3.1. Overview🔗

In [??] we reduced FLT, modulo a hard theorem from the 1970s, to theorem Theorem 2.7.2, the assertion that p-torsion in the Frey curve is reducible. In this chapter we deduce this assertion from three more complex claims about hardly ramified Galois representations. It is relatively straightforward to reduce one of these three claims to a result of Fontaine proved in the 1980s in his paper on the nonexistence of nontrivial abelian schemes over \mathbf{Z}. The other two claims lie deeper, and their proofs use techniques initially developed by Wiles in the 1990s.

3.2. Hardly Ramified Representations🔗

Let (a,b,c,p) be a Frey package, so in particular p\geq5 is prime and a^p+b^p=c^p, let E be the corresponding Frey curve over \Q, and let \rho : \GQ \to \Aut(E(\Qbar)[p]) be the 2-dimensional Galois representation on the p-torsion of E. Recall that our goal is to prove that \rho is reducible.

What we need to leverage is the fact that \rho has very little ramification. To give a toy example before we start: if K is a number field, that is, a finite extension of \Q, and if the extension K / \Q is unramified at all primes, then an old theorem of Minkowski tells us that K = \Q. We want to prove a theorem in a similar vein, namely that if a 2-dimensional mod p Galois representation is hardly ramified, then it is reducible. Below, we give a precise definition of what it means for a continuous 2-dimensional representation \GQ \to \GL_2(R) to be hardly ramified. Before we do that, we need to say precisely which topological rings R we will allow. We say that a topological ring is pro-Artinian if it is a projective limit of Artin local rings each equipped with the discrete topology, and if it has the projective limit topology. We are only concerned with local pro-Artinian rings with finite residue field; such things can be checked to be the same thing as topological local rings with finite residue field whose underlying topological space is profinite, and such that additive translates of open ideals form a basis for the topology. Let us call such rings coefficient rings for now.

We make some remarks to orient the reader.

• Any complete local Noetherian ring with finite residue field is a coefficient ring, if the ring is equipped with the \m-adic topology where \m is the maximal ideal. In this case, all powers of \m are open.

• In particular finite fields, and integer rings of finite extensions of \Q_p, are coefficient rings.

• If R is a coefficient ring then R is isomorphic to the projective limit of the finite rings R/I as I runs over the open ideals of R.

• A non-Noetherian example of a coefficient ring is the projective limit over n of the rings \Z/p\Z[\varepsilon_1,\ldots,\varepsilon_n]/(\forall i,j,\varepsilon_i\varepsilon_j=0); these rings are convenient to include as coefficient rings for technical reasons; they make representability theorems easier.

• The category of coefficient rings is equivalent to the pro-category of the category of finite local rings.

• A coefficient ring is pseudocompact in the sense of Grothendieck. A pseudocompact local ring is however a more general concept as such a thing may have an infinite residue field and would thus not be profinite.

• If R is a coefficient ring with residue field of characteristic \ell, then there is a unique continuous map \Z_\ell\to R. Indeed, it suffices to prove that there is a unique continuous map \Z_\ell\to R/I for each open ideal I, but R/I is a finite local ring with residue field of characteristic \ell. R/I is hence Artinian, so some power of the maximal ideal is zero by Nakayama. This means that \ell^N=0 for some sufficiently large N, and hence R/I is a \Z/\ell^N\Z-algebra and thus admits admits a unique map from \Z_\ell.

• It will be more convenient to fix once and for all the integer \mathcal{O} in a finite extension of \Q_\ell and consider coefficient \mathcal{O}-algebras, namely coefficient rings R equipped with a continuous map \mathcal{O}\to R which is a local homomorphism inducing an isomorphism on residue fields.

Because a coefficient ring R with residue field of characteristic \ell is naturally a \mathbf{Z}_\ell-algebra, we can talk about the \ell-adic cyclotomic character \GQ\to R^\times. We are now ready to define hardly ramified representations.

A well-known result, which basically goes back to Frey, is the following:

Note that irreducibility and absolute irreducibility for hardly ramified mod \ell representations are the same, because our assumptions that \ell\geq3 and that the determinant is cyclotomic imply that the image of complex conjugation has distinct eigenvalues defined over the ground field.

Note that this (deep) claim is a consequence of Serre's conjecture (Serre, 1987), now a theorem of Khare and Wintenberger (Khare and Wintenberger, 2009), and indeed we shall use methods introduced by Khare and Wintenberger to prove this special case of Serre's conjecture. Given this result, we can deduce theorem Theorem 2.7.2 (which we restate here) easily:

Our job of reducing FLT to theorems of the 1980s is hence reduced to proving theorem Theorem 3.2.3.