The Shimura-Taniyama conjecture (d’après Wiles)

Henri Darmon

May 23, 2000

Contents

1 Preliminaries 4 1.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Wiles’ strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Representability of AD_{Σ} and the ring R_{Σ} . . . . . . . . . . . . 12 1.4 Construction of a Hecke ring T_{Σ} . . . . . . . . . . . . . . . . 12 1.5 The map R_{Σ} ⟶ T_{Σ} . . . . . . . . . . . . . . . . . . . . . . . 15

2 Wiles’ isomorphism criterion 17

2.1 The invariants Φ and η . . . . . . . . . . . . . . . . . . . . . . 18 2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 Some examples . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Basic properties of Φ_{A} and η_{A} . . . . . . . . . . . . . . . . . . 20 2.3 Complete intersections . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Isomorphism theorems . . . . . . . . . . . . . . . . . . . . . . 27 2.5 A resolution lemma . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 A criterion for complete intersections . . . . . . . . . . . . . . 31 2.7 Proof of Wiles’ isomorphism criterion . . . . . . . . . . . . . . 32 2.8 The relative invariant η_{T′/T} . . . . . . . . . . . . . . . . . . . 33 2.9 Interpretation of η_{T′/T} in the Gorenstein case . . . . . . . . . 34 3 Interpretation of Φ_{R_{Σ}} and η_{T_{Σ}} 35

3.1 Interpretation of Φ_{R_{Σ}} . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 A formula for η_{T_{Σ}} . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 A map T_{Σ′} ⟶ T_{Σ} . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Computing η_{T′/T} . . . . . . . . . . . . . . . . . . . . . 41

3.3 Relation with the Bloch-Kato conjectures . . . . . . . . . . . . 44

4 Proof of the inequality #Φ_{R_{Σ}} ≤ #(Z_{5}/η_{T_{Σ}}) 45 4.1 Reduction to the case Σ = ∅ . . . . . . . . . . . . . . . . . . . 45 4.2 Proof of the inequality for Σ = ∅ . . . . . . . . . . . . . . . . 47

4.2.1 The group S_{Σ}^{mod}(Q, A) . . . . . . . . . . . . . . . . . . 49 4.2.2 Local Tate duality . . . . . . . . . . . . . . . . . . . . 51 4.2.3 Bounding S_{∅}(Q, A_{5}^{∗}) . . . . . . . . . . . . . . . . . . . 53 4.2.4 Bounding S_{∅}(Q, A_{5}) . . . . . . . . . . . . . . . . . . . 55

5 Wiles’ general strategy 58 The conjecture of Shimura and Taniyama that every elliptic curve over Q is modular has been described as a “Himalayan peak” [Mu] whose conquest is one of the great challenges of mathematics. Until recently, the summit was viewed as impregnable. Then, in June 1993, Andrew Wiles stunned the world by mapping out a means of ascent to those lofty peaks.

Wiles’ methods can be used to prove that there are infinitely many elliptic curves over Q with distinct j-invariants that are modular. In this survey we will give a fairly complete proof of this result. Our goal here has been to take as many short-cuts as possible, guiding the reader along a beginner’s trail which avoids the treacherous slopes of Euler Systems and p-adic Hodge theory, but still leads to a vantage point from which some of Wiles’ beautiful achievement can be contemplated.

In the general case, Wiles reduces the Shimura-Taniyama conjecture to a conjectural upper bound on the Selmer groups associated to certain motives of rank 3. He also gives a method for establishing this upper bound, assuming that certain rings of Hecke operators are local complete intersections. The fundamental work of Matthias Flach suggests that a proof of the upper bound might also be possible by using the ideas on “Euler systems” introduced by V.A. Kolyvagin. “Euler system” calculations have been performed before in many different settings, always with spectacular results, most notably by Thaine [Th], Kolyvagin [Ko], Rubin [Ru], Nekovar [Ne], and Flach [Fl].

In any case, Wiles’ reduction removes much of the mystery behind the Shimura-Taniyama conjecture – and, to the optimist, suggests that a proof must be within reach! We will very briefly indicate some of the features of Wiles’ general strategy at the end.

We do not say anything about the well-known connection between the Shimura-Taniyama conjecture and Fermat’s Last Theorem, which is amply documented, for example in the excellent survey articles [Co], [Gou], [Ri4], [RH], [RS].

This paper is entirely expository. It owes everything to the ideas of Wiles, and to a course he and his students gave at Princeton University in the Spring semester 1994. I would also like to thank Brian Conrad, Fred Diamond, Ravi Ramakrishna, Ken Ribet, Anna Rio, Richard Taylor and Larry Washington for making their notes available to me, answering my questions, and making comments and corrections on earlier versions of this manuscript. Some notes of H.W. Lenstra [Le] on the ring theoretic aspects of Wiles work were also tremendously helpful in writing chapter 2.

1 Preliminaries

1.1 The main results

We say that j ∈ Q is modular if it is the j-invariant of a modular elliptic curve over Q. It follows from the work of Shimura that if j is modular, then every elliptic curve with invariant j is modular.

Theorem 1.1 (Wiles) There are infinitely many j ∈ Q which are modular.

This result represents a major watershed: before Wiles’ announcement, a proof seemed to be nowhere in sight.

If E is an elliptic curve, then we denote by E_{p} the group of p-division points of E; it is a two-dimensional F_{p}-vector space, endowed with a natural action of G_{Q} = Gal(Q̄/Q).

The infinite set of modular j-invariants can be obtained via the following more precise statement:

Theorem 1.2 (Wiles) Let X be a semistable modular elliptic curve, let d be the degree of a minimal modular parametrization X_{0}(N) ⟶ X, and let p be an odd prime such that p does not divide d and X_{p} is absolutely irreducible as a Galois module.

If E is an elliptic curve over Q such that

E_{p} ≃ X_{p} as Galois modules,

and E has good or semistable reduction at p, then the curve E is modular.

Remarks:

1. There are many examples of pairs (X, p) which satisfy the assumptions of the theorem, for example:

X = X_{0}(11), p any odd prime ≠ 5,

X = X_{0}(14), p > 3,

X = X_{0}(15) or X_{0}(17), p any odd prime, etc...

2. Let X(p)_{/Q} be the modular curve over Q which classifies pairs (E, ϕ) where E is an elliptic curve and ϕ is a monomorphism ϕ : Z/pZ×Z/pZ ⟶ E. The group scheme X_{p} can be used to define the twisted modular curve X′(p)_{/Q}, which classifies pairs (E, ϕ) with ϕ : X_{p} ↪ E is an injection. When p = 3 or p = 5, the curves X(p) and X′(p) are a union of p − 1 curves of genus 0.

(For example, the curve X(5) is related to the classical “icosahedral curve” studied by Klein.) The pair (X, ϕ : X_{p} ↪ X) then defines a rational point x on one of the components of X′(p)_{/Q}. This component is a curve of genus 0 having a Q-rational point, and hence is isomorphic to P_{1/Q}. The rational points in X′(p)(Q) which are close to x in the p-adic topology, give rise to an infinite set of elliptic curves E satisfying the hypothesis of the theorem. (In fact, they yield a family of modular j-invariants which are the values at certain rational arguments of a rational function of degree 12 when p = 3, and 60 when p = 5.) For p > 5, thm. 1.2 is not as interesting, since the curves E satisfying the hypothesis for a fixed X correspond to rational points on a twist of X(p), which has genus strictly greater than 1; by Faltings’ proof of the Mordell conjecture, there are only finitely many such E.

We will explain the proof of theorem 1.2 only in the special case where X = X_{0}(17) and p = 5. This allows us to lighten the exposition by avoiding a number of technical issues, while still being sufficient to prove thm. 1.1.

Theorem 1.3 Let E be an elliptic curve over Q which has good reduction at 5 and satisfies

E_{5} ≃ X_{0}(17)_{5} as Galois modules.

Then E is modular.

Remarks:

1. The curve X = X_{0}(17) is a curve of genus 1, and hence is a modular elliptic curve; the degree d of the modular parametrization is 1. Also, as we will see in lemma 1.4, the module X_{5} gives a two-dimensional mod 5 representation of G_{Q} = Gal(Q̄/Q) which is irreducible, and in fact, surjective. Hence, thm. 1.3 is a special case of thm. 1.2.

2. The statement of thm. 1.3 would also be true with 5 replaced by 3; working with 5 instead of 3 allows us to avoid certain slight technical difficulties. Note, however, that the prime 3 plays a crucial role in Wiles’ general strategy for proving the Shimura-Taniyama conjecture; cf. sec. 5.

From now on, let ρ̄_{0} be the mod 5 representation arising from the 5 division points of X_{0}(17),

ρ̄_{0} : G_{Q} ⟶ Aut (X_{5}) ≃ GL_{2}(F_{5}).

Lemma 1.4 The representation ρ̄_{0} satisfies the following properties:

1. The character det(ρ̄_{0}) is the cyclotomic character ϵ̄: G_{Q} ⟶ F_{5}^{∗} giving the action of G_{Q} on the 5th roots of unity.

2. (Local behaviour at 5). The decomposition group D_{5} at 5 acts on X_{5} by preserving a one-dimensional subspace, and the inertia group I_{5} acts trivially on the quotient:

ρ̄_{0}|_{D_{5}} ≃ ( \begin{matrix}χ̄_{1} & ∗ \\ 0 & χ̄_{2} \end{matrix} , ) ρ̄_{0}|_{I_{5}} ≃ ( \begin{matrix}ϵ̄ & ∗ \\ 0 & 1 \end{matrix} ) .

Furthermore, the unramified character χ̄_{2} is of order 4.

3. (Local behaviour at 17):

ρ̄_{0}|_{D_{17}} ≃ ( \begin{matrix}ϵ̄ & Ψ̄ \\ 0 & 1 \end{matrix} ) ,

and Ψ̄|_{I_{17}} is non-trivial.

4. The representation ρ̄_{0} is absolutely irreducible - in fact, it is surjective.

Proof:

1. The Weil pairing gives a G_{Q}-equivariant isomorphism ∧^{2}X_{5} = μ_{5}, and hence det(ρ̄_{0}) = ϵ̄.

2. For each prime l, let a_{l} = l + 1 − #X(F_{l}). Since a_{5} = −2 ≠ 0, the curve X has good ordinary reduction at 5. Hence X_{5}(F̄_{5}) ≃ Z/5Z, and there is an exact sequence of I_{5}-modules

0 ⟶ μ_{5} ⟶ X_{5} ⟶ Z/5Z ⟶ 0

induced by the reduction map X(Q̄_{5}) ⟶ X(F̄_{5}). Let W = X_{5}(F̄_{5}) ≃ Z/5Z be the unramified quotient of X_{5}. Since

#X(F_{5}) = 5 + 1 − a_{5} = 8, #X(F_{25}) = 25 + 1 − a_{25} = 32,

the character χ̄_{2} cannot be of order 1 or 2; hence it is of order 4.

3. The curve X has split multiplicative reduction at 17, and is isomorphic over Q_{17} to a Tate curve G_{m}/q^{⟨Z⟩} where q ∈ 17Z_{17} is the Tate parameter.

From this explicit model, one sees that the representation ρ̄_{0}|_{I_{17}} is as given in the proposition. To see that Ψ̄ is non-trivial, let Δ be the minimal discriminant of X at 17. One can check from tables [Cr] that ord_{17}(Δ) ≠ 0 (mod 5). Hence

ord_{17}(q) = −ord_{17}(Δ) ≠ 0 (mod 5),

so that Ψ̄|_{I_{17}} ≠ 0.

4. Part 4 follows from prop. 21, p. 306, of [Sr1], using the fact that

ord_{17}(Δ) ≠ 0 (mod 5) and 2 + 1 − a_{2} ≠ 0 (mod 5).

A deformation of ρ̄_{0} is an equivalence class of representations

ρ : G_{Q} ⟶ GL_{2}(A)

where A is a complete Noetherian local Z_{5}-algebra with residue field F_{5}, such that the residual representation

G_{Q} ⟶ GL_{2}(A) ⟶ GL_{2}(F_{5})

is equal to ρ̄_{0}. These representations are taken modulo conjugation by matrices in GL_{2}(A) which reduce to the identity in GL_{2}(F_{5}).

The obvious example of a deformation of ρ̄_{0} is ρ_{0}, the 5-adic representation associated to the 5-adic Tate module of X_{0}(17).

We now define the two interesting classes of deformations that we wish to compare.

Definition 1.5 A deformation ρ : G_{Q} ⟶ GL_{2}(A) of ρ̄_{0} is admissible if

1. The character det(ρ) is the cyclotomic character ϵ : G_{Q} ⟶ Z_{5}^{∗} ⊂ A^{∗} giving the action of G_{Q} on the 5^{n}th roots of unity.

2. (Local behaviour at 5). The decomposition group D_{5} at 5 acts on the underlying rank 2 A-module by preserving a free submodule of rank 1 with free quotient, and the inertia group I_{5} acts trivially on this quotient:

ρ|_{D_{5}} ≃ ( \begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} ) , ρ|_{I_{5}} ≃ ( \begin{matrix}ϵ & ∗ \\ 0 & 1 \end{matrix} ) .

3. (Local behaviour at 17):

ρ|_{D_{17}} ≃ ( \begin{matrix}ϵ & Ψ \\ 0 & 1 \end{matrix} ) .

Part of the motivation for this definition, in the context of our problem, is suggested by the following proposition:

Proposition 1.6 If E is an elliptic curve satisfying the assumption of thm. 1.3, then the 5-adic representation ρ_{E} associated to E is admissible. (In particular, ρ_{0} is an admissible deformation of ρ̄_{0}.)

Proof: We check that the representation ρ_{E} of G_{Q} acting on the 5-adic Tate module T a_{5}(E) of E satisfies the three properties in the definition of admissibility.

