Why Modularity implies Fermat’s Last Theorem

In this post we shall sketch the reduction of Fermat’s Last Theorem to Wiles’ Theorem that every semistable elliptic curve over ${{\mathbb Q}}$ 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 ${p \geq 5}$ , that there are integers ${a,b,c}$ with ${a^p+b^p+c^p = 0}$ , ${abc \not=0}$ and, assuming wlog that ${a \equiv 3 \mod 4}$ and ${b}$ is even, form the elliptic curve

$\displaystyle E_0: y^2 = x(x-a^p)(x+b^p).$

Then, by a direct computation, ${E_0}$ is semistable with minimal discriminant ${\Delta_{E_0} = 2^{-8} (abc)^{2p}}$ and conductor ${N_{E_0}=\prod_{l|abc} l}$ .

Our strategy is to study the representation ${\rho_E: G_{\mathbb Q} \rightarrow GL_2(\mathbb{F}_p)}$ coming from the action of Galois on the ${p}$ -torsion points of ${E}$ over ${\bar{{\mathbb Q}}}$ (note that this is the same ${p}$ as in the previous paragraph).

In general, such Galois representations can be restricted to ${G_{{\mathbb Q}_l}}$ , considered as the decomposition group of the prime ${l}$ in ${G_{\mathbb Q}}$ (that this is only defined up to conjugacy, determined by a choice of embedding ${\bar{{\mathbb Q}} \rightarrow \bar{{\mathbb Q}}_l}$ 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 ${l \not= p}$ , that ${\rho_E}$ be unramified at ${l}$ (in the sense that the absolute inertia group acts trivially), and for ${l=p}$ a more general rather more technical condition which we describe by saying that ${\rho_E}$ is flat at ${p}$ .

Let us (nonstandardly) call a prime ${l}$ bad (for a representation ${\rho_E}$ ) if ${l \not= p}$ and ${\rho_E}$ is ramified at ${l}$ , or ${l=p}$ and ${\rho_E}$ fails to be flat at ${p}$ . 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 ${E}$ a semistable elliptic curve, ${l}$ is a bad prime for ${\rho_E}$ iff ${p}$ does not divide ${ord_l(\Delta_E)}$ .

In particular, this implies for our curve ${E_0}$ that the only bad prime for ${\rho_{E_0}}$ is ${2}$ . 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 ${\mathbb{F}_p}$ coming from torsion points on an elliptic curve, it is possible to construct similar representations coming from certain modular forms. Indeed, from a newform ${f}$ of weight 2, whose Fourier coefficients and character are rational, it is possible to form a representation ${\rho_f: G_{\mathbb Q} \rightarrow GL_2(\mathbb{F}_p)}$ . 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 ${f}$ is a weight two newform of conductor ${N}$ and ${\rho_f}$ is absolutely irreducible, then letting ${N'}$ be the factor of ${N}$ obtained by throwing away all the non-bad primes which divide ${N}$ at most once, it is possible to find a different weight two newform ${g}$ of conductor ${N'}$ such that ${\rho_f \cong \rho_g}$ .

Finally, we need some results about modular curves. In fact, we just need the following basic fact. If ${E}$ is a modular curve, then the representation ${\rho_E}$ is equal to ${\rho_f}$ for some newform ${f}$ of conductor ${N_E}$ .

Proposition 3 The curve ${E_0}$ above (constructed from a contradiction of Fermat’s Last Theorem) is not modular.

First, one shows that ${\rho_{E_0}}$ defined above is absolutely irreducible. Now, if ${E_0}$ it were modular, ${\rho_{E_0}}$ would be isomorphic to a representation ${\rho_f}$ coming from a newform of conductor ${N_{E_0}}$ . But looking at ${N_{E_0}}$ and the fact that ${2}$ is the only bad prime of ${\rho_f}$ , Ribet’s theorem implies that there is a newform ${g}$ of weight ${2}$ and conductor ${2}$ such that ${\rho_f \cong \rho_g}$ . But the space ${S_2(\Gamma_0(2))}$ has dimension ${0}$ , by the basic theory of modular forms, so no such newform exists.

Having proved proposition 3, recalling that ${E_0}$ was a semistable elliptic curve over ${{\mathbb Q}}$ , Wiles’ theorem that every semistable elliptic curve over ${{\mathbb Q}}$ is modular gives a contradiction and thus proves that there are no integers ${a,b,c}$ with ${a^p+b^p+c^p=0}$ , ${abc \not= 0}$ .