in~\cite\MYTHESIS.)


Localization of (7.9) at~$\m$ yields an exact
sequence of $\T[\Gal(\Qqbar/\Qq)]$-modules
$$
  0\to\Hom(Y_q/\ell Y_q,\mu_\ell)_\m \to J[\ell]_\m \to 
  (Y^\prime_q/\ell Y^\prime_q)_\m \to 0.
\leqno(7.10)$$
This sequence {\it splits\/} as a sequence of $\T$-modules.
Indeed, consider again a Frobenius element
$\Frob_q$ of $\Gal(\Qqbar/\Qq)$.  This operator commutes with the
action of~$\T$ and operates on each of
$\Hom(Y_q/\ell Y_q,\mu_\ell)_\m$ 
and $(Y^\prime_q/\ell Y^\prime_q)_\m $ as an element
of~$\T$.
Indeed, $\Frob_q$ operates on~$Y_q$ as the involution~$U=T_q$; therefore
it operates as $qU$ on $\Hom(Y_q/\ell Y_q,\mu_\ell)_\m$
and as $U$ on $(Y^\prime_q/\ell Y^\prime_q)_\m $. These
elements of~$\T$
are incongruent mod~$\m$, since $q$ is~$-1$
(mod~$\ell$).  This forces the desired splitting.

Since (7.10) splits, it remains exact after we 
take ``kernels of~$\m$."  In other words, we have a sequence of
$(\T/\m)[\Gal(\Qqbar/\Qq)]$-modules
$$
  0\to\Hom(Y_q/\ell Y_q,\mu_\ell)[\m] \to J[\m] \to 
  (Y^\prime_q/\ell Y^\prime_q)[\m] \to 0.
$$
Since $\Frob_q$ operates on $\Hom(Y_q/\ell Y_q,\mu_\ell)[\m]$ and
$ (Y^\prime_q/\ell Y^\prime_q)[\m] $ as homotheties, the
dimensions of each of these modules must be~$\mu$. 
  To see this, we observe
that the eigenvalues of~$\Frob_q$ on  the two-dimensional
representation $V$ are $+1$ and~$-1$.  Therefore, the characteristic
polynomial arising from the action
of~$\Frob_q$ on~$J[\m]$ is $(T-1)^\mu (T+1)^\mu$.
Suppose that $\Hom(Y_q/\ell Y_q,\mu_\ell)[\m]$ has dimension~$d_1$
and that $\Frob_q$ operates on~$\Hom(Y_q/\ell Y_q,\mu_\ell)[\m]$ as
the scalar~$\xi_1$. Define $d_2$ and $\xi_2$ similarly, using
$ (Y^\prime_q/\ell Y^\prime_q)[\m] $ in
place of $\Hom(Y_q/\ell Y_q,\mu_\ell)[\m]$.
Then we clearly have $$(T-1)^\mu (T+1)^\mu = (T-\xi_1)^{d_1}
(T-\xi_2)^{d_2}.$$  Since $-1$ and~$+1$ are distinct, we have
$d_1 = d_2 = \mu$.

Let $\Psi_q$ again be the group of components associated with the
mod~$q$ reduction of~$J$. As was recalled in~\S6,
$\Psi_q$ may be constructed from
the monodromy pairing on $Y_q$. 
Indeed, if we view this pairing as an
inclusion $Y_q^\prime\hookrightarrow Y_q^*$, where
$Y_q^*=\Hom(Y_q,\Z)$, then $\Psi_q$ is naturally
the cokernel of this inclusion. 
By considering the ``multiplication by $\ell$" maps in the exact
sequence $0\to Y_q^\prime\to 
  Y_q^* \to\Psi_q\to 0$, and using the Snake Lemma,
we obtain an inclusion 
$\Psi_q[\ell] \subseteq Y_q^\prime/\ell Y_q^\prime$.
In particular, we have 
$\Psi_q[\m]\subseteq (Y_q^\prime/\ell Y_q^\prime)[\m] $.
Thus, $\mu$ is bounded from below by the dimension of  $\Psi_q[\m]$.
By~(6.9), this dimension coincides with that of~$X_p/\m X_p$,
since
$\eta$ lies in~$\m$.
Thus we have the desired inequality
$\dim X_p/\m X_p \le \mu$.
\endproof

Using (7.4), Proposition~7.6, and Proposition~7.7, we obtain
$$ 2\mu \le \lambda,$$
which implies $\mu\le\lambda$,
since $\mu$ is a natural number.  From~(7.5)
and Propositions 7.6, 7.7, and~7.8,
we get
$$ 2\lambda \le 2\mu.$$
Clearly, these inequalities imply
$\lambda=\mu=0$.  As indicated above, this contradiction
completes the proof of Theorem~1.5.