1. The Weil pairing gives a G_{Q}-equivariant isomorphism ∧^{2}T a_{5}(E) = Z_{5}(1), and hence det(ρ_{E}) = ϵ.

2. If E has good reduction at 5, the assumption that ρ̄_{E} ≃ ρ̄_{0} forces E to have good ordinary reduction at 5, since X_{0}(17) does. For, otherwise the image of the inertia group I_{5} at 5 under ρ̄_{E} would be a non-split cartan subgroup, (cf. [Sr1], prop. 12, p. 275), and ρ̄_{E} could not be isomorphic to ρ̄_{0}. By the ordinariness of E, there is an exact sequence of Z_{5}[I_{5}]-modules

0 ⟶ Z_{5}(1) ⟶ T a_{5}(E) ⟶ Z_{5} ⟶ 0,

where the rightmost map comes from the reduction map E(Q̄_{5}) ⟶ E(F̄_{5}). If E has multiplicative reduction, then, after twisting by an unramified character, the curve E_{/Q_{5}} becomes isomorphic to a Tate curve G_{m}/q^{⟨Z⟩}, where q ∈ 5Z_{5}. This implies that the representation ρ_{E}|_{D_{5}} is of the form

ρ_{E}|_{D_{5}} ≃ ( \begin{matrix}α & ∗ \\ 0 & β \end{matrix} )

where β is a character of D_{5}/I_{5} of order 1 or 2. By part 2 of lemma 1.4, one has ρ̄_{E} ≠ ρ̄_{0}, and hence, the case where E has multiplicative reduction is ruled out.

3. The proof that the local condition at 17 is satisfied proceeds along similar lines. One notes that E necessarily has multiplicative reduction at 17, and then uses the Tate model to analyze the behaviour of ρ_{E} restricted to D_{17}.

The second class of deformations that we wish to consider are the modular deformations. We say that f is a cusp form of weight 2 (or simply a cusp form, since we will deal only with weight 2) if it is holomorphic on the extended upper half plane, vanishes at the cusps, and satisfies the usual transformation property under a congruence subgroup Γ_{0}(N) for some N. By the q-expansion principle, the space S_{2}(N, Z) of such cusp forms having integral Fourier expansions is a free Z-module of rank g = genus(X_{0}(N)), and

S_{2}(N, Z) ⊗ C = S_{2}(N, C),

the space on the right being the usual space of holomorphic cusp forms. This fact allows us to define the space S_{2}(N, A) of cusp forms with values in a ring A by

S_{2}(N, A) := S_{2}(N, Z) ⊗ A.

The module S_{2}(N, Z) is endowed with an action of the Hecke operators T_{p} for (p, N) = 1 and U_{q} for q|N. This action can be extended to S_{2}(N, A) by linearity.

An eigenform (with coefficients in A) is a form in S_{2}(N, A) which is an eigenfunction for all the Hecke operators T_{p} and U_{q}. We always assume that eigenforms are normalized so that their first Fourier coefficient is 1, so that f can be expanded about the cusp ∞ as

f = ∑_{n=1}^{∞} a_{n}q^{n}, a_{1} = 1, q = e^{2πiτ}.

Let O_{f} be the subring of A generated by the a_{p}, p |̸N. (Note that in general, O_{f} may be strictly contained in the ring generated by all the Fourier coefficients of f.)

Definition 1.7 A deformation ρ : G_{Q} ⟶ GL_{2}(A) of ρ̄_{0} is said to be modular if there exists a normalized eigenform f = ∑_{n} a_{n}q^{n} of weight 2 and some level N, with coefficients in A′ ⊃ A, such that the element ρ(Frob_{l}) has characteristic polynomial

x^{2} − a_{l}x + l,

for all l not dividing 5N. Futhermore, a modular deformation of ρ̄_{0} is said to be good if 5 does not divide N.

Proposition 1.8 Every good modular deformation of ρ̄_{0} is admissible.

This follows from the construction, due to Eichler and Shimura, of the representations attached to normalized cusp forms of weight 2 (cf. [Sh]), and the work of Carayol which, among other things, analyzes their basic properties at the bad places (cf. [Ca1]).

What Wiles shows is the converse of this statement.

Theorem 1.9 (Wiles) Every admissible deformation of ρ̄_{0} is modular.

This theorem implies theorem 1.3. For, if E is an elliptic curve satisfying the assumption of thm. 1.3, then the Galois representation on the 5-adic Tate module of E gives an admissible representation

ρ_{E} : G_{Q} ⟶ GL_{2}(Z_{5})

which is a deformation of ρ̄_{0}. Hence it is modular. It follows from the isogeny conjectures proved by Faltings that E appears in (is isogenous to a factor of) the jacobian of a modular curve, and hence, E itself is modular.

We will now concentrate on proving thm. 1.9.

1.2 Wiles’ strategy

Wiles’ proof of thm. 1.9 is a sophisticated counting argument. Roughly speaking, Wiles shows that the map

{Modular deformations of ρ̄_{0}} ⟶ {Admissible deformations of ρ̄_{0}}

is an isomorphism (of sets) by counting the respective orders of these sets. This is not quite true, because both of these sets are infinite. To cut down their sizes, one fixes a finite set of primes, Σ, which does not contain the “exceptional set” {5, 17}. One says that a deformation of ρ̄_{0} is Σ-admissible (Σmodular) if it is admissible (modular) and is unramified outside Σ ∪ {5, 17}.

One now considers the map

{Σ-Modular deformations of ρ̄_{0}} ⟶ {Σ-Admissible deformations of ρ̄_{0}}

and attempts to show it is an isomorphism; the set on the right is defined purely in terms of the properties of G_{Σ} = Gal(Q_{Σ}/Q), the Galois group of the maximal extension unramified outside Σ ∪ {5, 17}. As we will see later, the size of this rather subtle set is controlled by a subgroup of a Galois cohomology group – the Selmer group of the symmetric square associated to ρ_{0}. Wiles’ strategy is to bound the size of this Selmer group in terms of “modular” data related to the set on the left, and thus show that the set on the right cannot be bigger than the set on the left, so that all the admissible deformations are forced to be modular.

However, the sets above, even with the restrictions associated to Σ, are still infinite. To obtain finite sets one needs to fix a Z_{5}-algebra, A, and restrict one’s attention to the (equivalence classes of) deformations of ρ̄_{0} with values in A. A better-posed problem is thus to show that the map

\{ \begin{matrix}Σ-Modular deformations \\ of & ρ̄_{0} \\ with values in A \end{matrix} \} ⟶ \{ \begin{matrix}Σ-Admissible deformations \\ of & ρ̄_{0} \\ with values in A \end{matrix} \}

is an isomorphism for all local Noetherian Z_{5}-algebras A. Let MD_{Σ}(A) and AD_{Σ}(A) denote these two sets. The assignments A ↦ MD_{Σ}(A) and A ↦ AD_{Σ}(A) are functors from the category of local Z_{5}-algebras to sets. With these new notions, one can reformulate the problem as that of showing that the “natural” natural transformation

MD_{Σ} ⟶ AD_{Σ}

is a natural equivalence of functors.

A key fact is that the functor AD_{Σ} is representable, i.e., there exists a local Noetherian Z_{5}-algebra R_{Σ} such that

AD_{Σ} = Spec(R_{Σ}).

(Here we are identifying a scheme with its functor of points.)

From the work of Ribet and others, one also expects that the modular deformation functor MD_{Σ} should be representable by a finite flat local Z_{5}- algebra T_{Σ} constructed by completing an appropriate ring of Hecke operators acting on S_{2}(N_{Σ}, Z), for some level N_{Σ} depending on Σ.

The natural transformation MD_{Σ} ⟶ AD_{Σ} would then translate into a ring homomorphism

ϕ : R_{Σ} ⟶ T_{Σ}

which can be studied by using the machinery of commutative algebra.

1.3 Representability of AD_{Σ} and the ring R_{Σ}

In this section we state the result on the representability of AD_{Σ}.

Theorem 1.10 (Mazur): The functor of Σ-admissible deformations of ρ̄_{0} is equal to hom(R_{Σ}, −), where R_{Σ} is a finitely generated local Z_{5}-algebra with residue field F_{5}.

The ring R_{Σ} is called the universal deformation ring associated to the representation ρ̄_{0} and the set Σ. Its construction is based on a formal representability criterion of Schlessinger, and is described in [Mz2] (or, see also [MT]). Remark: A more general result of Ramkrishna [Ra] establishes the existence of the universal deformation ring in certain cases where the residual representation is not ordinary. We will not need this for our application, however.

By definition, there is a “universal admissible deformation”

ρ_{Σ,univ} : G_{Q} ⟶ GL_{2}(R_{Σ})

such that for any Σ-admissible deformation ρ : G_{Q} ⟶ GL_{2}(A) of ρ̄_{0}, there is a unique homomorphism α : R_{Σ} ⟶ A such that ρ_{Σ,univ} ⊗_{α} A = ρ. In particular, the deformation ρ_{0} corresponding to T a_{5}(X_{0}(17)) gives rise to a natural “base point map”

π_{R_{Σ}} : R_{Σ} ⟶ Z_{5},

which will play an important role later on.

1.4 Construction of a Hecke ring T_{Σ}

We are unable to establish a priori that the modular deformation functor is representable. However, we do have:

Expectation 1.11 The set of good Σ-modular deformations of ρ̄_{0} is equal to hom(T_{Σ}, −), where T_{Σ} is a finite flat local Z_{5}-algebra with residue field F_{5}.

In our situation, Wiles proposes a precise construction of T_{Σ}. We briefly describe this construction here. Let

N_{Σ} = 17 ∏_{p∈Σ} p^{2},

let Γ_{0}(N_{Σ}) be the usual congruence subgroup of level N_{Σ}, and let

X_{0}(N_{Σ}) = Γ_{0}(N_{Σ})\H ∪ {cusps}

be the corresponding curve, compactified in the usual way by adjoining the cusps.

Now let T(Σ) be the ring of Hecke operators acting on S_{2}(N_{Σ}, Z), the space of cusp forms of weight 2 on X_{0}(N_{Σ}) with integer Fourier coefficients. This ring is generated over Z by the Hecke operators T_{l}, for l not diving N_{Σ}, and by the Hecke operators U_{q} for q ∈ Σ ∪ {17}.

Let

f = ∑_{n} a_{n}q^{n}

be the normalized eigenform of level 17 corresponding to X = X_{0}(17). Its L-function is equal to

L(f, s) = ∑_{n} a_{n}n^{−s} = (1 − 17^{−s})^{−1} ∏_{p≠17}(1 − a_{p}p^{−s} + p^{1−2s})^{−1}.

The form f is an eigenform for all the Hecke operators in T(∅) acting on S_{2}(17, Z), but not for T(Σ) when viewed as a form in S_{2}(N_{Σ}, Z), since it is no longer an eigenform for the Hecke operators U_{q}, with q ∈ Σ. We remedy this problem by defining a cusp form f_{Σ} on X_{0}(N_{Σ}) which is an eigenform for T(Σ) by the inductive formula:

f_{∅} = f, f_{Σ∪{q}} = f_{Σ}(τ) − a_{q}f_{Σ}(qτ) + qf_{Σ}(q^{2}τ).

The reader can check that f_{Σ} is a modular form on Γ_{0}(N_{Σ}) which satisfies

T_{l}f_{Σ} = a_{l}f_{Σ} (l ∉ Σ ∪ {17}), U_{q}f_{Σ} = 0 (q ∈ Σ), U_{17}f_{Σ} = f_{Σ},

and hence is an eigenform for all of T(Σ). The construction of f_{Σ} corresponds to “removing the Euler factors at the primes of Σ”, since:

L(f_{Σ}, s) = (1 − 17^{−s})^{−1} ∏_{p∉Σ∪{17}}(1 − a_{p}p^{−s} + p^{1−2s})^{−1}.

It follows that the ideal

M_{Σ} = ⟨5, T_{l} − a_{l}, U_{q}, U_{17} + 1⟩ ⊂ T(Σ)

is a proper ideal of T(Σ). Let T_{Σ} denote the completion of T(Σ) at M_{Σ},

T_{Σ} = lim_{←} T(Σ)/M_{Σ}^{n}.

The ring T_{Σ} is a local Z_{5}-algebra with residue field F_{5}. It is finite and flat (i.e., finitely generated and free as a Z_{5}-module). If O is any Z_{5}-algebra with residue field F_{5}, the Z_{5}-algebra homomorphisms T_{Σ} ⟶ O correspond to 5adic eigenforms on Γ_{0}(N_{Σ}) which are congruent to f_{Σ} mod 5. In particular, the form f_{Σ} defines a canonical homomorphism

π_{T_{Σ}} : T_{Σ} ⟶ Z_{5}.

The fundamental construction of Eichler and Shimura, which associates a Galois representation to a cusp form of weight 2, completed by the work of Carayol, gives the following theorem:

Theorem 1.12 There is a (unique, up to conjugation) Galois representation

ρ_{Σ,mod} : G_{Q} ⟶ GL_{2}(T_{Σ})

which is admissible, and satisfies

trace(ρ_{mod}(Frob_{l})) = T_{l}, det(ρ_{mod}(Frob_{l})) = l, ∀l ∉ Σ ∪ {5, 17}.

Furthermore, this representation is admissible, and the operator U_{17} is the eigenvalue of Frob_{17} acting on the one-dimensional unramified quotient at 17.

Proof: See for example [Sh], [Ca1]. Remark: It is not clear a priori that the ring T_{Σ} represents the functor MD_{Σ} as we have defined it, although such a result is suggested by the work of Ribet [Ri2], and it will eventually follow a posteriori from the proof of thm. 1.9. A crucial ingredient in Wiles’ proof of thm. 1.3 is to first establish this for certain sets Σ.

