\magnification=\magstep1
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 .3333emplus.0555em\relax}

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\let\newfoil\newpage
\def\foilhead#1{\bigbreak\hrule\vskip2pt
\hrule\medskip\centerline{\bf #1}\medskip}
\centerline{\bf Galois Representations and Modular Forms}
\vskip2pt
\centerline{Progress in Mathematics Talk, August 17, 1994}
\vskip2pt
\centerline{Professor Kenneth A. Ribet, UC Berkeley}
                                 
\def\hugeleftrightarrow{{\leftarrow\joinrel\relbar\joinrel
\relbar\joinrel\rightarrow}}
\let\phi\varphi
\def\seteq{\mathrel{:=}}

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\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\Q{{\bf Q}}
\def\F{{\bf F}}
\def\T{{\bf T}}
\def\Fp{\F_p}
\def\SL#1{{\bf SL}(2,#1)}
\def\GL#1{{\bf GL}(2,#1)}
\def\rhobar{\overline{\rho}}

\def\seteq{\mathrel{:=}}
\def\Qbar{\overline{\Q}}
\def\GL#1{{\bf GL}(2,#1)}
\def\go#1{\Gamma_0(#1)}

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\hbox{/}\kern-0.75pt}}\mathrel{\box1}}

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\def\newop#1
{\expandafter\def\csname #1\endcsname{\mathop{{\rm #1}}\nolimits}}
\newop Aut
\newop Gal
\newop tr









\foilhead{}
This lecture is about:
\item{$\bullet$}Elliptic curves and abelian varieties over~$\Q$
\item{$\bullet$}Clasical modular forms in one complex variable
\item{$\bullet$}Representations of Galois groups over~$\Q$
\item{$\bullet$}Fermat's Last Theorem

\newpage
Concerning Fermat:

In January, I lectured at the AMS Winter Meeting.
At that time, there was a
``gap'' in 
Andrew Wiles's announced
proof of
Fermat's Last Theorem.

To the best of my knowledge, there has
been no significant progress since last
winter.

\newfoil
The link between Fermat's Last Theorem
and the other topics stems from a
construction of G. Frey and from
subsequent work by Serre, Mazur, and myself.

The idea is to work by contradiction:
You assume that $a^\ell+b^\ell=c^\ell$
and write down Frey's equation
$$ y^2 = x(x-a^\ell)(x+b^\ell).$$
Here, $\ell$ is
a prime different from 2 and~3, and
$a$, $b$, and~$c$ are non-zero relatively
prime integers.
\newfoil
Frey's equation
defines an {\it elliptic
curve}, a non-singular curve in the
projective plane.

\proclaim Theorem.
Frey's elliptic curve does not satisfy the
Taniyama-Shimura conjecture.


The theorem means that Frey's curve cannot be
linked with modular forms.
If you prove that all elliptic curves over~$\Q$
are ``modular,'' then you achieve a contradiction
and settle Fermat's Last Theorem.

\foilhead{Modular elliptic curves}

Suppose that $A$ is an elliptic curve over~$\Q$.
For all primes~$p$, we can ``reduce $A$ mod~$p$,''
or at least attempt to.  Usually we succeed, and
get an elliptic curve over~$\Fp$.  Occasionally,
we fail and say that the elliptic curve has ``bad
reduction at~$p$.''

For instance, under mild conditions on $A$ and~$B$,
$y^2=x(x-A)(x+B)$ has good reduction for all $p\notdivide ABC$,
where $A+B+C=0$.
\newfoil
For each good~$p$, let $t_p$ be the number of points on~$A$
over~$\Fp$, counting the extra ``point at infinity.''

Set $a_p \seteq p{+}1 - t_p$.

We get a sequence $a_2, a_3, a_5, \ldots$,
where the indices run through all primes of
good reduction for~$A$.

For example, consider $$A: y^2 +y = x^3 -x^2.$$
Only $p=11$ is bad.
For small $p$:
$$\vbox{
\halign{$\hfil#$\tvi\vrule\hskip 2pt&&$\vrule\hfil\,#\,$\cr
p\,&
2& 3& 5& 7& 13& 17&19&23&29\cr
\noalign{\hrule}
a_p\,&
-2& -1& 1& -2& 4& -2&0&-1&0\cr}}$$
\newfoil
 

Incidentally, $a_p\equiv p+1$ mod~5
for all~$p$ in this case; equivalently, $5|t_p$ for all~$p$.
The divisibility stems from the fact that $A(\Q)$ is a
cyclic group of order~5.  In this remark, we allude for the
first time to the {\it group structure\/} of~$A$.

