3 Reducibility of p-torsion of the Frey curve
3.1 Overview
In chapter 2 we reduced FLT, modulo a hard theorem from the 1970s, to Theorem 2.8, 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 claims to a result of Fontaine proved in the 1980s in his paper on the nonexistence of nontrivial abelian schemes over \(\mathbb {Z}\). 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 \(p\geq 5\) is prime and \(a^p+b^p=c^p\)), let \(E\) be the corresponding Frey curve over \(\mathbb {Q}\), and let \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{Aut}(E(\overline{\mathbb {Q}})[p])\) 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 (i.e., a finite extension of \(\mathbb {Q}\)) and if the extension \(K/\mathbb {Q}\) is unramified at all primes, then an old theorem of Minkowski tells us that \(K=\mathbb {Q}\). We want to prove a theorem in a similar vein; 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 2-dimensional representation \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(R)\) to be hardly ramified. Before we do that, we need to say precisely which rings \(R\) we will allow. We recall the following definition, due to Grothendieck:
A commutative topological ring \(R\) is pseudocompact if it satisfies the following properties:
If \(U\subseteq R\) is any open set containing 0, then there’s an open ideal \(I\subseteq U\).
If \(I\subseteq R\) is an open ideal, then \(R/I\) is Artinian.
The natural map \(R\to \lim _{\leftarrow } R/I\) is a topological and algebraic isomorphism, where \(I\) runs over the open ideals of \(R\). Here \(R/I\) has the discrete topology and the projective limit has the projective limit topology.
A coefficient ring is a local pseudocompact topological ring \(R\) with finite residue field.
Equivalently, a coefficient ring is a local profinite topological ring with finite residue field. Examples of coefficient rings include finite fields, and integer rings of finite extensions of \(\mathbb {Q}_p\). There are also non-Noetherian examples, for example the projective limit over \(n\) of the rings \(\mathbb {Z}/p\mathbb {Z}[\varepsilon _1,\ldots ,\varepsilon _n]/(\forall i,j,\varepsilon _i\varepsilon _j=0)\); these rings are convenient to include for technical reasons. If \(R\) is a coefficient ring with maximal ideal \(\mathfrak {m}\) and residue field \(k\) of characteristic \(\ell {\gt}0\), then for any open ideal \(I\) there is a ring homomorphism \(\mathbb {Z}/\ell ^n\mathbb {Z}\to R/I\) for all sufficiently large naturals \(n\), and thus a continuous ring homomorphism \(\mathbb {Z}_{\ell }\to R\). The \(\ell \)-adic cyclotomic character is a continuous representation \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \mathbb {Z}_{\ell }^\times \) and it thus induces a continuous representation \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to R^\times \) which we also refer to as the cyclotomic character.
Let \(R\) be a coefficient ring with finite residue field of characteristic \(\ell \geq 3\). Let \(V\) be a finite free \(R\)-module of rank 2, equipped with the product topology. A continuous representation \(\rho : \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_R(V)\) is said to be hardly ramified if it satisfies the following four conditions:
\(\det (\rho ):\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to R^\times \) is the cyclotomic character;
\(\rho \) is unramified outside \(2\ell \);
The restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) is reducible (more precisely, there is a short exact sequence \(0\to R\to V\to R\to 0\) which is stable under the action of \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\)) and the Galois action on the 1-dimensional quotient is an unramified representation of \(\operatorname{Gal}(\overline{\mathbb {Q}}_2/\mathbb {Q}_2)\) whose square is trivial;
The restriction of \(\rho \) to \(\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\) is flat, by which we mean that for all open ideals \(I\) of \(R\), the (finite image) representation \(\rho \) mod \(I:\operatorname{Gal}(\overline{\mathbb {Q}}_\ell /\mathbb {Q}_\ell )\to \operatorname{GL}_{R/I}(V/I)\) comes from a finite flat group scheme.
A well-known result, which basically goes back to Frey, is the following:
The \(\ell \)-torsion \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) in the Frey curve associated to a Frey package \((a,b,c,\ell )\) is hardly ramified.
This was well-known in the 1980s. A proof sketch is as follows. First note that \(\ell \geq 5{\gt}3\) by definition of a Frey package. Let \(\rho \) denote the Galois representation on the \(\ell \)-torsion of the Frey curve. The fact that \(\rho \) is 2-dimensional is Corollary III.6.4(b) of [ 11 ] , and the fact that its determinant is cyclotomic is Proposition III.8.3 of the same reference. These results hold for elliptic curves in general. The remaining claims are specific to the Frey curve and lie deeper. The fact that \(\rho \) is unramified outside \(2\ell \) is a consequence of (4.1.12) and (4.1.13) of [ 9 ] . The fact that \(\rho \) at 2 has an unramified 1-dimensional quotient of order at most 2 follows from the fact that the Frey curve is semistable at 2 (see (4.1.5) of [ 9 ] ) and the theory of the Tate curve. Finally, the claim that \(\rho \) is flat at \(\ell \) is Proposition 5 and (4.1.13) of [ 9 ] .