Naturally, the representability of MD_{Σ} as it was defined naively before is not logically necessary for Wiles’ argument, so that our argument is not circular. One just takes a different tack, and defines the functor MD_{Σ} to be hom(T_{Σ}, −).

1.5 The map R_{Σ} ⟶ T_{Σ}

By thm. 1.12 and the universality property of the universal deformation ring R_{Σ}, there is a canonical ring homomorphism

ϕ_{Σ} : R_{Σ} ⟶ T_{Σ}.

Moreover, this homomorphism is compatible with the base point maps, i.e., the diagram

\begin{matrix}R_{Σ} & ⟶ & T_{Σ} \\ ↓ & & ↓ \\ Z_{5} & = & Z_{5} \end{matrix}

commutes, where the vertical arrows are the base point maps π_{R_{Σ}} and π_{T_{Σ}}. For, it follows directly from the Chebotarev density theorem that the maps π_{R_{Σ}} and π_{T_{Σ}} ∘ ϕ_{Σ} induce equivalent (i.e., equal) deformations, so that these maps are the same, by the universality property of R_{Σ}.

Now, Wiles’ theorem 1.9 can be restated (yet again) as

Theorem 1.13 (Wiles) The map ϕ_{Σ} : R_{Σ} ⟶ T_{Σ} is an isomorphism.

From now on we will concentrate on the proof of thm. 1.13.

In this section we will show that ϕ_{Σ} is surjective, which is (by far!) the easy half of thm. 1.13.

Theorem 1.14 The map ϕ_{Σ} is surjective.

Let T^{0}(Σ) ⊂ T(Σ) be the subring generated by the “good” Hecke operators T_{l}, for l ∉ {5, 17} ∪ Σ, and let T_{Σ}^{0} be the closure of the image of T^{0}(Σ) in T_{Σ}.

Lemma 1.15 The image of ϕ_{Σ} contains T_{Σ}^{0}.

Proof: For all l ∉ {5, 17} ∪ Σ,

T_{l} = Tr (ρ_{Σ,mod}(Frob_{l})) = ϕ_{Σ}(Tr (ρ_{Σ,univ}(Frob_{l}))),

so that T_{l} ∈ ϕ_{Σ}(R_{Σ}), for all l ∉ {5, 17} ∪ Σ. The result follows.

Lemma 1.16 If q ∈ Σ, then the image of the Hecke operator U_{q} in T_{Σ} is 0. In particular, U_{q} ∈ T_{Σ}^{0}.

Proof: The operator U_{q} in T(Σ) satisfies the relation:

U_{q}(U_{q}^{2} − 1) ∏_{g}(U_{q}^{2} − a_{q}(g)U_{q} + q) = 0

where the product is taken over all the newforms of level dividing N_{Σ}/q^{2} (cf. [Kn], prop. 9.27, p.289). Since U_{q} belongs to M, its image in T_{Σ} is topologically nilpotent. It follows that the expression (U_{q}^{2} − 1) ∏(U_{q}^{2} − a_{q}(g)U_{q} + q) maps to a unit in T_{Σ}, so that the image of U_{q} in T_{Σ} is 0. The result follows.

To see that the operators U_{17} and T_{5} belong to ϕ(R_{Σ}) requires a better understanding of ρ_{Σ,mod}.

Lemma 1.17 The representation ρ_{Σ,mod} : G_{Q} ⟶ GL_{2}(T_{Σ}) is conjugate to a representation with values in GL_{2}(T_{Σ}^{0}).

Proof: This follows directly from the following more general statement. (I am grateful to J-P. Serre for explaining this to me.)

Proposition 1.18 Let A^{0} ⊂ A be an inclusion of local rings (i.e., if m_{A} and m_{A^{0}} are the maximal ideals of A and A^{0}, then m_{A^{0}} = m_{A} ∩ A^{0}, and A^{0} and A have the same residue fields), and suppose that the residue field k of A has trivial Brauer group, H^{2}(k, k̄^{∗}) = 0. Let ρ : G ⟶ GL_{n}(A) be a representation of a group G over A, and suppose that

1. The residual representation ρ̄: G ⟶ GL_{n}(k) is absolutely irreducible.

2. Tr (ρ(σ)) ∈ A^{0} for all σ ∈ G. Then ρ is conjugate to a representation with values in GL_{n}(A^{0}).

Proof: Let B be the A^{0}-subalgebra of M_{n}(A) generated by ρ(G). Since ρ̄ is absolutely irreducible, the image of B in M_{n}(k) is an Azumaya algebra over k, and hence is the full matrix ring M_{n}(k) since H^{2}(k, k̄^{∗}) = 0. Let e_{1}, …, e_{n^{2}} be elements of B that reduce to a standard basis of M_{n}(k). We claim that e_{1}, …, e_{n^{2}} is an A^{0} basis for B. By Nakayama’s lemma, it is an A-basis for M_{n}(A), and hence every element b of B can be written uniquely as a combination

b = ∑ a_{i}e_{i}, with a_{i} ∈ A.

We claim that the a_{i} actually belong to A^{0}. For, multiplying the above relation by the transpose e_{j}^{t} of e_{j} and taking traces, we find the system:

Tr (be_{j}^{t}) = ∑ a_{i}Tr (e_{i}e_{j}^{t}), j = 1, …, n^{2}.  (1)

The matrix (c_{ij}) = (Tr (e_{i}e_{j}^{t})) is an n^{2} × n^{2}-matrix with coefficients in A^{0}, by assumption. Since the e_{i} reduce to the standard basis of M_{n}(k), the image of (c_{ij}) in M_{n^{2}}(k) is the identity matrix. Hence, (c_{ij}) is invertible in M_{n^{2}}(A^{0}), so that the system (1) can be solved for the a_{i}, and hence a_{i} ∈ A^{0}. It follows that B is an algebra of rank n^{2} over A^{0}, generated by the e_{i}. Let V ⊂ A^{n} be the A^{0}-module generated by the first columns (say) of elements of B. Then V is a free A^{0}-module of rank n, and the action of B on V gives a map B ⟶ end(V) ≃ M_{n}(A^{0}). By Nakayama B ≃ M_{n}(A^{0}), and the result follows.

Lemma 1.19 The image of U_{17} in T_{Σ} belongs to T_{Σ}^{0}.

Proof: By lemma 1.17, the representation ρ_{Σ,mod} is conjugate to a representation with values in GL_{2}(T_{Σ}^{0}). But U_{17} can be recovered as the eigenvalue of Frob_{17} acting on the unique unramified one-dimensional quotient of the underlying T_{Σ}^{0}-module, by thm. 1.12. Therefore, U_{17} ∈ T_{Σ}^{0}.

Lemma 1.20 The image of T_{5} in T_{Σ} belongs to T_{Σ}^{0}.

Proof: One recovers T_{5} as α_{5} + 5/α_{5}, where α_{5} ∈ T_{Σ}^{0} is the eigenvalue of Frob_{5} acting on the unique unramified rank 1 quotient of ρ_{Σ,mod}, by thm. 1.12. Hence, T_{5} belongs to T_{Σ}^{0}, as before.

Corollary 1.21 The natural inclusion T_{Σ}^{0} ⊂ T_{Σ} is an isomorphism.

Proof: This follows immediately from lemmas 1.16, 1.19, and 1.20.

Proof of thm. 1.14: Combine lemma 1.15 and cor. 1.21.

2 Wiles’ isomorphism criterion

Wiles gives a beautiful criterion, based entirely on commutative algebra, to show that certain surjective maps of local rings are isomorphisms.

As we saw in the previous section, the rings that come up in Wiles’ situation are finitely generated complete local Z_{p}-algebras A and are equipped with a natural surjective map A ⟶ Z_{p}. Thus it is natural to work in the category C whose objects are pairs (A, π), where A is a finitely generated local Z_{p}-algebra and π : A ⟶ Z_{p} is a surjective ring homomorphism (called the “base point map”). Morphisms in this category are local ring homomorphisms which are compatible in the obvious way with the base point maps. By abuse of notation we often omit mentioning the base point map π when we talk of objects in C, and simply use A to denote (A, π), when this causes no confusion.

2.1 The invariants Φ and η

2.1.1 Definition

Let A = (A, π) be an object of C. One associates to such an object two fundamental invariants:

Φ_{A} = (ker π)/(ker π)^{2};  (2)
η_{A} = π(Ann_{A} ker π).  (3)

The invariant Φ_{A} can be thought of as a tangent space for the object A. (More precisely, it is the cotangent space for the scheme spec(A) at the point ker π.) It is a finitely generated Z_{p} module.

The invariant η_{A} is called the congruence ideal. The reason for this terminology should become clear in section 2.1.2.

We can now state Wiles’ isomorphism criterion:

Theorem 2.1 (Wiles) Let R and T be objects of C such that T is a finitely generated torsion-free Z_{p}-module, and let ϕ : R ⟶ T be a surjective morphism. If

#Φ_{R} ≤ #(Z_{p}/η_{T}) < ∞,

then ϕ is an isomorphism.

This beautiful result is the engine which is at the heart of Wiles’ proof of thm. 1.13; the reader is invited on a first reading to skip to sec. 3 to see how thm. 1.13 is deduced from the isomorphism criterion.

The proof we will give of Wiles’ isomorphism criterion follows closely a presentation of Lenstra [Le].

2.1.2 Some examples

Before going further, it is good to pause and consider some examples of objects of C and the invariants associated to them. While logically independent of the proof, the examples should help the reader develop some intuition. It is therefore a good idea to work out these examples. (For an indication on how to compute the tangent spaces Φ_{A}, see the paragraph at the beginning of sec. 2.3.)

Example 1:

A = {(a, b) ∈ Z_{p} × Z_{p}, a ≡ b (mod p^{n})}
≃ Z_{p}[[T]]/(T (T − p^{n})),
π(a, b) = a.
Φ_{A} ≃ Z/p^{n}Z, η_{A} = (p^{n}).

Example 2:

A = {(a, b, c) ∈ Z_{p} × Z_{p} × Z_{p}, a ≡ b ≡ c (mod p)}
≃ Z_{p}[[X, Y]]/(X(X − p), Y (Y − p), XY),
π(a, b, c) = a.
Φ_{A} ≃ Z/pZ × Z/pZ, η_{A} = (p).

Example 3:

A = Z_{p}[[X]]/(X^{2}),
π(f) = f(0).
Φ_{A} ≃ Z_{p}, η_{A} = 0.

Example 4:

A = \{(a, b, c, d) ∈ Z_{p} × ⋯ × Z_{p}, \begin{matrix}a ≡ b ≡ c ≡ d (mod p), \\ a + d ≡ b + c (mod p^{2}) \end{matrix} \}
≃ Z_{p}[[X, Y]]/(X(X − p), Y (Y − p)),
π(a, b, c, d) = a.
Φ_{A} ≃ Z/pZ × Z/pZ, η_{A} = (p^{2}).

Example 5:

A = Z_{p}[[X_{1}, …, X_{n}]],
π(f) = f(0).
Φ_{A} ≃ (Z_{p}^{n}), η_{A} = (0).

Example 6: (p ≡ −1 (mod 4)).

A = {(a, b + ci) ∈ Z_{p} × Z_{p}[i], a ≡ b (mod p^{2}), c ≡ 0 (mod p)}
= Z_{p}[[X]]/(X(X^{2} + p^{2})).
π(a, b + ci) = a.
Φ_{A} ≃ Z/p^{2}Z, η_{A} = (p^{2}).

Example 7:

A = Z_{p}[[T]]/(pT) = Z_{p} ⊕ F_{p}T ⊕ F_{p}T^{2} ⊕ ⋯
π(f) = f(0).
Φ_{A} ≃ (Z/pZ), η_{A} = (p).

2.2 Basic properties of Φ_{A} and η_{A}

In this section we collect some of the basic properties of the invariants Φ_{A} and η_{A}. The first two concern the behaviour of these invariants under morphisms (particularly surjective morphisms).

1. The assignment A ↦ Φ_{A} is a functor from the category C to the category of finitely generated Z_{p}-modules. Hence a morphism A ⟶ B in C induces a homomorphism Φ_{A} ⟶ Φ_{B} of Z_{p}-modules. Moreover, if A ⟶ B is surjective, then so is the induced map on the tangent spaces. Therefore, when A maps surjectively onto B we have

#Φ_{A} ≥ #Φ_{B}.  (4)

2. Unlike the assignment A ↦ Φ_{A}, the assignment A ⟶ η_{A} is not functorial, but it does have a nice behaviour under surjective morphisms: namely, if ϕ : A ⟶ B is surjective, then

η_{A} ⊂ η_{B}, i.e., #(Z_{p}/η_{A}) ≥ #(Z_{p}/η_{B}).  (5)

This is simply because in that case ϕ induces a map

Ann ker π_{A} ⟶ Ann ker π_{B}.

3. The third general property gives a relation between the invariants Φ_{A} and η_{A}. Observe that in all the examples of the previous section, we have #Φ_{A} ∈ η_{A}, whenever #Φ_{A} is finite. In fact, this is true in general:

#Φ_{A} ≥ #(Z_{p}/η_{A}).  (6)

The key behind proving this identity is to interpret #Φ_{A} in terms of Fitting ideals. Namely, #Φ_{A} (if it is finite) gives a generator for the Fitting ideal, Fit_{Z_{p}}(Φ_{A}) of Φ_{A} as a Z_{p} module. Hence

#Φ_{A} ∈ Fit_{Z_{p}}(Φ_{A})
= π_{A}(Fit_{A}(ker π_{A}))
⊂ π_{A}(Ann_{A} ker π_{A}) = η_{A}.

The first equality follows from the fact that, if M is any A module, then

π_{A}(Fit_{A}(M)) = Fit_{Z_{p}}(M ⊗_{A} Z_{p}),

where the tensor product is taken with respect to π_{A}. Applying this to M = ker π_{A}, and observing that ker π_{A} ⊗_{A} Z_{p} = (ker π_{A})/(ker π_{A})^{2}, one finds the desired equality.

