费马定理及其证明

The Proof of Fermat's Last Theorem

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]

Contents:

 Frey curves

 Proof of Fermat's Last Theorem from the Taniyama-Shimura conjecture

 Proof of the semistable case of the Taniyama-Shimura conjecture

Frey curves

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.

Proof of Fermat's Last Theorem from the

Taniyama-Shimura conjecture

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,