\head 8. Mazur's Principle \endhead\noindent
In this \S, $\ell$ is an odd prime, and $\F$ is again a fixed
algebraic closure of~$\Fell$.  Consider an odd
irreducible representation $\rho\:\GalQ\to\GLTWO\F$, and a prime
number~$q$ 
different from~$\ell$.
(The case $q=\ell$ may be included
if one introduces the concept of ``finiteness" of a representation;
see \cite{\DUKE, p.~189} and \cite{\FERMAT, Lemma~6.2}.)

The following result is a variant of~(4.7):

\proclaim{(8.1) Theorem}
Suppose that $\rho$ arises from the space of weight-two cusp forms
on  $\Gamma_1(N)\cap\Gammao(q)$, where $N$ is prime to~$q$.
Assume that $\rho$ is unramified at~$q$ and that~$q\not\equiv1$ mod~$\ell$.
Then $\rho$ arises from the
space of weight-two cusp forms on~$\Gamma_1(N)$.
\endproclaim

\noindent
The space $\Cal S$ of weight-two cusp forms on~$\Gamma_1(N)\cap\Gammao(q)$
contains (as its ``$q$-old" subspace) a direct sum ${\Cal S}_o$
of two copies of
$S_2(\Gamma_1(N))$. This subspace is stable under the action of the
Hecke operators $T_n$ (for $n\ge1$) on~$\Cal S$.  Let $\T$ be the
subring of~$\End\Cal S$ generated by the~$T_n$, and let $\T_o$ be
the $q$-old quotient of~$\T$, i.e., the image of~$\T$ in
$\End {\Cal S}_o$.  These algebras contain the
diamond bracket operators
$\diamond d$ for $d\in(\Z/N\Z)^*$.
The assumption that $\rho$ arises from~$\Cal S$ implies that there
is a homomorphism
	$$\varphi\:\T\to\F$$
satisfying
$$ \varphi(T_r) = \trace(\rho(\Frob_r)), \qquad \varphi(\diamond r r) =
   \det(\rho(\Frob_r))$$
for all prime numbers  $r$ prime to $\ell q N$.  We view $\varphi$
as an embedding $k\hookrightarrow\F$, where $k\seteq\T/\m$, and
observe that there exists a model for~$\rho$ over the field~$k$.
This model is a two-dimensional $k$-vector space $V$
furnished with an action of~$\GalQ$.  The $\F$-vector space
$V\otimes_k\F$ affords the representation~$\rho$.
As in~\S5,  to prove that $\rho$ arises from~${\Cal S}_o$, it suffices to
show that $\m$ arises by pullback from the quotient $\T_o$ of~$\T$.
We will sketch a proof that this is so; a more detailed proof, 
but with $\Gamma_1(N)$ replaced by~$\Gammao(N)$, is
given in~\cite{\FERMAT, \S6}.

In the discussion below, we view $V$ as a $\T$-module in the obvious
way: $\T$ acts through its quotient~$k$.

Consider the modular curve $C$ attached to the group
$\Gamma_1(N)\cap\Gammao(q)$.  The operators $T_n$  and~$\diamond d$
act on~$C$ as correspondences.  The correspondences $T_n$ and~$\diamond d$
of~$C$ induce a faithful action
of~$\T$ on the Jacobian $J\seteq\Pico C$.  The endomorphisms of~$J$
induced by the $T_n$ and~$\diamond d$ will be denoted, as usual, by
the same symbols $T_n$ and~$\diamond d$.
The kernel $J[\m]$ of~$\m$ on~$J(\Qbar)$, which is
non-zero, is a
direct sum of copies of~$V$ (cf.~\cite\BLR).  In particular, there exist
$\T[\GalQ]$-equivariant maps
$V\buildrel\tilde\iota\over\hookrightarrow J(\Qbar)$.
Because $V$ is unramified at~$q$, and in view of the result of Serre-Tate
used above, each choice $\tilde\iota$
determines a map of $\T[\GalQq]$-modules
   $$ V \buildrel\iota\over\hookrightarrow \underline J(\Fqbar).$$
Here, $\underline J$ represents the fiber over~$\Fq$ of the N\'eron
model of~$J$.  Fix one~$\tilde\iota$, and use the resulting
map~$\iota$ to view
$V$ as a subgroup of~$\underline J(\Fqbar)$.