4. Computing tangent spaces: Any object (A, π_{A}) in C can be expressed as a quotient of the object U = Z_{p}[[X_{1}, …, X_{n}]] of example 5 with base point map given by π_{U}(f) = f(0). For, one can take a_{1}, …, a_{n} to be A-module generators of the finitely generated A-module ker π_{A}, and obtain the desired quotient map by sending X_{i} to a_{i}.

The tangent space Φ_{U} of U is a free Z_{p}-module of rank n which can be written down canonically as

Z_{p}X_{1} ⊕ Z_{p}X_{2} ⊕ ⋯ ⊕ Z_{p}X_{n},

the map (ker π_{U})/(ker π_{U})^{2} being simply the map which sends a power series f ∈ U with no constant term to its degree 1 term, which we will denote by f̄.

If A is expressed as a quotient U/(f_{1}, …, f_{r}), then one has

Φ_{A} = Φ_{U}/(f̄_{1}, …, f̄_{r}).

This formula gives the most natural way of computing tangent spaces.

2.3 Complete intersections

We say that A is a complete intersection if it can be expressed as a quotient

A ≃ Z_{p}[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n})

where there are as many relations as there are variables.

The rings of examples 1, 3, 4, 6 and 7 are complete intersections (according to our definition), and the others are not. Observe that in the seven examples that are worked out, a ring A is a complete intersection precisely when #Φ_{A} = #(Z_{p}/η_{A}).

An object A in C is said to be Gorenstein if

Hom(A, Z_{p}) ≃ A as A-modules.

Proposition 2.2 If A is a complete intersection, and A is a finitely generated Z_{p}-module, then A is Gorenstein. (In particular, it is torsion-free as a Z_{p}-module.)

Remark: The condition that A be finitely generated is essential. For example, the ring A of example 7 is a complete intersection, but it is certainly not Gorenstein, since it is not even free as a Z_{p}-module.

The remainder of this section will be devoted to proving prop. 2.2.

We recall some definitions from commutative algebra that we will need. An ideal I of a local ring R is said to be primary if I ≠ R and every zero divisor in R/I is nilpotent. If (x_{1}, …, x_{n}) generates a primary ideal of R, and n = dim R, then (x_{1}, …, x_{n}) is called a system of parameters for R.

Lemma 2.3 The sequence (f_{1}, …, f_{n}, p) is a system of parameters for U = Z_{p}[[X_{1}, …, X_{n}]].

Proof: The quotient ring U/(f_{1}, …, f_{n}, p) is local and is finitely generated as an F_{p}-vector space; therefore every element in its maximal ideal is nilpotent.

A sequence (x_{1}, …, x_{n}) in a ring R is said to be a regular sequence if x_{i} is not a zero-divisor in R/(x_{1}, …, x_{i−1}) for i = 1, …, n.

Lemma 2.4 The sequence (f_{1}, …, f_{n}) is a regular sequence for U.

Proof: The ring U is Cohen Macaulay, since p, X_{1}, …, X_{n} is a system of parameters of U which is also a regular sequence. Hence, by theorem 17.4 (iii) of [Mat], (f_{1}, …, f_{n}, p) is a regular sequence in U. A fortiori, the sequence (f_{1}, …, f_{n}) is also a regular sequence.

To go further, we will introduce the Koszul complex

K(\underline{x}, R) := ⊕_{p=0}^{n}K_{p}(\underline{x}, R)

associated to a local ring R and a sequence \underline{x} = (x_{1}, …, x_{n}) of elements in its maximal ideal. This complex is defined to be the free graded differential algebra generated by symbols u_{1}, …, u_{n}:

K_{p}(\underline{x}, R) := ⊕_{i_{1}<i_{2}<···<i_{p}}R · u_{i_{1}} ∧ ⋯ ∧ u_{i_{p}},

with differential d : K_{p} ⟶ K_{p−1} defined by

d(u_{i_{1}} ∧ ⋯ ∧ u_{i_{p}}) = ∑_{t=1}^{p}(−1)^{t}x_{t} · u_{i_{1}} ∧ ⋯ ∧ u_{i_{t−1}} ∧ u_{i_{t+1}} ∧ ⋯ ∧ u_{i_{p}}.

We denote by H_{p}(\underline{x}, R) the homology groups of this complex. We record here the main properties of this complex that we will use.

Proposition 2.5 1. H_{0}(\underline{x}, R) = R/(\underline{x}).

2. There is a long exact homology sequence

⋯ ⟶ H_{p}(\underline{x}, R) ⟶ H_{p}(\underline{x}, x_{n+1}, R) ⟶ H_{p−1}(\underline{x}, R) ⟶^{±x_{n+1}}

H_{p−1}(\underline{x}, R) ⟶ H_{p−1}(\underline{x}, x_{n+1}, R) ⟶ H_{p−2}(\underline{x}, R) ⟶ ⋯

3. H_{p}(\underline{x}, R) is annihilated by the ideal (\underline{x}), i.e., it has a natural R/(\underline{x})- module structure.

4. If \underline{x} is a regular sequence, then H_{p}(\underline{x}, r) = 0 for all p > 0 (i.e., the complex K_{p}(\underline{x}, R) is a free resolution of R/(\underline{x}).)

Proof: The first assertion follows directly from the definition. For 2 and 3, see [Mat], th. 16.4. The assertion 4 can be proved by a direct induction argument on n, using the long exact homology sequence: For p > 1, this sequence becomes

0 ⟶ H_{p}(\underline{x}, x_{n+1}, R) ⟶ 0,

and for p = 1, it is

0 ⟶ H_{1}(\underline{x}, x_{n+1}, R) ⟶ H_{0}(\underline{x}, R) ⟶^{x_{n+1}} H_{0}(\underline{x}, R).

But the assumption that \underline{x}, x_{n+1} is a regular sequence means that multiplication by x_{n+1} is injective on H_{0}(\underline{x}, R) = R/(\underline{x}). Hence, the assertion 4 follows. For more details on the Koszul complex and its relation to regular sequences, the reader can consult [Mat], especially §16.

Now, we turn to the proof of prop. 2.2, following a method of Tate which is explained in the appendix of [MRo]. For any ring R, write R[[\underline{X}]] := R[[X_{1}, …, X_{n}]]. Let a_{1}, …, a_{n} be the images in A of X_{1}, …, X_{n} by the natural map

α : Z_{p}[[\underline{X}]] ⟶ A = Z_{p}[[\underline{X}]]/(f_{1}, …, f_{n}),

and let

β : A[[\underline{X}]] ⟶ A

be the natural map which sends X_{i} to a_{i}. The sequence (g_{i}) = (X_{i} − a_{i}) generates the kernel of β. Since the f_{i}, viewed as polynomials in A[[\underline{X}]], also belong to ker β, we have:

(f_{1}, …, f_{n}) = (g_{1}, …, g_{n})M,

where M is an n × n matrix with coefficients in A[[\underline{X}]]. Let

D = det(M) ∈ A[[\underline{X}]].

Our goal is to construct an A-module isomorphism

hom_{Z_{p}}(A, Z_{p}) ⟶ A.

We begin by constructing a surjective (Z_{p}-linear) map

hom_{Z_{p}[[\underline{X}]]}(A[[\underline{X}]], Z_{p}[[\underline{X}]]) ⟶ A.

Lemma 2.6 (Tate): The function Φ(f) = α(f(D)) induces an isomorphism of Z_{p}[[\underline{x}]]-modules

hom_{Z_{p}[[\underline{X}]]}(A[[\underline{X}]], Z_{p}[[\underline{X}]])/(g_{1}, …, g_{n}) ⟶ A.

Proof: By lemma 2.4, the sequence (\underline{f}) = (f_{1}, …, f_{n}) is a regular Z_{p}[[\underline{X}]]- sequence. One can see directly from the definition that the sequence (\underline{g}) = (g_{i}) = (X_{i} − a_{i}) is a regular A[[\underline{X}]]-sequence. Let K(\underline{f}) and K(\underline{g}) be the Koszul complexes associated to these two sequences. It follows from prop. 2.5 that the Koszul complex K(\underline{f}) is a resolution of A by free Z_{p}[[\underline{X}]]-modules, and the Koszul complex K(\underline{g}) is a resolution of A by free A[[\underline{X}]]-modules, and hence a fortiori, by free Z_{p}[[\underline{X}]]-modules. We define a map Φ : K(\underline{f}) ⟶ K(\underline{g}) of complexes by letting

Φ_{0} : K_{0}(\underline{f}) ⟶ K_{0}(\underline{g})

be the natural inclusion of Z_{p}[[\underline{x}]] into A[[\underline{x}]], and letting

Φ_{1} : K_{1}(\underline{f}) ⟶ K_{1}(\underline{g})

be the map defined by

(Φ_{1}(u_{1}), …, Φ_{1}(u_{n})) = (v_{1}, …, v_{n})M,

and extending it by skew-linearity a map of exterior algebras. One can check that the resulting map Φ is a morphism of complexes which induces the identity map A ⟶ A, and satisfies

Φ_{n}(u_{1} ∧ ⋯ ∧ u_{n}) = D · v_{1} ∧ ⋯ ∧ v_{n}.

Applying the functor hom_{Z_{p}[[\underline{X}]]}(−, Z_{p}[[\underline{X}]]) to these two free resolutions, and taking the homology of the resulting complexes, we find that Φ induces an isomorphism on the cohomology, and in particular, on the nth cohomology:

Φ_{n} : hom_{Z_{p}[[\underline{X}]]}(A[[\underline{X}]], Z_{p}[[\underline{X}]])/(g_{1}, …, g_{n}) ⟶^{≃}

hom(Z_{p}[[\underline{X}]], Z_{p}[[\underline{X}]])/(f_{1}, …, f_{n}) = A,

which is given explicitly by the formula:

Φ_{n}(f) = α(f(D)).

We finally come to the proof of prop. 2.2, which we can state in a more precise form.

Lemma 2.7 The map Ψ : hom_{Z_{p}}(A, Z_{p}) ⟶ A defined by Ψ(f) = α(f̃(D)), where f̃ : A[[\underline{X}]] ⟶ Z_{p}[[\underline{X}]] is the base change of f, is an A-module isomorphism, and hence, A is Gorenstein.

Proof: The key point is to show that Ψ is A-linear. By definition, if

a = α(a′) ∈ A, with a′ ∈ Z_{p}[[\underline{X}]],

then

Ψ(af) = α(f̃(aD)) = α(f̃((a − a′)D)) + α(f̃(a′D)).

Since a − a′ ∈ ker β, it can be written as a A[[\underline{X}]]-linear combination of the g_{i}. By multiplying the relation

(f_{1}, …, f_{n}) = (g_{1}, …, g_{n})M

by the matrix D ·M^{−1} ∈ M_{n}(A[[\underline{X}]]), one sees that the Dg_{i} can be written as A[[\underline{X}]]-linear combinations of the f_{i}’s. Hence, so can the expression (a−a′)D. By the Z_{p}[[\underline{X}]]-linearity of f̃, and the fact that each f_{i} belongs to ker α, it follows that

α(f̃((a − a′)D)) = 0.

Therefore,

Ψ(af) = α(a′ f̃(D)) = aΨ(f).

This shows that Ψ is A-linear. To show that Ψ is surjective, observe that if f_{1}, …, f_{r} is a Z_{p}-basis of hom_{Z_{p}}(A, Z_{p}), then f̃_{1}, …, f̃_{r} is a Z_{p}[[\underline{X}]]-basis for hom_{Z_{p}[[\underline{X}]]}(A[[\underline{X}]], Z_{p}[[\underline{X}]]). Hence, for all a ∈ A, there exist p_{1}, …, p_{r} such that

Φ_{n}(p_{1}f̃_{1} + ⋯ + p_{r}f̃_{r}) = a.

But this means that

Ψ(α(p_{1})f_{1} + ⋯ + α(p_{r})f_{r}) = a,

so that Ψ is surjective. Finally, since hom_{Z_{p}}(A, Z_{p}) and A are free Z_{p}-modules of the same finite rank, and Ψ is surjective, it must also be injective. Hence Ψ is an isomorphism, as was to be shown.

Examples:

1. Let A ⊂ Z_{p} × Z_{p} be the ring of example 1 in section 2.1.2. Then, f = T^{2} − p^{n}T, and g = T − (0, p^{n}). Hence, f = (T − (p^{n}, 0))g, so that D = T − (p^{n}, 0). It follows that

Φ(h) = α(h̃(T − (p^{n}, 0))) = α(T h(1, 1) − h(p^{n}, 0)) = (0, p^{n})h(1, 1) − h(p^{n}, 0).

The reader can check that Φ is indeed an A-linear isomorphism from the A-module hom_{Z_{p}}(A, Z_{p}) to A.

2. Let A = Z_{p}[ϵ]/(ϵ^{2}) be the ring of example 3 in section 2.1.2. Then, f = T^{2}, and g = T − ϵ. Hence, f = (T + ϵ)g, so that D = T + ϵ. It follows that

Φ(f) = α(f̃(T + ϵ)) = α(T f(1) + f(ϵ)) = ϵf(1) + f(ϵ).

A good way to gain insight into prop. 2.2 is to work out the isomorphism hom_{Z_{p}}(A, Z_{p}) for the complete intersection ring A of example 4 of sec. 2.1.2.

2.4 Isomorphism theorems

The usefulness of the notion of complete intersections comes from the following two (vaguely stated) principles:

1. Isomorphisms to complete intersections can often be recognized by looking at their effects on the tangent spaces.

2. Isomorphisms from complete intersections can often be recognized by looking at their effects on the invariants η.

These vague principles are made precise in theorems 2.8 and 2.9 respectively.

