For a quick definition of many of the terms used here, you may refer to the Glossary.
External references for this section: [Gou], [Rib], [Rih]
Suppose there were a nontrivial solution of the Fermat equation for some number n, i. e. nonzero integers a, b, c, n such that
Then we recall that around 1982 Frey called attention to the elliptic curve
Call this curve E. Frey noted it had some very unusual properties, and guessed it might be so unusual it could not actually exist.
To begin with, various routine calculations enable us to make some useful
simplifying assumptions, without loss of generality. For instance, n may
be supposed to be prime and 
5. b can be assumed to be
even, a 
3 (mod 4), and c
1 mod 4. a, b, and c can be assumed
relatively prime.
The "minimal discriminant" of E, can be computed to be
- a power of 2 times a perfect prime power. One
unusual thing about E is how large the discriminant is.
The conductor is a product of primes at which E has bad reduction, which
is the same as the set of primes that divide the minimal discriminant. However,
the exact power of each prime occurring in the conductor depends on
what type of singularity the curve possesses modulo the primes of
bad reduction. The definition of the conductor provides that p divides
the conductor only to the first power if
x(x-a
)(x+b) has only a
double root rather than a triple root mod p. Now, any prime can divide
only a or b but not both, since otherwise it would also divide c, and we
have assumed a, b, and c are relatively prime. Hence the the polynomial
will have the form x
(x+d) mod p,
where (p,d) = 1. Hence there is only at most
a double root modulo any prime, and therefore the conductor
is square-free.
In other words, E is semistable.
There are other odd things about E, which have to do with specific properties of its Galois representations. Because of these, Ribet's results allow us to conclude that E cannot be modular.
After Frey drew attention to the unusual elliptic curve which would result if there were actually a nontrivial solution to the Fermat equation, Jean-Pierre Serre (who has made many contributions to modern number theory and algebraic geometry) formulated various conjectures which, sometimes alone and sometimes together with the Taniyama-Shimura conjecture, could be used to prove Fermat's Last Theorem.
Kenneth Ribet quickly found a way to prove one of these conjectures. The conjecture itself doesn't really talk about either Frey curves or FLT. Instead, it simply states that if the Galois representation associated with an elliptic curve E has certain properties, then E cannot be modular. Specifically, it cannot be modular in the sense that there exists a modular form which gives rise to the same Galois representation.
We need to introduce a little additional notation and terminology
to explain this more
precisely. Let S(N) be the (vector) space of cusp forms for
(N) of
weight 2. "Classical" theory of modular forms shows that S(N) can be
identified with the space of "holomorphic differentials" on the
Riemann surface X
(N). Furthermore, the dimension of
S(N) is finite
and equal to the "genus" of X(N). "Genus" is a
standard topological
property of surfaces, which is intuitively the number of holes in the
surface. (E. g. a torus, such as an elliptic curve, has genus 1.)
But there are relatively simple explicit formulas for the genus of
X(N). These formulas, developed long ago by Hurwitz
in the theory of
Riemann surfaces, involve the index of (N) in G.
A fact of crucial
importance is that for N <.; 11, the genus of X(N),
and hence the
dimension of S(N), is zero. In other words, S(N) contains only the
constant form 0 in that case. We shall use this fact about S(2) very
soon.
There are certain operators called Hecke operators, after Erich Hecke,
on spaces of modular forms, and for the subspace S(N) in particular,
since they preserve the weight of a form.
Hecke operators can be defined concretely in various
ways. There is a Hecke operator T(n) for all n 1.
There are formulas that relate T(n) for composite n to T(p) where p
is a prime dividing n, so T(p) for prime p determine all T(n).
All T(n) are linear operators on S(N). If there is an f in S(N) that
is a simultaneous eigenvector of all T(n),
i. e. T(n)(f) = 
(n)f, where (n)
C,
f is called an eigenform.
(Nontrivial eigenforms need not exist, e. g. if S(N) has dimension 0.)
f is said to be normalized if its leading Fourier series coefficient is 1.
In that case, the eigenvalues (n) turn out to be
the Fourier series coefficients in the expansion
It can be shown that if f(z) is a cusp form which is a normalized eigenfunction for all T(p), then there is an Euler product decomposition for the L-function L(f,s). This is obviously of great technical usefulness in relating L-functions of forms and those of elliptic curves (which are Euler products by definition).
If f S(N) is a normalized eigenform of all
Hecke operators,
it can in fact be shown that the coefficients in the Fourier expansion
are all algebraic numbers and that they generate a finite extension K
of Q.