The key theorem about hardly ramified representations is the following.
If \(\ell \geq 3\) is a prime and \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) is hardly ramified, then \(\rho \) is reducible.
Note that this (deep) claim is a consequence of Serre’s conjecture [ 9 ] , now a theorem of Khare and Wintenberger [ 6 ] , 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 2.8 (which we restate here) easily:
If \(\overline{\rho }\) is the mod \(p\) Galois representation associated to a Frey package \((a,b,c,p)\) then \(\overline{\rho }\) is reducible.
Our job, in the first phase of this project, is hence reduced to proving Theorem 3.5.
3.2.1 Hardly ramified mod \(p\) representations are reducible
In this section we will state three theorems, from which Theorem 3.5 easily follows.
Firstly, we claim that an irreducible hardly ramified mod \(\ell \) representation lifts to an \(\ell \)-adic representation.
If \(\ell \geq 3\) is prime and \(\overline{\rho }:\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) is hardly ramified and irreducible, then there exists a finite extension \(K\) of \(\mathbb {Q}_\ell \) with integer ring \(\mathcal{O}\) and maximal ideal \(\mathfrak {m}\) and a hardly ramified representation \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathcal{O})\) whose reduction modulo \(\mathfrak {m}\) is isomorphic to \(\rho \).
Next we claim that a hardly ramified \(\ell \)-adic representation “spreads out” to a compatible family of hardly ramified \(q\)-adic representations for all primes \(q\).
If \(\ell \geq 3\) is prime, \(K\) is a finite extension of \(\mathbb {Q}_\ell \) with integers \(\mathcal{O}\) and if \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathcal{O})\) is a hardly ramified representation whose reduction is irreducible, then there exists a number field \(M\) and, for each finite place \(\mu \) of \(M\) of characteristic prime to 2, with completion \(M_\mu \) having integer ring \(R_\mu \), a hardly representation \(\rho _\mu :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(R_\mu )\), with the following properties:
There is some \(\lambda \mid \ell \) of \(M\) such that \(\rho _\lambda \cong \rho \), the isomorphism happening over some appropriate local field containing a copy of \(M_\lambda \) and a copy of \(K\);
If \(\mu _1\) and \(\mu _2\) are two finite places of \(M\) with odd characteristics \(m_1\) and \(m_2\), and if \(p\nmid 2m_1m_2\) is prime, then \(\rho _{\mu _1}\) and \(\rho _{\mu _2}\) are both unramified at \(p\) and the characteristic polynomials \(\rho _{\mu _1}(\operatorname{Frob}_p)\) and \(\rho _{\mu _2}(\operatorname{Frob}_p)\) lie in \(M[X]\) and are equal.
In particular, we can “move” from an irreducible hardly ramified mod \(\ell \) representation to a hardly ramified 3-adic representation. However, we can essentially completely classify the hardly ramified 3-adic Galois representations.
Suppose \(L/\mathbb {Q}_3\) is a finite extension, with integer ring \(\mathcal{O}_L\), and suppose \(\rho _3:\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathcal{O}_L)\) is hardly ramified. Then \(\rho _3^{ss}=1\oplus \chi _3\) where \(1\) is the trivial character and \(\chi _3\) is the 3-adic cyclotomic character.
Theorem 3.5 (if \(\ell \geq 3\) is a prime and \(\overline{\rho }:\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) is hardly ramified, then \(\overline{\rho }\) is reducible) is an easy consequence of these theorems, as we now show.
Assume for a contradiction that \(\overline{\rho }\) is irreducible. By theorem 3.7, \(\overline{\rho }\) lifts to a hardly ramified \(\ell \)-adic reprepresentation \(\rho \). By theorem 3.8, \(\rho \) is part of a compatible family of \(q\)-adic Galois representations. By theorem 3.9, any 3-adic member \(\rho _3\) of this family has semisimplification \(1\oplus \chi _3\) and in particular for \(p\nmid 6\) we have that the characteristic polynomial of \(\rho _3(\operatorname{Frob}_p)=(X-p)(X-1).\) But by compatibility of the family we know that for \(p\nmid 6\ell \) the characteristic polynomial of \(\rho (\operatorname{Frob}_p)\) is \((X-p)(X-1)\), and thus the characteristic polynomial of \(\overline{\rho }(\operatorname{Frob}_p)\) is \((X-p)(X-1)\). By the Brauer-Nesbitt theorem, \(\overline{\rho }\) is reducible, the contradiction we seek.
What remains then (modulo several results which were known in the 1980s), is to prove the three theorems 3.7, 3.8 and 3.9. By far the easiest is theorem 3.9; this follows from old estimates of Fontaine (ultimately relying on bounds for root discriminants due to Odlyzko and Poitou), originally developed to prove that there was no nontrivial abelian scheme over \(\mathbb {Z}\). The other two theorems are deeper, and both use modern variants of Wiles’ \(R=T\) machinery.
The story stops here for now; as we write more LaTeX it will become clear that here the proof basically bifurcates, with different techniques used to prove these three theorems.