Theorem 2.8 Let ϕ : A ⟶ B be a surjective morphism of the category C with B a complete intersection. If ϕ induces an isomorphism Φ_{A} ⟶ Φ_{B}, and these modules are finite, then ϕ is an isomorphism.

Remarks:

1. Let A = Z_{p}[[X, Y]]/(X(X − p), Y (Y − p)) be the ring of example 4, let B = Z_{p}[[X, Y]]/(X(X − p), Y (Y − p), XY) be the ring of example 2, and let ϕ : A ⟶ B be the natural projection. The map ϕ induces an isomorphism Φ_{A} ⟶ Φ_{B}, even though ϕ is not an isomorphism. The assumption that B be a complete intersection is crucial for concluding that ϕ is an isomorphism.

2. The natural map

Z_{p}[[X]]/(X^{3}) ⟶ Z_{p}[[X]]/(X^{2})

is a surjective morphism inducing an isomorphism on tangent spaces, and the target ring is a complete intersection. Yet this map is not an isomorphism. This shows that the assumption on the finiteness of the tangent spaces cannot be dispensed with.

Proof: Let

ν_{B} : Z_{p}[[X_{1}, …, X_{n}]] ⟶ B

be a surjective C-morphism with ker ν_{B} = (f_{1}, …, f_{n}). Let b_{1}, …, b_{n} ∈ ker π_{B} denote the images of X_{1}, …, X_{n} by ν_{B}, and let a_{1}, …, a_{n} ∈ ker π_{A} denote inverse images of b_{1}, …, b_{n} by ϕ. Since ϕ is an isomorphism on tangent spaces, the elements a_{i} generate (ker π_{A})/(ker π_{A})^{2}. Hence they generate ker π_{A} as an A-module, by Nakayama’s lemma. It follows (again by Nakayama) that the C-morphism

ν_{A} : Z_{p}[[X_{1}, …, X_{n}]] ⟶ A

defined by ν_{A}(X_{i}) = a_{i} is surjective. We claim that ker ν_{B} ⊂ ker ν_{A} (and hence, ker ν_{B} = ker ν_{A}). For, let g_{1}, …, g_{n} be elements of ker ν_{A} whose linear terms ḡ_{1}, …, ḡ_{n} generate the kernel of

ν̄_{A} : Φ_{U} ⟶ Φ_{A}.

Since ker ν_{A} ⊂ ker ν_{B}, it follows that there exists an n×n matrix M ∈ M_{n}(U) with entries in U such that

(g_{1}, …, g_{n}) = (f_{1}, …, f_{n})M.

Let M̄ be the matrix of constant terms of the matrix M. Then we have

(ḡ_{1}, …, ḡ_{n}) = (f̄_{1}, …, f̄_{n})M̄.

Since (ḡ_{1}, …, ḡ_{n}) and (f̄_{1}, …, f̄_{n}) generate the same submodules of rank n and finite index in Φ_{U}, it follows that det M̄ is a unit in Z_{p}. Hence, M is invertible, and therefore the f_{i} can be expressed as a U-linear combination of the g_{j}. This implies that ker ν_{B} ⊂ ker ν_{A}. Now we see that ν_{A}ν_{B}^{−1} gives a well-defined inverse to ϕ, so that ϕ is an isomorphism.

Theorem 2.9 Let ϕ : A ⟶ B be a surjective morphism of the category C, with A being a complete intersection, and A and B finitely generated torsion free Z_{p}-modules. If η_{A} = η_{B} ≠ 0, then ϕ is an isomorphism.

Remark: The torsion-freeness assumption on B is essential. For if A is a finitely generated complete intersection with η_{A} ≠ (0), and n is large enough, then B = A/p^{n}A satisfies η_{A} = η_{B}, although the natural map A ⟶ B is not surjective.

Proof: By proposition 2.2, A is Gorenstein, i.e., it satisfies

A^{∗} = hom_{Z_{p}}(A, Z_{p}) ≃ A as A-modules.

Now, we observe that

ker π_{A} ∩ Ann_{A} ker π_{A} = 0,  (7)

and likewise for B. For, let x be a non-zero element of η_{A}, and let x′ ∈ Ann_{A} ker π_{A} satisfy π_{A}(x′) = x. For all a ∈ ker π_{A} ∩ Ann_{A} ker π_{A}, we have

0 = a(x − x′) = ax,

the first equality because a belongs to Ann_{A} ker π_{A} and (x − x′) belongs to ker π_{A}, the second equality because a belongs to ker π_{A} and x′ belongs to Ann_{A} ker π_{A}. Hence a belongs to the Z_{p}-torsion submodule of A, and therefore is 0.

It follows from (7) that π_{A} (resp. π_{B}) induces an isomorphism from Ann_{A} ker π_{A} (resp. Ann_{B} ker π_{B}) to η_{A} (resp. η_{B}). Since η_{A} = η_{B}, it follows that ϕ induces an isomorphism from Ann_{A} ker π_{A} to Ann_{B} ker π_{B}, i.e.,

ϕAnn_{A} ker π_{A} = Ann_{B} ker π_{B}.

From (7) it also follows a fortiori that

ker ϕ ∩ Ann_{A} ker π_{A} = 0,

hence there is an exact sequence of A-modules:

0 ⟶ ker ϕ ⊕ Ann_{A} ker π_{A} ⟶ A.  (8)

The cokernel of the last map is

A/(ker ϕ ⊕ Ann_{A} ker π_{A}) ≃ B/(ϕAnn_{A} ker π_{A}) ≃ B/(Ann_{B} ker π_{B}),

which is torsion-free, since there is a natural injection

B/(Ann_{B} ker π_{B}) ↪ end_{Z_{p}}(ker π_{B}).

Hence, the exact sequence (8) splits over Z_{p}. Taking Z_{p} duals in (8) and using the Gorenstein condition for A, we thus get an exact sequence of A-modules:

A ⟶ (ker ϕ)^{∗} ⊕ (Ann_{A} ker π_{A})^{∗} ⟶ 0.

Applying the functor − ⊗_{A} F_{p} (relative to the map A ⟶ F_{p}), we find

1 = dim_{F_{p}}(A⊗_{A} F_{p}) ≥ dim_{F_{p}}((ker ϕ)^{∗} ⊗_{A} F_{p})+dim_{F_{p}}((Ann_{A} ker π_{A})^{∗} ⊗_{A} F_{p}).

Since η_{A} ≠ 0, it follows that (Ann_{A} ker π_{A})^{∗} ⊗_{A} F_{p} ≠ 0, and hence we must have

(ker ϕ)^{∗} ⊗_{A} F_{p} = 0.

Therefore by Nakayama’s lemma and duality, ker ϕ = 0, which proves the theorem.

2.5 A resolution lemma

Although objects in C need not be complete intersections, they always can be “resolved” (in a weak sense) by a complete intersection, namely,

Theorem 2.10 Let A be an object of C. Then there is a morphism Ã ⟶ A of C which induces an isomorphism ΦÃ ⟶ Φ_{A} and such that Ã is a complete intersection. Moreover, if A is finitely generated over Z_{p}, then Ã can be chosen so as well.

Proof: Write A as a quotient of U = Z_{p}[[X_{1}, …, X_{n}]] (with π_{U} : U ⟶ Z_{p} the map which sends each X_{i} to 0.) Let f_{1}, …, f_{n} ∈ ker π_{U} be such that f̄_{1}, …, f̄_{n} generate the kernel of Φ_{U} ⟶ Φ_{A}. Now letting

Ã = U/(f_{1}, …, f_{n})

gives the desired ring Ã.

It remains to show that, if A is a finitely generated Z_{p}-module, then Ã can be chosen to be finitely generated. This is not generally true of the ring Ã constructed above, of course: we must take some care so that the n relations f_{1}, …, f_{n} we leave in when defining Ã are nice enough, so that Ã is finitely generated.

Let a_{1}, …, a_{n} be Z_{p}-module generators of the finitely generated module ker π_{A}, and define a homomorphism ϕ from the polynomial ring

V = Z_{p}[X_{1}, …, X_{n}]

to A by sending X_{j} to a_{j}. Clearly ϕ is surjective. Let f_{1},… f_{n} be elements of ker ϕ chosen in the same way as before, and let m denote their maximal degree. Since the elements a_{i}^{2} belong to ker π_{A}, we may write

a_{i}^{2} = h_{i}(a_{1}, …, a_{n}),

where h_{i}(X_{1}, …, X_{n}) is a linear polynomial. Now, replacing the relations f_{i} by the relations

f_{i} + X_{i}^{m}h_{i} − X_{i}^{m+2},

and viewing these relations as belonging to the power series ring U instead of the polynomial ring V, we find that the ring

Ã = U/(f_{1}, …, f_{n})

has the desired properties:

1. The natural homomorphism from Ã to A induces an isomorphism on the tangent spaces, since the linear terms of the f_{i} generate the kernel of the induced map Φ_{U} ⟶ Φ_{A} on the tangent spaces.

2. The quotient Ã is a finitely generated Z_{p}-module, generated by the images of the monomials of degree ≤ n(m + 1), since the relations allow us to rewrite any monomial of higher degree in terms of ones of lower degree.

2.6 A criterion for complete intersections

The results we have accumulated so far allow us to give an important criterion for an object A to be a complete intersection:

Theorem 2.11 Let A be an object of C which is a finitely generated torsionfree Z_{p}-module. If #Φ_{A} ≤ #(Z_{p}/η_{A}) < ∞, then A is a complete intersection.

Proof: Let ϕ : Ã ⟶ A be the surjective morphism given by the resolution theorem (thm. 2.10). Then we have

#(Z_{p}/η_{A}) ≥ (#Φ_{A}) = (#ΦÃ) ≥ #(Z_{p}/ηÃ),

where the first inequality is by assumption, the second by the choice of Ã, and the third is by the equation (6). On the other hand, by equation (5), we have

#(Z_{p}/ηÃ) ≥ #(Z_{p}/η_{A}).

It follows that

η_{A} = ηÃ,

so that ϕ is an isomorphism by thm. 2.9. It follows that A is a complete intersection.

2.7 Proof of Wiles’ isomorphism criterion

Let us now recall the statement of Wiles’ isomorphism criterion:

Theorem 2.12 (Wiles) Let R and T be objects of C such that T is a finitely generated torsion-free Z_{p}-module, and let ϕ : R ⟶ T be a surjective morphism. If

#Φ_{R} ≤ #(Z_{p}/η_{T}) < ∞,

then ϕ is an isomorphism.

Proof: We have:

#(Z_{p}/η_{T}) ≤ #Φ_{T} ≤ #Φ_{R} ≤ #(Z_{p}/η_{T}),

where the first inequality is by equation (6), the second follows from the surjectivity of ϕ, and the third follows from the assumption of the theorem. Therefore,

#Φ_{T} = #(Z_{p}/η_{T}),

and hence T is a complete intersection. Since the orders of Φ_{R} and Φ_{T} are the same, ϕ induces an isomorphism between them. Hence ϕ is an isomorphism R ⟶ T, by thm. 2.8. This completes the proof.

2.8 The relative invariant η_{T′/T}

.

Let π : T′ ⟶ T be a map of objects of C. We will generalize slightly the invariant η of the previous sections, and introduce a relative invariant η_{T′/T} defined by

η_{T′/T} = π(Ann_{T′} ker π),

which is an ideal of T.

We now describe a set-up in which the relative congruence ideal can be computed, in the case where T and T′ are finite flat and reduced.

Suppose that Λ′ is a free T′-module and Λ is a free T -module of the same rank, k. The module Λ can also be viewed as a T′-module via the map T′ ⟶ T. By choosing isomorphisms Λ′ ≃ T^{′k} and Λ ≃ T^{k}, the map π induces a map α : Λ′ ⟶ Λ of T′-modules.

Proposition 2.13 If β : Λ ⟶ Λ′ is an injective map of T′-modules such that Λ′/βΛ has no Z_{p}-torsion, then

η_{T′/T} = Ann_{T} (Λ/αβΛ).

Proof: The map

hom_{T′}(T, T′) ⟶ Ann_{T′} ker π
f ↦ f(1)

is an isomorphism, and hence hom_{T′}(Λ, Λ′) ≃ M_{n}(Ann_{T′} ker π). It follows that after choosing an isomorphism of Λ′ with T^{′k}, we have β(Λ) ⊂ (Ann_{T′} ker π)^{k}. Consider the exact sequence

0 ⟶ (Ann_{T′} ker π)^{k}/βΛ ⟶ Λ′/βΛ ⟶ Λ′/(Ann_{T′} ker π)^{k} ⟶ 0. (9)

Let r(M) = dim_{Q_{p}}(M ⊗ Q_{p}) denote the Z_{p}-rank of a finitely generated Z_{p}- module M. Since T′ is finite flat and reduced, we have

r(Ann_{T′} ker π) + r(ker π) = r(T′),

so that r(Ann_{T′} ker π) = r(T). Since β is injective, it follows that

r((Ann_{T′} ker π)^{k}) = r(Λ) = r(βΛ),

and therefore the leftmost module in the exact sequence (9) is Z_{p}-torsion. Therefore, it is trivial, since Λ′/βΛ is assumed to be torsion-free. It follows that

Λ′/βΛ ≃ Λ′/(Ann_{T′} ker π)^{k}.

Applying α yields the proposition:

Λ/αβΛ ≃ Λ/η_{T′/T} Λ.

2.9 Interpretation of η_{T′/T} in the Gorenstein case

Assume now that the rings T′ and T, in addition to being finite flat and reduced, are Gorenstein. Let π^{∨} : \underline{T}^{∨} ⟶ T^{′∨} be the dual map. By using the identifications of T and T′ with their duals, we obtain an element

π ∘ π^{∨} : T ≃ T^{∨} ⟶ T^{′∨} ≃ T′ ⟶ T,

which gives an element of T which is well-defined up to a unit.

