In this post we shall sketch the reduction of Fermat’s Last Theorem to Wiles’ Theorem that every semistable elliptic curve over is modular. I believe the ideas are due mainly to Frey, with a helpful big black box courtesy of Serre and Ribet. My main reference is the article of Stephens from the 1995 conference.
Suppose, for a fixed prime , that there are integers with , and, assuming wlog that and is even, form the elliptic curve
Then, by a direct computation, is semistable with minimal discriminant and conductor .
Our strategy is to study the representation coming from the action of Galois on the -torsion points of over (note that this is the same as in the previous paragraph).
In general, such Galois representations can be restricted to , considered as the decomposition group of the prime in (that this is only defined up to conjugacy, determined by a choice of embedding is not important for us – we’ve already picked a basis for the torsion points anyway). The typical `good’ local behaviour at these primes are, for , that be unramified at (in the sense that the absolute inertia group acts trivially), and for a more general rather more technical condition which we describe by saying that is flat at .
Let us (nonstandardly) call a prime bad (for a representation ) if and is ramified at , or and fails to be flat at . The important fact we need is that bad primes for such representations coming from elliptic curves are all flagged up by the minimal discriminant.
Proposition 1 For a semistable elliptic curve, is a bad prime for iff does not divide .
In particular, this implies for our curve that the only bad prime for is . People suspected that this possibility of such an `arithmetically simple’ representation coming from an elliptic curve should be pretty unlikely.
Now, as well as these representations on coming from torsion points on an elliptic curve, it is possible to construct similar representations coming from certain modular forms. Indeed, from a newform of weight 2, whose Fourier coefficients and character are rational, it is possible to form a representation . Serre made a series of conjectures about which representations could be obtained from newforms, one of which was the following result, a hard theorem proved by Ribet.
Theorem 2 (Serre’s Epsilon Conjecture) If is a weight two newform of conductor and is absolutely irreducible, then letting be the factor of obtained by throwing away all the non-bad primes which divide at most once, it is possible to find a different weight two newform of conductor such that .
Finally, we need some results about modular curves. In fact, we just need the following basic fact. If is a modular curve, then the representation is equal to for some newform of conductor .
Proposition 3 The curve above (constructed from a contradiction of Fermat’s Last Theorem) is not modular.
First, one shows that defined above is absolutely irreducible. Now, if it were modular, would be isomorphic to a representation coming from a newform of conductor . But looking at and the fact that is the only bad prime of , Ribet’s theorem implies that there is a newform of weight and conductor such that . But the space has dimension , by the basic theory of modular forms, so no such newform exists.
Having proved proposition 3, recalling that was a semistable elliptic curve over , Wiles’ theorem that every semistable elliptic curve over is modular gives a contradiction and thus proves that there are no integers with , .