Prime ideals of the ring of integers of K are
the analogues of prime numbers of Q. In the case that f is a normalized
eigenform it is possible to carry out the construction of a Galois
representation 
(f,
) of
Gal(
/Q)
for any prime ideal of the ring of integers of K.
At last we can describe what Ribet proved. Suppose E is a semistable elliptic
curve with conductor N and that its associated Galois representation
(E,p) for some prime p has certain properties.
Suppose 2 divides N (which is true for Frey curves).
If E is modular, then there is a normalized eigenform f and a prime ideal
lying over p (i. e. one of the prime factors of p in
the extension field
generated by the Fourier coefficients of f) such that the Galois representation
(f,) is (E,p).
Ribet showed that
it is possible to find an odd prime q 
p which divides N
such that there is another f' S(N/q), and a
corresponding prime ideal ' of the ring of integers
in the field generated by the coefficients of f' such that
(f',') gives essentially
the same Galois representation. This is known as the "level lowering"
conjecture since it asserts that under the right conditions there is an
eigenform of a lower level that gives essentially the same representation.
But this process can be repeated as long as N has any odd prime factors. It is important that the curve E is semistable so that N is square-free. This means that all odd prime factors of N can been eliminated, so there must be a nontrivial eigenform of level 2, i. e. in S(2), that gives essentially the same Galois representation. And that is a contradiction, since S(2) has dimension 0, hence contains no non-trivial forms. The contradiction means that E can't be modular.
Now we invoke the "unusual" properties of the Frey curve resulting from a solution of FLT. These properties allow it to be shown that the associated Galois representation has the properties required to apply Ribet's result. Hence the Frey curve can't be modular.
But the Frey curve is semistable, so the semistable case of the Taniyama-Shimura conjecture, which Wiles proved, implies the curve is modular. This contradiction means that the assumption of the existence of a nontrivial solution of the Fermat equation must be wrong, and so FLT is proved.
Not very surprisingly (since it was such hard work), the proof is quite technical. However, the outline of it is relatively simple. In the following, we assume that E is a semistable elliptic curve with conductor N. We have to prove E is modular.
We know we can construct a Galois representation
(E,p
):
G->GL
(Z
)
for any prime p.
To show that E is modular, we have to show this representation is modular
in a suitable sense. The wonderful thing is, this needs to be done for
only one prime p, and we can "shop around" for whatever prime is easiest
to work with.
To show (E,p)
is modular involves finding a normalized eigenform f in S(N)
with appropiate properties. The properties required are that the
eigenvalues of f, which are its Fourier series coefficients, should be
congruent mod q to
trace((E,p)
(
))
for all but a finite number of
prime q. (
G is the "Frobenius element".)
We know that the trace is, for q prime to pN, the coefficient
a = q + 1 - #(E(F))
of the Dirichlet series of L(E,s).
The longest and hardest part of Wiles' work was to prove a general
result which is roughly that if (E,p)
is modular then so is (E,p).
In other words, to show that E is modular, it is
actually sufficient just to show that
(E,p): G->GL(Z/pZ)
is modular. This is called the "modular lifting problem".
The problem boils down to assuming that (E,p) is modular and attempting
to "lift" the representation to
(E,p). This is done mainly by
working with the theory of representations as much as possible, without
specific reference to the curve E. The proof uses a concept called
"deformation", which suggests intuitively what goes on in the process of
lifting.
The outcome of this part of Wiles work is:
Theorem: Suppose that E is a semistable elliptic curve over Q. Let p be an odd prime. Assume that the representation
At this point, all we have to do is find a single prime p such that
(E,p)
is irreducible and modular. But Langlands and Tunnell had already proven
in 1980-81 that (E,3) is modular.
Unfortunately, this isn't quite enough. If (E,3) is
irreducible, we are done. But otherwise, one more step is required. So
suppose (E,3) is reducible. Wiles then considered
(E,5). That may be either reducible or irreducible as
well. If it is reducible, Wiles proved directly that E is modular.
So the last case is if (E,5) is irreducible. Wiles
showed that there is another semistable curve E' such that (E',3) is irreducible, and hence E' is modular by the
above theorem. But Wiles could also arrange that the representations
(E',5) and (E,5) are isomorphic.
Hence (E,5) is irreducible and modular, so E is
modular by the theorem.
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Copyright © 1996 by Charles Daney, All Rights Reserved
Last updated: March 13, 1996