Proposition 2.14 The ideal of T generated by the image of π ∘ π^{∨} is equal to η_{T′/T}.

Proof: Let f be a T′-module generator of T^{′∨}. The image of T^{∨} by π^{∨} is the set of all functions in T^{′∨} of the form x ↦ f(λx), for some λ ∈ Ann(ker π). The proposition follows.

This result allows us to give a relation between the relative congruence ideal and the absolute ones in the case where the rings T and T′ are Gorenstein, namely:

Corollary 2.15 If the rings T and T′ are finite flat and reduced and satisfy the Gorenstein condition, then

η_{T′} = η_{T} · π_{T} (η_{T′/T}).

This formula allows us to compute the invariant η_{T′} in terms of the relative invariant, and will be used later in studying the variation of η_{T} for Hecke rings when one increases the level.

3 Interpretation of Φ_{R_{Σ}} and η_{T_{Σ}}

To show theorem 1.13 Wiles must show that the inequality

#Φ_{R_{Σ}} ≤ #(Z_{5}/η_{T_{Σ}})  (10)

is satisfied for all finite sets Σ.

In section 3.1 we study the left hand side and show how to interpret Φ_{R_{Σ}} as the (dual of) a subgroup of a Galois cohomology group. We give an explicit formula for the right hand side in 3.2.

3.1 Interpretation of Φ_{R_{Σ}}

In this section we explain how one defines the symmetric square motive and the Selmer group associated to it, and how this Selmer group is (dual to) the tangent space Φ_{R_{Σ}}.

Let X = X_{0}(17), and let T = end^{0}(T_{5}(X)) denote the module of trace zero endomorphisms of the 5-adic Tate module T_{5}(X); it is a free Z_{5}-module of rank 3, equipped with a natural continuous G_{Q}-action, defined by

σ(t) = ρ_{0}(σ)tρ_{0}(σ)^{−1},

for all t ∈ T and σ ∈ G_{Q}. Define

V = T ⊗ Q_{5}, A = T ⊗ (Q_{5}/Z_{5}).

The vector space V and the Z_{5}-divisible module A inherit a Galois action from T in the obvious way, and there is an exact sequence of G_{Q}-modules

0 ⟶ T ⟶ V ⟶ A ⟶ 0.

Local behaviour at 5: Since X = X_{0}(17) is ordinary at 5, the module T_{5}(X) is equipped with a two-step filtration

0 ⊂ Z_{5}(1) ⊂ T_{5}(X)

preserved by the action of the inertia group I_{5}. Now, T contains a natural 1-dimensional I_{5}-stable subspace, consisting of the nilpotent endomorphisms which preserve this filtration. Call this subspace T_{(5)}^{o}. Likewise, define V_{(5)}^{o} and A_{(5)}^{o} by

V_{(5)}^{o} = T_{(5)}^{o} ⊗ Q_{5}, A_{(5)}^{o} = T_{(5)}^{o} ⊗ Q_{5}/Z_{5}.

A more careful study shows that T is an ordinary representation, i.e., it is equipped with a 3-step filtration with 1-dimensional quotients on which I_{5} acts by powers of the cyclotomic character. But we will not need this here.

Local behaviour at 17: Likewise, the fact that X_{0}(17) has split multiplicative reduction at 17 implies that T a_{5}(X) is equipped with a two-step filtration

0 ⊂ Z_{5}(1) ⊂ T a_{5}(X)

preserved by the action of the decomposition group D_{17}; this allows one to define as before a rank one D_{17}-stable submodule of T, consisting of the nilpotent endomorphisms preserving this filtration. Call this submodule T_{(17)}^{o}, and define as before V_{(17)}^{o} and A_{(17)}^{o} by

V_{(17)}^{o} = T_{(17)}^{o} ⊗ Q_{5}, A_{(17)}^{o} = T_{(17)}^{o} ⊗ Q_{5}/Z_{5}.

The Selmer group S_{Σ}(Q, A): Given a set of primes Σ not containing {5, 17}, we define a system J_{r} of subgroups of the local Galois cohomology groups

J_{r} ⊂ H^{1}(Q_{r}, A),

for each prime r of Q:

1. If r ∉ Σ ∪ {5, 17}, then

J_{r} = H^{1}(Q_{r}^{nr}/Q_{r}, A) := ker (H^{1}(Q_{r}, A) ⟶ H^{1}(I_{r}, A)) .

2. If r ∈ Σ, then

J_{r} = H^{1}(Q_{r}, A).

3. If r = 17, then

J_{17} = ker (H^{1}(Q_{17}, A) ⟶ H^{1}(Q_{17}, A/A_{(17)}^{o})) .

4. If r = 5, then

J_{5} = ker (H^{1}(Q_{5}, A) ⟶ H^{1}(I_{5}, A/A_{(5)}^{o})) .

Define the Selmer group (relative to Σ) by the formula

S_{Σ}(Q, A) = ker (H^{1}(Q, A) ⟶ ∏_{r} H^{1}(Q_{r}, A)/J_{r}) .

One can also define the Selmer groups S_{Σ}(Q, A_{5^{n}}) ⊂ H^{1}(Q, A_{5^{n}}), in the obvious way. It follows from the fact that H^{0}(Q, A_{5}) = 0 that S_{Σ}(Q, A_{5}) = S_{Σ}(Q, A)_{5}, and that

S_{Σ}(Q, A) = lim_{⟶} S_{Σ}(Q, A_{5^{n}}).

The universal deformation class: Recall that R_{Σ} is the universal deformation ring associated to the set of primes Σ, and let P_{R_{Σ}} be the kernel of the base point map R_{Σ} ⟶ Z_{5}.

There is a natural split exact sequence

1 ⟶ M_{2}(Φ_{R_{Σ}}) ⟶ GL_{2}(R_{Σ}/P_{R_{Σ}}^{2}) ⟶ GL_{2}(Z_{5}) ⟶ 1,

where the action of GL_{2}(Z_{5}) on the space of matrices M_{2}(Φ_{R_{Σ}}) is by conjugation. By the standard cohomological construction, the Galois representation

ρ′ : G_{Q} ⟶ GL_{2}(R_{Σ}/P_{R_{Σ}}^{2})

coming from the universal deformation gives rise to a “universal deformation class”

u_{Σ} ∈ H^{1}(Q, M_{2}(Φ_{R_{Σ}})).

The class u_{Σ} is constructed explicitly as follows. Let ρ̃_{0} be the natural lift of ρ_{0} to GL_{2}(R_{Σ}/P_{R_{Σ}}^{2}). Then

u_{Σ}(σ) = ρ̃_{0}(σ)ρ′(σ)^{−1}.

Since det(u_{Σ}(σ)) = 1, it follows that u_{Σ}(σ) can be viewed as belonging to the Lie algebra sl_{2}(Φ_{R_{Σ}}) = T ⊗ Φ_{R_{Σ}}. The class u_{Σ} sets up a homomorphism

ϕ_{Σ} : hom(Φ_{R_{Σ}}, Q_{5}/Z_{5}) ⟶ H^{1}(Q, A)

through the natural rule

ϕ_{Σ}(f) = f(u_{Σ}).

Claim 3.1 The homomorphism ϕ_{Σ} gives an isomorphism

ϕ_{Σ} : hom(Φ_{R_{Σ}}, Q_{5}/Z_{5}) ⟶^{≃} S_{Σ}(Q, A).

Proof: First, one must show that ϕ_{Σ} actually maps to the Selmer group. This follows from the definitions. At the prime 5, the assumption that the deformation ρ′ is admissible means that

ρ′|_{I_{5}} = ( \begin{matrix}ϵ & Ψ′ \\ 0 & 1 \end{matrix} ) , ρ̃_{o}|_{I_{5}} = ( \begin{matrix}ϵ & Ψ̃_{o} \\ 0 & 1 \end{matrix} ) ,

and hence the restriction of the universal class u_{Σ} to I_{5} belongs to

H^{1}(I_{5}, T_{(5)}^{o} ⊗ Φ_{R_{Σ}}).

Likewise, one checks that the restriction of the universal class u_{Σ} to the decomposition group D_{17} belongs to H^{1}(Q_{17}, T_{(17)}^{o} ⊗ Φ_{R_{Σ}}), so that Φ_{Σ}(f) does belong to S_{Σ}(A). The fact that ϕ_{Σ} is an isomorphism is a formal consequence of the universality of R_{Σ}. For, distinct elements of hom(Φ_{R_{Σ}}, Q_{5}/Z_{5}) giving rise to the same element in S_{Σ}(Q, A) would give distinct maps R_{Σ} ⟶ R (for some local ring R) corresponding to equivalent (i.e., equal) deformations, contradicting the uniqueness clause in the definition of the universal deformation ring. This proves injectivity. Surjectivity is proved in exactly the same way.

3.2 A formula for η_{T_{Σ}}

Recall that

f(q) = ∑_{n} a_{n}q^{n}

is the normalized eigenform with integer coefficients corresponding to X = X_{0}(17). The following gives a precise formula for #(Z_{5}/η_{T_{Σ}}).

Theorem 3.2 Up to units in Z_{5}^{∗}, we have:

#(Z_{5}/η_{T_{Σ}}) = ∏_{q∈Σ}(q − 1)(a_{q}^{2} − (q + 1)^{2}).

Observe that when Σ = ∅, then T_{∅} is the Z_{5}-algebra of Hecke operators acting on S_{2}(17, Z_{5}), and hence, since X_{0}(17) is of genus 1,

T_{∅} ≃ Z_{5}.

The base point map T_{∅} ⟶ Z_{5} is the identity map; hence, in that case

η_{T_{∅}} = (1).

We prove theorem 3.2 by induction on the size of Σ. Assume the theorem is true for Σ, and let Σ′ = Σ ∪ {q}, where q is a prime not in Σ. We view these rings, equipped with the base point maps arising from the forms f_{Σ} and f_{Σ′}, as objects in the category C introduced in section 2. To apply induction, one needs a surjective map T_{Σ′} ⟶ T_{Σ} between these objects. We being by constructing such a map.

3.2.1 A map T_{Σ′} ⟶ T_{Σ}

Since N_{Σ′} = q^{2}N_{Σ}, there is a natural “degeneracy map”

ξ : X_{0}(N_{Σ′}) ⟶ X_{0}(N_{Σ})^{3},

given by the formula:

ξ(τ) = (τ, qτ, q^{2}τ).

This map induces by Pic functoriality a map i:

i : J_{0}(N_{Σ})^{3} ⟶ J_{0}(N_{Σ′}).

Define a map β_{0}:

β_{0} : J_{0}(N_{Σ}) ⟶ J_{0}(N_{Σ})^{3}

by the formula

β_{0}(P) = (qP, −T_{q}P, P).

The map β_{0} is a map of T(Σ)-modules, and i commutes with the obvious action of T^{0}(Σ′) - in fact i respects the natural action of all the Hecke operators, except U_{q} which does not act on J_{0}(N_{Σ})^{3} in any obvious way. However, we do have:

Lemma 3.3 The image of J_{0}(N_{Σ})^{3} in J_{0}(N_{Σ′}) under i is stable under the action of the Hecke operator U_{q}, and U_{q} induces the endomorphism

( \begin{matrix}T_{q} & q & 0 \\ −1 & 0 & q \\ 0 & 0 & 0 \end{matrix} )

acting on J_{0}(N_{Σ})^{3}.

Proof: This is verified by a direct calculation. It is convenient to view elements of J_{0}(N_{Σ}) as formal sums of degree 0 of objects of the form [C], where C is an elliptic curve equipped with the appropriate N_{Σ} level structure, and elements of J_{0}(N_{Σ′}) as sums of objects of the form [C_{1} → C_{2} → C_{3}], where the C_{i} are curves with level N_{Σ} structure and the arrows are cyclic q-isogenies whose composite is cyclic of degree q^{2}. The formula for i on such objects is:

i([C_{1}], [C_{2}], [C_{3}]) = ∑_{A,B}[C_{1} → A → B] + ∑_{A,B}[A → C_{2} → B] +  (11)

+ ∑_{A,B}[A → B → C_{3}],

where the sums in each case are taken over all possible A, B so that the composite arrow is cyclic of degree q^{2}. (Hence, each sum contains q(q + 1) distinct terms.) The formula for U_{q} on J_{0}(N_{Σ′}) is

U_{q}([A → B → C]) = ∑_{D}[B → C → D],  (12)

where again the sum is taken over the q possible D such that the composite isogeny is cyclic of degree q^{2}. Using equations (11) and (12), one checks directly that i(J_{0}(N_{Σ})^{3}) is stable under U_{q}, and in fact that

U_{q}(i([C_{1}], [C_{2}], [C_{3}])) = i(T_{q}[C_{1}] + q[C_{2}], −[C_{1}] + q[C_{3}], 0),

so that U_{q} acts by the matrix

( \begin{matrix}T_{q} & q & 0 \\ −1 & 0 & q \\ 0 & 0 & 0 \end{matrix} ) ,

as was to be shown.

Let now β = iβ_{0}. From lemma 3.3, one sees that image of β is stable under U_{q}, and in fact, is killed by U_{q}. Hence, one can define a natural ring homomorphism

π : T(Σ′) ⟶ T(Σ)

sending an operator to its restriction to β(J_{0}(N_{Σ})).

Lemma 3.4 The map π : T(Σ′)⊗Z_{5} ⟶ T(Σ)⊗Z_{5} is surjective, and maps the ideal M_{Σ′} to the ideal M_{Σ}.