By results of Deligne-Rapoport \cite\DERA\ and Raynaud,
$\underline J$ is semiabelian.
In other words, $\underline J$ is an extension of its component
group $\Phi$ by a connected group ${\underline J}^o$, which in turn
is an extension of
an abelian variety $A$ by a torus~$T$.  More precisely, $A$ is
a product of two copies of~$J_1(N)_{/\Fq}$, while the character group
$X$ of~$T$ may be interpreted as the group of degree-zero
divisors on $X_1(N)_{/\Fq}$ which are concentrated at the set
of supersingular points of~$X_1(N)_{/\Fqbar}$.
The ring $\T$ acts naturally on $\Phi$, $T$ and~$A$ by
functoriality.
  To show that $\m$ arises by pullback from~$\T_o$, it suffices to
verify the following three points:
\item{---} The ring $\T$ acts on~$A$ through its quotient~$\T_o$.
\item{---} The $\T$-module $\Phi$ is prime to~$\m$ in the sense
that $\m$ does not lie in the support of~$\Phi$.
\item{---} The submodule $V$ of~$\underline J(\Qqbar)$ cannot be
contained in the torus~$T$.

\noindent For the first point, the standard degeneracy maps
$\alpha,\beta\:C\doublemap X_1(N)$ induce (in characteristic zero)
a map on Picard varieties $\delta\:J_1(N){\times} J_1(N) \to J$
which is known to be injective (cf.~\cite\DIA\ or \cite\ICM).
It is clear that $\T$ acts on the abelian subvariety 
$J_1(N){\times} J_1(N)$ of~$J$ through its old quotient~$\T_o$.
Indeed, if we dualize $\delta$ and then pass to cotangent
spaces, we obtain the canonical inclusion ${\Cal S}_o\hookrightarrow
{\Cal S}$.  The map $\delta$ induces in characteristic~$q$ a map
$\delta_{/\Fq}\: J_1(N)_{/\Fq} {\times} J_1(N)_{/\Fq} \to {\underline J}^o$,
and the point to be checked is that $\pi\circ\delta_{/\Fq}$ is
an isogeny, where $\pi$ is the structural map
${\underline J}^o\to A$. 
If we regard $A$ as~$J_1(N)_{/\Fq} {\times} J_1(N)_{/\Fq}$,
then $\pi\circ\delta_{/\Fq}$ becomes a two-by-two matrix of
endomorphisms of~$J_1(N)_{/\Fq}$.  This matrix, as computed by
Deligne-Rapoport \cite{\DERA, p.~287}, has diagonal components equal to
the identity endomorphism, and off-diagonal components equal to the
Verschiebung endomorphism (dual of the Frobenius).
This matrix is then visibly an isogeny, since it acts on the cotangent
space of~$J_1(N)_{/\Fq} {\times} J_1(N)_{/\Fq}$ as the identity map.

The following circumstance is
perhaps worth stressing.
In the functorial action of~$\T$ on~$A$, the Hecke operators
$\diamond d$ and $T_n$ with $n$ prime to~$q$ act ``as expected."
Namely, they are induced by their namesakes
on~$J_1(N)$, which operate diagonally on $J_1(N) {\times} J_1(N)$
and then by reduction (mod~$q$) on~$A$.  However, the functorial
action 
of~$T_q$ on~$A$ belongs naturally to characteristic~$q$.


For the second point, 
we show that $\Phi$ is {\it Eisenstein\/} in the 
strong sense that we have on~$\Phi$ the equations
$T_p = 1+p$ and $\diamond p =1$ for all $p$ prime to $qN$.
These equations imply immediately that $\m$ is prime to
the support of~$\Phi$ because $\rho$ is irreducible; cf.~\cite{\FERMAT,
Theorem~5.2c}.
That $\Phi$ is Eisenstein was proved in~\cite\BX\ for the abelian
variety~$J_o(qN)$.  The result we seek then follows for $N\le3$ because
the natural map $J_o(qN) \to J$ is an isomorphism.

Assume that $N\ge4$.
Recall that $\Phi$ is the cokernel of
the map $X\to\Hom(X,\Z)$ coming from the monodromy pairing on~$X$.
View $X$ as the kernel of the degree map $X\tilde{\null}\to\Z$,
where $X\tilde{\null}$ is the free abelian group on the set $\Sigma$ of
supersingular points of~$X_1(N)_{/\Fqbar}$.  The monodromy pairing
on~$X$ is the restriction to~$X$ of a diagonal pairing on~$X\tilde{\null}$.
This pairing takes the value $e(\sigma)$ on~$(\sigma,\sigma)$, where
$e(\sigma)$ is essentially the number of automorphisms of any pair
$(E,P)$ which represents~$\sigma$; we understand that
$E$ is a supersingular elliptic curve over~$\Fqbar$ and $P\in E(\Fqbar)$
is a point of order~$N$.
The precise formula for $(\sigma,\sigma)$ may be derived from \cite{\DERA,
p.~286, Th\'eor\`eme~6.9}: we have
$$ e(\sigma) = \cases
{1\over2}\#\Aut(E,P) &\text{if $-1\in\Aut\sigma$,} \\
\#\Aut(E,P) &\text{if $-1\not\in\Aut\sigma$.}\endcases   $$
The assumption $N\ge4$ implies that $e(\sigma)=1$ by
\cite{\KATZMAZUR, Corollary~2.7.4}.  Hence the monodromy pairing on~$X$
is just the restriction to~$X{\times} X$ of the standard diagonal
pairing on~$X\tilde{\null}$.
It follows   that the natural map
$$\xi\: X\tilde{\null}/X \longrightarrow \Phi=\Hom(X,\Z)/X, \qquad
\sum n_\sigma \sigma
 \mapstochar\relbar\joinrel\rightarrow
\left\langle\sum n_\sigma \sigma,
           \,\cdot\,\right\rangle$$
is surjective. Since
 $X\tilde{\null}/X$ is mapped isomorphically to~$\Z$
by the degree map, $\Phi$ is a finite cyclic group.