When $A$ is regarded as an algebraic group, the unique point
at infinity $O$ is the origin, and any three colinear points sum
to~$O$.
\newfoil
The curve $A: y^2 {+}y = x^3 {-}x^2$
 is {\it modular} because
 the sequence $a_p$
occurs in
the Fourier expansion of a cusp form of
weight two on the group of integral $2\times2$ matrices
which are upper-triangular modulo~11 and which have determinant~1.

Such cusp forms
form a
one-dimensional $\C$-vector space, generated by
a unique (``normalized'') element
$$ f = \sum_{n=1}^\infty c_n e^{2\pi i n z}, \qquad c_1=1.$$
{\it For
all $p\ne11$, we have $a_p=c_p$.}
\newfoil

\noindent
More generally, for each $N\ge1$, let $\go N$
be
the group of $2\times2$ integral matrices
which are upper-triangular modulo~$N$ and which
have determinant~1.
Consider the holomorphic $f(z)$ on the upper half-plane
which satisfy
$$f\left({az+b\over cz+d}\right)= (cz+d)^2 f(z)$$
for all
$\pmatrix{ a & b\cr c& d\cr}\in\go N$
and which ``vanish at infinity.''
These form a finite dimensional space $S(N)$.

\newfoil
Each $f(z)$ may be expanded as a Fourier series
$$ \sum_{n=1}^\infty c_n q^n, \qquad q\seteq e^{2\pi i z}.$$
Eigenforms are series $\sum c_n q^n$
which satisfy
$$ \displaystyle
\sum_{n=1}^\infty {c_n\over n^s} = 
\prod_{p\notdivide N} (1-c_p p^{-s} + p^{1-2s})^{-1}
\cdot \prod_{p|N}(1-c_p p^{-s})^{-1}.$$

\newfoil
The Taniyama-Shimura conjecture states that for each elliptic
curve $A$ over~$\Q$ there is some eigenform $f=\sum c_nq^n$
($c_n\in\Z$)
satisfying
$$ a_p = c_p \hbox{ for all good primes $p$.}$$

Every curve $A$ has a discriminant $\Delta_A$ and
a conductor $N_A$, both of which are divisible precisely
by the bad primes. For the curve $A:y^2+y=x^3-x^2$,
$\Delta_A=-11$ and $N_A=11$.  For
$y^2+xy+y=x^3-x^2-3002x+63929$,
the discriminant is $6561000000=2^6 3^8 5^6$
and the
conductor is~90.

\newfoil
In more precise form, the T-S conjecture requires
eigenform $f$ for~$A$
to come from~$S(N_A)$.  

In the opposite
direction, for each eigenform
$f\in S(N)$
with integral coefficients, one can manufacture an~$A=A_f$
whose sequence $(a_p)$ is derived from the coefficients of~$f$.
The conductor of $A_f$ is $N$, whenever $f$ is a {\it newform\/}
in the sense that it
doesn't really belong to $S(M)$ for some smaller~$M|N$.
The T-S conjecture states that the construction can be reversed:
\newfoil
$$\hbox{Curves of conductor $N$ }
\buildrel\hbox{T-S Conjecture} \over\hugeleftrightarrow
 \hbox{ Newforms in $S(N)$ with $\Z$ coeffs.}$$

Since there is a construction in one direction,
it is tempting to try to prove the conjecture by ``counting.''
Although we know the dimension of~$S(N)$, it seems impossible
to know in advance the number of eigenforms with {\it integral
coefficients}.  Also, the elliptic curves have to be counted
up to {\it isogeny}.


\newfoil
When $N\le10$, $S(N)=0$ and it is possible to check
that there are no elliptic curves of conductor~$N$.
Similarly, when
$N=11$, there is exactly one newform, and one can find
three elliptic curves, all isogenous, which correspond to it.

To verify the conjecture for $N=11$ is to prove that there
are no further elliptic curves of
conductor~11.  This was done in the mid-1970s by four
authors who sweated to do it.
Nowadays, a technique of Faltings and Serre makes
fast work of this.

\newfoil
One possible way of making the counting easier is to
include all newforms, not just those with integral
coefficients.  Instead of elliptic curves, one has to
take abelian varieties of arbitrary dimension $d\ge1$.
If $f=\sum c_n q^n$ is an eigenform, then the $c_n$
are all algebraic integers, and the field they generate
$$ E = \Q(\ldots, c_n, \ldots)$$
has finite degree over~$\Q$.  

\newfoil
Shimura associates to~$f$
an abelian variety $A_f$ over~$\Q$ whose dimension is
the degree of~$E$ over~$\Q$.  The abelian variety $A_f$ has
extra structure: an action of~$E$ up to isogeny.