Proof: To show surjectivity, since T(Σ) ⊗ Z_{5} is generated by the Hecke operators T_{l} and U_{l} for l ∈ {17} ∪ Σ, it suffices to show that these operators are in the image of π. The operators T_{l} with l ≠ q and U_{l} are just the images of the corresponding operators in T(Σ′) ⊗ Z_{5}. As for the operator T_{q}, it follows directly from the Chebotarev density theorem (using the fact that T(Σ) ⊗ Z_{5} is 5-adically topologically complete) that it is in the ring generated by the good Hecke operators T_{l}, l ≠ q. Finally, π maps M_{Σ} to M_{Σ′}, since it sends T_{l} − a_{l} to T_{l} − a_{l} (l ∉ {17} ∪ Σ) and sends U_{l} to U_{l} if l ≠ q, and to 0 otherwise.

Hence, by completing at the ideals M_{Σ′} and M_{Σ}, we get a surjective map (which by abuse of notation we also call π):

π : T_{Σ′} ⟶ T_{Σ},

and we can define a relative congruence ideal associated to this map, as in sec. 2.8.

3.2.2 Computing η_{T′/T}

Let T = T_{Σ} and T′ = T_{Σ′} be the Hecke rings at level Σ and Σ′. By applying the induction hypothesis and cor. 2.15 of sec. 2.8, we are reduced to showing that

π_{T}(η_{T′/T}) = ((q − 1)(a_{q}^{2} − (q + 1)^{2}).

In fact, we will show:

η_{T′/T} = ((q − 1)(T_{q}^{2} − (q + 1)^{2})),

where T_{q} ∈ T is the qth Hecke operator.

Let the maps i, β_{0}, and β be as above, and let j, α_{0} and α denote their duals.

The maps i, j, α_{0} and β_{0} induce maps on the 5-adic Tate modules of the associated abelian varieties, which, by abuse of notation, we will denote by the same letters, and we have a sequence of maps:

\begin{matrix}T a_{5}(J_{0}(N_{Σ}))^{3} & ⟶^{i} T a_{5}(J_{0}(N_{Σ′})) & ⟶^{j} & T a_{5}(J_{0}(N_{Σ}))^{3} \\ β_{0} ↑ & & & ↓ α_{0} \\ T a_{5}(J_{0}(N_{Σ})) & & & T a_{5}(J_{0}(N_{Σ})) \end{matrix} .

Now let

Λ := T a_{5}(J_{0}(N_{Σ})) ⊗_{T(Σ)} T_{Σ}, Λ′ := T a_{5}(J_{0}(N_{Σ′})) ⊗_{T(Σ′)} T_{Σ′}.

The maps α and β induce maps on these tensored modules, which, by abuse of notation, we will again call α and β. Hence, we have a diagram of morphisms:

Λ ⟶^{β} Λ′ ⟶^{α} Λ.

Note that β and α are maps of T_{Σ′}-modules when we endow Λ with its T_{Σ′}- module structure coming from the map T_{Σ′} ⟶ T_{Σ}. Furthermore, we have the following key propositions:

Theorem 3.5 The rings T_{Σ} and T_{Σ′} are Gorenstein, and Λ ≃ T_{Σ}^{2} and Λ′ ≃ T_{Σ′}^{2}.

This theorem was proved first by Mazur in [Mz1] in the case of J_{0}(N), N prime, and has since been extended by a number of people, including Wiles in [Wi]; cf. also [Ed].

Theorem 3.6 The module Λ′/βΛ has no Z_{5}-torsion.

Since T a_{5}(J_{0}(N_{Σ})^{3})/β_{0}(T a_{5}(J_{0}(N_{Σ}))) is torsion free, it suffices to show that T a_{5}(J_{0}(N_{Σ′})/i(T a_{5}(J_{0}(N_{Σ})^{3}) is torsion free, at least after tensoring with T_{Σ′}. This was proved by Ribet [Ri1] in the case of the map

i : J_{0}(N)^{2} ⟶ J_{0}(Nq), gcd(q, N) = 1

induced by the obvious two degeneracy maps, using some results of Ihara. Building on Ribet’s result, Wiles has extended it to cover the case he needs; cf. [Wi].

Thanks to thm. 3.5 and thm. 3.6, we are in the situation of prop. 2.13, so that it suffices now to compute the modules Λ/αβΛ.

Lemma 3.7 The map j ∘ i, viewed as an endomorphism of T a_{5}(J_{0}(N_{Σ}))^{3} ⊗ T_{Σ} in M_{3}(T_{Σ}), is equal to

( \begin{matrix}q(q + 1) & qT_{q} & T_{q}^{2} − (q + 1) \\ qT_{q} & q(q + 1) & qT_{q} \\ T_{q}^{2} − (q + 1) & qT_{q} & q(q + 1) \end{matrix} ) .

Proof: One proceeds as in the proof of lemma 3.3, using the explicit formula for i given there, combined with the formula

j([A → B → C]) = ([A], [B], [C]).

Finally we come to:

Proposition 3.8 αβ = −q(q − 1)(T_{q}^{2} − (q + 1)^{2}).

Proof: One notes that α_{0}, the dual of β_{0}, is given by the formula

α_{0}([C_{1}], [C_{2}], [C_{3}]) = q[C_{1}] − T_{q}[C_{2}] + [C_{3}].

αβ([C]) = α_{0}jiβ_{0}([C]) = α_{0}ji(q[C], −T_{q}[C], [C])
= α_{0}(−(q − 1)(T_{q}^{2} − (q + 1)^{2})[C], 0, 0)
= −q(q − 1)(T_{q}^{2} − (q + 1)^{2})[C].

Corollary 3.9 η_{T′/T} = ((q − 1)(T_{q}^{2} − (q + 1)^{2})).

Proof: By prop. 3.8, we have

Ann_{T}(Λ/αβΛ) = ((q − 1)(T_{q}^{2} − (q + 1)^{2})),

since q ≠ 5 is a unit in T_{Σ}. The proof follows from prop. 2.13.

The proof of thm. 3.2 now follows from cor. 3.9 and cor. 2.15.

3.3 Relation with the Bloch-Kato conjectures

In this section we mention briefly the relation between η_{T_{Σ}} and special values of L-functions, and the relation between Wiles’ inequality and the Bloch-Kato conjecture. This section is not logically needed for the proof of thm. 1.13.

Let us drop for this section our running assumption that X is the modular elliptic curve X_{0}(17). Let X be an arbitrary (semistable, to simplify) modular elliptic curve, and define algebraic numbers α_{q} and β_{q} for each prime q by the conditions

α_{q} + β_{q} = a_{q}, α_{q}β_{q} = q, if X has good reduction at q,
α_{q} = ±1, β_{q} = 0 otherwise,

where α_{q} is 1 precisely when X has split multplicative reduction at q. Now define the local L-function associated to T by

L_{q}(T, s) = (1 − α_{q}^{2}q^{−s})^{−1}(1 − β_{q}^{2}q^{−s})^{−1}(1 − α_{q}β_{q}q^{−s})^{−1}.

Finally, we define the global L-function associated to T and the set Σ by

L_{Σ}(T, s) = ∏_{q∉Σ} L_{q}(T, s).

This function extends to an entire function on the complex plane, and satisfies a functional equation interchanging s and 3 − s.

By using Rankin’s method and some techniques of Shimura, Sturm [St] has shown that L_{Σ}(T, 2) is a rational multiple of a transcendental period Ω,

Ω = ∫_{X(C)} ω ∧ ω̄,

where ω is a Néron differential attached to X.

If x ∈ Q, let [x]_{5} = 5^{ord_{5}(x)} denote the 5-part of x. Hida, using the fundamental work of Ribet and others, has succeeded in relating the special value L_{Σ}(T, 2) to congruences between modular forms in a very precise way (usually in the case where Σ is minimal). His ideas, combined with the calculations of sec. 3.2, lead to the following relation:

Theorem 3.10

#(Z_{5}/η_{T_{Σ}}) = [\frac{L_{Σ}(T, 2)}{Ω}] .

(Of course, one has a similar formula with any prime p replacing 5.) Thanks to thm. 3.10, Wiles conjectured inequality (10) can be reformulated as:

#S_{Σ}(Q, A) ≤ [\frac{L_{Σ}(T, 2)}{Ω}] .

This fits into the framework of the general conjectures of Bloch and Kato on the special values of the L-functions attached to motives which are a vast generalization of Dirichlet’s class number formula and the Birch Swinnerton Dyer conjecture.

Such a reformulation is striking, since it gives a justification for the Shimura-Taniyama conjecture from a rather unexpected point of view. On the other hand, a proof of the inequality could be a priori quite difficult - one just has to think of how elusive the Birch Swinnerton-Dyer conjecture remains, even for modular elliptic curves!

Fortunately, the Bloch-Kato conjecture for the symmetric square is “easier” than the Birch Swinnerton-Dyer conjecture, in one important respect: the special value of the symmetric square L-function is always non-zero at the point of interest. Hence the difficulties which are caused by the unpredictable behaviour of the order of vanishing at s = 1 for the L-function of an elliptic curve, do not arise in this situation.

Furthermore, the ideas of Wiles and Flach allow us to come to grips with the Selmer groups of the symmetric square in many cases, (and, for example, to prove their finiteness). Naturally, one hopes that these techniques can be pushed further to prove the analogue of the class number formula in this setting, and hence, the full Shimura-Taniyama conjecture.

4 Proof of the inequality #Φ_{R_{Σ}} ≤ #(Z_{5}/η_{T_{Σ}})

4.1 Reduction to the case Σ = ∅

Wiles first shows that the inequality

#(Φ_{R_{Σ}}) ≤ #(Z_{5}/η_{T_{Σ}})

follows from the corresponding inequality for the minimal set Σ = ∅, by a simple induction argument.

Assume the inequality is true for Σ, and let Σ′ = Σ ∪ {q} for some prime number q. The exact sequence

0 ⟶ S_{Σ}(Q, A) ⟶ S_{Σ′}(Q, A) ⟶ H^{1}(I_{q}, A)^{D_{q}}

implies that

#Φ_{Σ′} ≤ (#Φ_{Σ})(#H^{1}(I_{q}, A)^{D_{q}}),  (13)

and our explicit formula for η_{T_{Σ}} of thm. 3.2 implies that

#(Z_{5}/η_{T_{Σ′}}) = #(Z_{5}/η_{T_{Σ}})((q + 1)^{2} − a_{q}^{2})(q − 1).  (14)

Combining equations (13) and (14), we are reduced to showing that

#H^{1}(I_{q}, A)^{D_{q}} ≤ [((q + 1)^{2} − a_{q}^{2})(q − 1)].

This follows from an explicit calculation; since I_{q} acts trivially on A,

H^{1}(I_{q}, A)^{D_{q}} = hom_{D_{q}}(I_{q}, A) = hom_{D_{q}}(Z_{5}(1), A).

Since I_{q} acts trivially on Z_{5}(1) and A, the group D_{q} acts on both via D_{q}/I_{q}, which is topologically generated by the Frobenius element Frob_{q}. The eigenvalue of Frob_{q} on Z_{5}(1) is q, and on A the eigenvalues are

1, α_{q}^{2}/q, β_{q}^{2}/q.

Hence, we find

#H^{1}(I_{q}, A)^{D_{q}} = [(q − 1)(q^{2} − α_{q}^{2})(q^{2} − β_{q}^{2})].  (15)

Using the identity α_{q}^{2} + β_{q}^{2} = a_{q^{2}} = a_{q}^{2} − 2q, we find that the right hand side is [(q − 1)((q + 1)^{2} − a_{q}^{2})], as was to be showed.

Remark: In the end, all the inequalities that Wiles proves turn out to be actual equalities. Hence, the above calculation indicates that the reduction map

S_{Σ′}(Q, A) ⟶ H^{1}(I_{q}, A)^{D_{q}}

is surjective for all choices of Σ′. This assertion, in the case Σ′ = {q} and Σ = ∅ for certain primes q, turns out to be a key ingredient in bounding the order of S_{∅}(Q, A), as we will explain in the next section.

4.2 Proof of the inequality for Σ = ∅

We are reduced to showing the inequality (10) for the minimal set Σ = ∅.

Theorem 4.1 #S_{∅}(Q, A) = 1.

There are at least three ways in which this can be proved:

1. The ad hoc method: Taking cohomology of the exact sequence

0 ⟶ A_{5} ⟶ A ⟶ A ⟶ 0,

and using the fact that H^{0}(Q, A) = 0, gives the isomorphism:

H^{1}(Q, A)_{5} ≃ H^{1}(Q, A_{5}).

Let K/Q be the splitting field of A_{5}, and let G = Gal(K/Q) ≃ GL_{2}(F_{5}) be its Galois group. As we will see later (lemma 4.9) the cohomology group H^{1}(G, A_{5}) vanishes, and hence the inflation-restriction sequence gives an injection

H^{1}(Q, A_{5}) ⟶ hom_{G}(Gal(K_{(5)}/K), A_{5}),

where K_{(5)} is the maximal abelian extension of K of exponent 5, and the action of G is the natural one. Let K(5)^{′} be the abelian extension of exponent 5 which is unramified outside of 5 and 17. Then S_{∅}(Q, A_{5}) injects into

hom_{G}(Gal(K(5)^{′}/K), A_{5}),

and one can therefore hope to control S_{∅}(Q, A_{5}) by controlling the size of this group, which is (dual to) a piece of the 5-part of a well-defined class group of K. Since the field K is of degree 480 over Q, this calculation seems rather daunting. It would be interesting to see if it can be carried out in practice, (which would probably require a more careful description of the image of S_{∅}(Q, A) as well as some theoretical insights into how the class group of the field K behaves.) At any rate, it is worth noting that at this stage we have reduced the proof of thm 1.1 to a finite amount of (machine) calculation.

2. Flach’s approach: In [Fl], Flach shows that if the module A arises from the symmetric square of any (semi-stable) modular elliptic curve X, then deg ϕ annihilates #S_{Σ}(Q, A), where ϕ is a minimal degree of a modular parametrization from X_{0}(N) to X. In our case, this modular parametrization is the identity map X_{0}(17) ⟶ X_{0}(17) of degree 1. Hence S_{Σ}(Q, A) is trivial.