Fix $p$ prime to~$Nq$, and let
$T$ denote the usual $p$th Hecke operator on~$X\tilde{\null}$: this
is the $\Z$-linear map of~$X\tilde{\null}$ which sends (the class of)
$(E,P)$ to 
$\sum (E/{\Cal C}, P\hbox{ mod }\Cal C)$.  Here,
 the sum ranges
over the $p{+}1$   subgroups $\Cal C$ of~$E[p]$ whose order is~$p$.
Let $T'$ denote the
composite $T\circ\diamond p^{-1}$, where $\diamond p\:(E,p)\mapsto
(E,pP)$.  One checks the formula 
$\langle Tx,y\rangle = \langle x,T'y\rangle$,
where $\langle \,,\,\rangle$ is the Euclidean pairing on~$X\tilde{\null}$.
Also,
it is clear that both $T$ and~$T'$ preserve the subgroup $X$
of~$X\tilde{\null}$ and induce the map ``multiplication by $p{+}1$"
on the quotient $X\tilde{\null}/X$.

The operator $T_p$ on~$\Phi$ is the map on
$\Hom(X,\Z)/X$ induced by the operators $T$ on~$X$ and
$\Hom(T',\Z)$ on~$\Hom(X,\Z)$.
The formula relating $T$, $T'$, and~$\langle \, , \, \rangle$
shows that
$\xi$ is $T_p$-equivariant if $T_p$ acts naturally on~$X\tilde{\null}/X$
by multiplication by~$p{+}1$.  Hence $T_p = p+1$ on~$\Phi$.
In a similar manner, we find that the diamond-bracket operators
are trivial on~$\Phi$.




For the last point, we must determine the action of~$\Frob_q$
on~$T(\Fqbar)$. 
Let $w$ be the automorphism of~$C$ which corresponds to the map 
$$(E,P,{\Cal C}) \mapsto (E/{\Cal C}, P~{\roman{mod}}~{\Cal C},E[q]/{\Cal C})$$
on triples consisting of an elliptic curve $E$, a point $P$ on~$E$ of
order~$M$, and a cyclic subgroup $\Cal C$ of~$E$ which has order~$q$.
A short computation shows that we have $w^2=\diamond q$.
One checks as well
that the correspondence $T_q+w_q$ of~$C$ may be written
$\beta'\circ\alpha$ where $\alpha$ and~$\beta$ are again the degeneracy
maps $C\doublemap X_1(N)$.  It follows from this that $T_q+w_q$
induces an endomorphism of~$\underline J$ which kills the torus~$T$,
since this endomorphism factors through a map $\underline J\to J_1(N)$.
On the other hand, $w$ coincides with the Frobenius
map on~$\Sigma$ but permutes the two components of~$C_{/\Fq}$.  Therefore
$\Frob_q=-w=T_q$ on~$X$, since $X$ is canonically the first
integral homology group of the dual graph associated with~$C_{/\Fq}$.
Thus $\Frob_q=qT_q$ on~$T(\Fqbar)$.
Therefore,
if $V\hookrightarrow T(\Fqbar)$, then $\Frob_q$ acts on~$V$ as
a scalar, i.e., as an element of~$k$.  Since the square of this
scalar is~$q^2\diamond q$, the determinant the action 
of~$\Frob_q$ on~$V$ is~$q^2\diamond q$.

On the other hand, it is clear from the \v Cebotarev Density Theory
that the determinant of~$V$ is the character $\chi\epsilon\:\GalQ\to k^*$,
where $\epsilon$ is the Dirichlet character $\diamond{\phantom d}$,
i.e., where $\epsilon(\Frob_r)=\diamond r$ for every prime number
$r$ prime to~$M$.  In particular, the determinant of
the action of~$\Frob_q$ on~$V$ is $q\diamond q$.  Therefore, we cannot
have $V\hookrightarrow T(\Fqbar)$ unless $q\equiv1$ mod~$\ell$.

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\enddocument