The T-S conjecture admits a natural generalization
$$
\hbox{Newforms } \hugeleftrightarrow
\hbox{ Abelian vars.\ with extra structure.}$$

\newfoil
This seems to ``help,'' because all newforms get considered.
The new complication is that $A_f$ corresponds not only to~$f$,
but also to the conjugates of~$f$.

For example, take $N=23$.  Then $S(N)$ has dimension~two.
There are two newforms, with coefficients in $\Z[{1+\sqrt5\over2}]$.
Changing the sign of~$\sqrt5$ moves between
the two forms.  
Up to isogeny,
there is exactly one abelian variety in the
correspondence.  Known as $J_0(23)$,
it is a simple two dimensional abelian variety.

\foilhead{Serre's Conjecture}
An outgrowth of Frey's elliptic curve
construction was a close
analysis of the conjectured relation between elliptic
curves and modular forms.  Serre, who was one of the analyzers,
was led to make a new conjecture linking 
$$
\hbox{Newforms mod~$\ell$ }\hugeleftrightarrow
\hbox{ Galois representations mod~$\ell$.}$$

\newfoil
The new conjecture\emrule actually a series of conjectures\emrule
has to do with two-dimensional representations of the
Galois group of~$\Q$: it relates them to modular forms.

Two dimensional representations of the type envisaged
arise naturally from elliptic curves over~$\Q$ and more
generally from the abelian varieties with ``extra structure''
which appeared above.  Modular forms give rise to such
abelian varieties, and thereby to representations.  The
conjecture proposes a suitable converse.

\newfoil
In 1987, Serre conjectured a
series of
refined statements with striking consequences: 
the Taniyama-Shimura conjecture,
Fermat's Last Theorem,
and a host
of other results.  

My work showing that T-S implies
Fermat, and subsequent work of other authors, reduced the
``refined'' Serre conjecture to a single ``coarse'' statement
which proposes
a necessary and sufficient condition for
irreducible representations
to be modular.  %This will be explained below.

\newfoil 
Since  Serre's conjecture
does not involve elliptic curves, we can say that it
is different {\it in spirit\/} from the T-S conjecture.
Hence it might turn out to shed light on the T-S
conjecture and other related conjectures.
\newfoil
To introduce the representations which come into play, we
review the case of elliptic curves.  If $A$ is an elliptic
curve over~$\C$, we can write it as a complex torus $\C/L$.
The lattice $L\approx\Z^2$ appears canonically as
$H_1(A(\C),\Z)$.  

In studies of elliptic curves, the construction $A\mapsto L$
seems to play a central role.
\newfoil

In the 1940s and 1950s, A.~Weil 
studied elliptic curves and abelian varieties over finite
fields.  He needed a replacement for~$L$
which
did not rely on the topology of~$\C$.  He noticed that $L/nL=
{1\over n}L/L$ has the %algebraic 
definition
$$A[n]\seteq \{\,P\in A(\C)\,|\, nP=O\,\}.$$
The collection of these groups can give the same information
as~$L$.

In practice, one studies $L/nL$ when $n$
is a prime~$\to\infty$, or when $n=\ell^\nu$ runs through
the powers of a single prime.
\newfoil
Weil used the definition of~$A[n]$ with $\C$ replaced by
the algebraic closure of a finite field and $n$ prime to
the characteristic of this field.

However, mathematicians including Mumford, 
Serre, Taniyama and Tate realized that the construction
$$A\mapsto A[n]$$ is fruitful in characteristic~0, for example
when $A$ is defined over~$\Q$.
\newfoil
In this case, $A[n]$ acquires an action of the
Galois group $\GalQ$.  The points in~$A[n]$, which have algebraic
coordinates, are permuted around by~$\GalQ$; it
acts on~$A[n]$ by group automorphisms:
$$ \rho_{A,n}\:\GalQ\to\Aut(A[n]).$$

Concretely, the image of~$\rho_{A,n}$ is a 
finite Galois group $\Gal(K/\Q)$ embedded in the matrix
group $$\Aut(A[n])=\GL{\Z/n\Z}.$$
\newfoil
The arithmetic of~$A$ emerges from the~$\rho_{A,n}$
because the numbers $a_p$ mod~$n$
(for good primes $p\notdivide N$) can be seen as matrix
traces of distinguished (``Frobenius'')
elements $\sigma_p$ in the image of~$\rho_{A,n}$.

Qualitatively: if you know the $\rho_{A,n}$ well enough
for lots of~$n$ (e.g., whenever $n=\ell$
is prime), then
you know the numbers $a_p$ (because you know them mod~$n$
for many~$n$) and you
can decide whether or not $A$ is a
modular elliptic curve.
\newfoil

Galois representations are attached to newforms~$f$
(on groups somewhat more general than $\go N$ and of
weights $k\ge2$).  If $f=\sum c_nq^n$ is such a newform,
and if $E$ is the field generated by the~$a_n$, then there's
a representation $\rho_{f,\lambda}$ for each prime (ideal)
$\lambda$ in the ring of integers of~$E$:
$$\rho_{f,\lambda}\:\GalQ\to\GL{\F_\lambda},$$
where $\F_\lambda$ is the residue field of~$\lambda$.
The $\rho_{f,\lambda}$ can be made from ``division points''
of abelian varieties with extra structure, so they are like
$\rho_{A,\ell}$'s.
\newfoil
The connection between $f$ and the $\rho_{f,\lambda}$s is given
by the congruence
$$\tr\left(\rho_{f,\lambda}(\sigma_p)\right)\equiv 
   a_p\quad \hbox{mod }\lambda,$$
which is valid for all $p$ not dividing the product of the
level of~$f$ with the prime number associated to~$\lambda$.

$\bullet$
The {\it determinant\/} of~$\rho_{f,\lambda}$ is easy
to calculate.  This determinant takes the value~$-1$
on ``complex conjugation'' $c\in\GalQ$.

$\bullet$
The representation $\rho_{f,\lambda}$ is typically
irreducible, so we stick to this case.
\newfoil
Serre's conjecture, in its coarse form, states that an
irreducible %representation
$$\rho\:\GalQ\to\GL{\F}$$
over a finite field is isomorphic to a~$\rho_{f,\lambda}$
if and only if $\rho(c)=-1$.

The coarse form has been shown equivalent to a refined
form of the conjecture\emrule it's this which implies
Fermat, Taniyama-Shimura and more.

The conjecture doesn't involve any ``geometry'' in its
statement\emrule no abelian varieties, no elliptic curves.

\newfoil
I think that Serre's conjecture is fundamental, and may
ultimately be the way to prove that elliptic curves are
modular.  But for the moment, no one seems to know how
to manufacture a form~$f$, given a representation~$\rho$.

We've surely made ``Progress in Mathematics'' with
Serre's conjecture, but the big payoff is not yet
in sight.
\newfoil
This brings us to Wiles's idea for proving T-S.
This idea is ``orthogonal'' to the Serre conjecture method.
Start with an elliptic
curve $A$ over~$\Q$.  Because the finite
group $\GL{\F_3}$ is {\it solvable}, it is possible to
show that $\rho_{A,3}$ is of the form $\rho_{f,\lambda}$.
This is already a highly non-trivial theorem of Langlands
and Tunnell.
\newfoil
Wiles tries to show that all $\rho_{A,3^\nu}$
come from modular forms by a sort of bootstrapping
process.  Using the language of deformations, Wiles
produces a surjection of $\Z_3$-algebras $\phi\:R\to\T$
and shows that the $\rho_{A,3^\nu}$ are modular if
$\phi$ is an {\it isomorphism}.


Here $\T$ is a subtly defined Hecke ring and $R$ is a
universal deformation ring that classifies 
modular-looking
liftings
of~$\rho_{A,3}$.  
The definitions of~$\T$ and~$R$ are due to Wiles and
to Ramakrishna, respectively.
\newfoil
\noindent
Wiles has exhibited several interrelated criteria, each 
practical
enough
to prove that $\phi$ is an isomorphism
in many cases.  

Using his criteria,
Wiles has proved the T-S
conjecture for several infinite classes of elliptic curves\emrule
this is a first.
\newfoil
One criterion involves a Selmer group for the
symmetric squares of the representations $\rho_{A,n}$.
This group is analogous to classical ideal class groups.
Wiles believed
that he could verify this criterion by adapting the
technique of M.~Flach's recent
Cambridge PhD thesis.

It now appears that the construction 
of Flach
does not
generalize as easily as Wiles expected.  On the other
hand, some people feel that Flach's thesis alone contains
information that has yet to be ``mined.''
\newfoil
Wiles gave the last plenary
talk at the ICM in Zurich last week.  According to reports,
Wiles likened our current 
state of knowledge
of the symmetric
square Selmer group to 
the state of knowledge early in the last century
when
Jacobi
studied the class group $C_k$ of an imaginary quadratic
field~$k$.  Jacobi 
wrote down an integer $h_k$ which he conjectured to be
the order of the group.  He could prove only that ${h_k}$
annihilated all elements of~$C_k$.
Six years later, Dirichlet proved that
$h_k$ is indeed the order of~$C_k$.
\bye