Flach’s argument proceeds by constructing explicit “Kolyvagin-type” cohomology classes

c(l) ∈ H^{1}(Q, T^{∗}),

where T^{∗} = hom(T, Z_{5}(1)) is the Kummer dual of T. Roughly speaking, these classes are constructed as follows: the degeneracy maps of section 3.2 give a map of X_{0}(17l) into X_{0}(17) × X_{0}(17) whose image is the Hecke correspondance T_{l} on the surface X_{0}(17)×X_{0}(17). Flach defines a certain modular unit u on X_{0}(17l) whose divisor becomes trivial on the image T_{l}. The divisor T_{l}, together with the unit u, gives rise to a class in a certain algebraic Kgroup H^{1}(X_{0}(17) × X_{0}(17), K_{2}), which maps to the étale cohomology group

H_{et}^{3}(X_{0}(17) × X_{0}(17), Z_{5}(2)).

The Hochshild-Serre spectral sequence, combined with the fact that

H^{0}(Q, H_{et}^{3}(X_{0}(17) × X_{0}(17), Z_{5}(2))) = 0

allows Flach to transfer this class to the group

H^{1}(Q, H_{et}^{2}(X_{0}(17) × X_{0}(17), Z_{5}(2))),

which maps, via the Künneth projections, to

H^{1}(Q, H_{et}^{1}(X_{0}(17), Z_{5}(1))^{⊗2}).

By projecting to the symmetric tensors, Flach obtains his class

c(l) ∈ H^{1}(Q, T^{∗}).

By a careful and delicate analysis, Flach shows that c(l) is ordinary at 5 and ramified only at l, and that its restriction to the inertia group I_{l}, on the other hand, generates H^{1}(I_{l}, T^{∗})^{D_{l}}, at least for sufficiently many l, (those for which 5 does not divide ((l + 1)^{2} − a_{l}^{2})). An application of the local Tate duality and the global reciprocity law of class field theory shows that for all these l, and for all s ∈ S_{Σ}(Q, A), the restriction of s to the decomposition group D_{l} is trivial. Such stringent local conditions force the Selmer group S_{Σ}(Q, A) to be trivial, by an application of the Chebotarev density theorem. All of this is very well explained in Flach’s paper [Fl].

Flach’s method is inspired by the work of Kolyvagin, who constructed ramified classes c(l) in different contexts to obtain annihilators of the appropriate Selmer groups [Ko1], thereby establishing some striking cases of the Birch Swinnerton Dyer conjecture. In his seminal paper [Ko] on Euler systems, Kolyvagin showed how to strengthen this argument to obtain actual upper bounds on the Selmer groups, by constructing systems of cohomology classes c(l_{1}, …, l_{k}), extending his classes c(l), satisfying a subtle compatibility property relating the restriction to D_{l_{k}} of c(l_{1}, …, l_{k}) to that of the previous class c(l_{1}, …, l_{k−1}).

It is a tantalizing question to see if Flach’s cohomology classes c(l) can be likewise extended to an Euler system, and whether this approach can yield a proof of the inequality (10) in more general contexts.

3. Wiles’ approach: Wiles proves the inequality (10) for X_{0}(17) by a completely different route, which relies crucially on the fact that the Hecke ring T_{∅} is a local complete intersection.

We will explain Wiles’ proof here in some detail.

4.2.1 The group S_{Σ}^{mod}(Q, A)

For any finite set Σ not containing 5 and 17, let

S_{Σ}^{mod}(Q, A) := Φ_{T_{Σ}}^{∨},

where M^{∨} = hom(M, Q_{5}/Z_{5}) denotes the Pontrjagin dual of a Z_{5}-torsion module. The surjective map Φ_{R_{Σ}} ⟶ Φ_{T_{Σ}} gives, upon passing to the duals, an inclusion

S_{Σ}^{mod}(Q, A) ⊂ S_{Σ}(Q, A).

The group S_{Σ}^{mod}(Q, A) should be thought of as the part of S_{Σ}(Q, A) which “comes from modular forms”.

If Σ_{1} ⊂ Σ_{2} are finite sets of primes not containing 5 and 17, then, by definition there is an exact sequence

0 ⟶ S_{Σ_{1}}(Q, A) ⟶ S_{Σ_{2}}(Q, A) ⟶ ⊕_{Σ_{2}−Σ_{1}}H^{1}(I_{q}, A)^{D_{q}}.

One could expect a similar statement for the modular Selmer groups, i.e., that the sequence

0 ⟶ S_{Σ_{1}}^{mod}(Q, A) ⟶ S_{Σ_{2}}^{mod}(Q, A) ⟶ ⊕_{Σ_{2}−Σ_{1}}H^{1}(I_{q}, A)^{D_{q}}

is also exact. Injectivity of the first map follows from the fact that T_{Σ_{2}} ⟶ T_{Σ_{1}}, and hence, Φ_{T_{Σ_{2}}} ⟶ Φ_{T_{Σ_{1}}}, is surjective.

Exactness at the second stage asserts that a modular deformation of level N_{Σ_{2}} which is unramified at the primes in Σ_{2} −Σ_{1}, already arises from a modular form of level N_{Σ_{1}}. This is very much in the spirit of Ribet’s celebrated “lowering the level” result [Ri2]. (Ribet only concerns himself with mod p modular forms, whereas what is needed here is some version for modular forms mod p^{k}.)

Say that a prime q is good if

(q + 1)^{2} − a_{q}^{2} ≠ 0 (mod 5).

Proposition 4.2 (Wiles) If Σ_{2} − Σ_{1} consists only of good primes, then the sequence

0 ⟶ S_{Σ_{1}}^{mod}(Q, A) ⟶ S_{Σ_{2}}^{mod}(Q, A) ⟶ ⊕_{Σ_{2}−Σ_{1}}H^{1}(I_{q}, A)^{D_{q}}

is exact.

This proposition is the main technical ingredient in Wiles’ proof of the inequality (10). The proof relies in an essential way on the work of Ribet and on its subsequent refinements and extensions. For details, the reader may consult [Wi].

Now, let q be a good prime. In our particularly simple situation, we know that S_{∅}^{mod}(Q, A) = 0, and hence prop. 4.2 gives an injective map

S_{{q}}^{mod}(Q, A) ↪ H^{1}(I_{q}, A)^{D_{q}}.

On the other hand

#S_{{q}}^{mod}(Q, A) = #Φ_{T_{{q}}} ≥ #(Z_{5}/η_{T_{{q}}}) = [(q − 1)((q + 1)^{2} − a_{q}^{2})],

where the first inequality follows from equation (6), and the last equality is thm. 3.2. Hence by using equation (15) and counting orders, we find that the map S_{{q}}^{mod}(Q, A) ⟶ H^{1}(I_{q}, A)^{D_{q}} is an isomorphism. Hence, so is the map S_{{q}}^{mod}(Q, A)_{5} ⟶ H^{1}(I_{q}, A)_{5}^{D_{q}}. Since S_{{q}}^{mod}(Q, A)_{5} injects into S_{{q}}(Q, A)_{5}, and since S_{{q}}(Q, A_{5}) surjects onto S_{{q}}(Q, A)_{5}, we have shown:

Lemma 4.3 If q is a good prime, then the map

S_{{q}}(Q, A_{5}) ⟶ H^{1}(I_{q}, A_{5})^{D_{q}} = hom_{D_{q}}(I_{q}, A_{5})

is surjective.

4.2.2 Local Tate duality

In this section (and this section only!) let A be a finite G_{Q}-module of p-power order, and let A^{∗} = hom(A, G_{m}) be the Kummer dual of A, equipped with the natural Galois action.

Proposition 4.4 (Tate) For all primes q cup-product induces a canonical non-degenerate pairing

⟨ , ⟩_{q} : H^{1}(Q_{q}, A) × H^{1}(Q_{q}, A^{∗}) ⟶ Q_{p}/Z_{p}.

The local Tate pairing is defined by composing the natural maps

H^{1}(Q_{q}, A) × H^{1}(Q_{q}, A^{∗}) ⟶^{∪} H^{2}(Q_{q}, A ⊗ A^{∗}) = H^{2}(Q_{q}, G_{m}) = Q/Z,

where the first is given by cup product and the last equality follows from local class field theory [CF].

A simple description of the local Tate pairing can be given when q ≠ p and the module A is unramified at q. In this case, the inflation-restriction sequence gives an exact sequence:

0 ⟶ H^{1}(Q_{q}^{nr}/Q_{q}, A) ⟶ H^{1}(Q_{q}, A) ⟶ H^{1}(I_{q}, A)^{D_{q}} ⟶ 0.

Since A is unramified, the term on the left can be identified with the module A_{D_{q}} of D_{q}-coinvariants of A, and the term on the right can be identified with

hom(I_{q}, A)^{D_{q}} = A(−1)^{D_{q}}.

One can show that the submodules A_{D_{q}} ⊂ H^{1}(Q_{q}, A) and A_{D_{q}}^{∗} ⊂ H^{1}(Q_{q}, A^{∗}) are the orthogonal complements of each other under the local Tate pairing, and that the induced pairings on A_{D_{q}} × A^{∗}(−1)^{D_{q}} and A_{D_{q}}^{∗} × A(−1)^{D_{q}} are the obvious ones.

If a and a′ are classes in H^{1}(Q, A) and H^{1}(Q, A^{∗}) respectively, let a_{v} and av^{′} denote their images in the local groups H^{1}(Q_{v}, A) and H^{1}(Q_{v}, A^{∗}).

Proposition 4.5 (Tate) For all a and a′, we have

∑_{v}⟨a_{v}, av^{′}⟩_{v} = 0.

Proof: We have

⟨a, a′⟩_{v} = inv_{v}(a ∪ a′),

where a ∪ a′ ∈ H^{2}(Q, G_{m}) is an element in the global Brauer group of Q. Hence the proposition follows directly from the global reciprocity law of class field theory [CF] which states that

∑_{v} inv_{v}(x) = 0, ∀x ∈ H^{2}(Q, G_{m}).

Now, let J_{q} ⊂ H^{1}(Q_{q}, A) be a choice of subgroups of the local cohomology groups, satisfying

J_{q} = H^{1}(D_{q}/I_{q}, A^{I_{q}}) for almost all q,

and for any such system of (J_{q}), define the generalized Selmer group:

S_{(J_{q})}(Q, A) =:= {s ∈ H^{1}(Q, A) | s_{q} ∈ J_{q} ∀q }.

If we define J_{q}^{∗} ⊂ H^{1}(Q_{q}, A^{∗}) to be the orthogonal submodules under the local Tate pairings, then J_{q}^{∗} also satisfies the condition:

J_{q}^{∗} = H^{1}(D_{q}/I_{q}, A^{∗I_{q}}) for almost all q.

We define the dual Selmer group of S_{(J_{q})}(Q, A) to be the Selmer group S_{(J_{q}^{∗})}(Q, A^{∗}).

The following Euler characteristic formula compares the orders of the Selmer group S_{(J_{q})}(Q, A) and its dual S_{(J_{q}^{∗})}(Q, A^{∗}).

Proposition 4.6 The Selmer groups S_{(J_{q})}(Q, A) and S_{(J_{q}^{∗})}(Q, A^{∗}) are finite, and

\frac{#S_{(J_{q})}(Q, A)}{#S_{(J_{q}^{∗})}(Q, A^{∗})} = \frac{#H^{0}(Q, A)}{#H^{0}(Q, A^{∗})}h_{∞} ∏_{q} h_{q},

where

h_{∞} = #H^{0}(R, A^{∗}), h_{q} = #H^{0}(Q_{q}, A^{∗})/[H^{1}(Q_{q}, A) : J_{q}].

For a proof of the general formula, see [Gre]. Note that when A is unramified at q ≠ p, then

J_{q} = H^{1}(D_{q}/I_{q}, A) ⇒ #H^{0}(D_{q}/I_{q}, A^{∗}) = [H^{1}(Q_{q}, A) : J_{q}],

and h_{q} = 1. Hence, h_{q} = 1 for almost all q, so that the above product makes sense.

4.2.3 Bounding S_{∅}(Q, A_{5}^{∗})

Define S_{Σ}(Q, A_{5}^{∗}) to be the dual Selmer group of S_{Σ}(Q, A_{5}), using the general construction of the previous section. Our goal in this section is to show

Proposition 4.7 #S_{∅}(Q, A_{5}^{∗}) = 1.

Proof: We begin by showing that any s ∈ S_{∅}(Q, A_{5}^{∗}) is locally trivial for many q. More precisely, if s ∈ S_{∅}(Q, A_{5}^{∗}), and q is any prime, let s_{q} be the image of s in H^{1}(Q_{q}, A_{5}^{∗}) by restriction. If q is not 5 or 17, then s_{q} belongs to H^{1}(D_{q}/I_{q}, A_{5}^{∗}).

Lemma 4.8 If s belongs to S_{∅}(Q, A_{5}^{∗}) and if q is a good prime, then s_{q} = 0.

Proof: For all γ ∈ S_{{q}}(Q, A_{5}), we have, by prop. 4.5,

∑_{v}⟨s_{v}, γ_{v}⟩_{v} = 0.  (16)

But if v ≠ q, then by definition ⟨s_{v}, γ_{v}⟩_{v} = 0. Hence, equation (16) reduces to:

⟨s_{q}, γ_{q}⟩_{q} = 0, ∀γ ∈ S_{{q}}(Q, A_{5}).

By lemma 4.3 (which is the main actor in this proof), it follows that

⟨s_{q}, α⟩_{q} = 0, ∀α ∈ H^{1}(I_{q}, A_{5})^{D_{q}}.

But this proves that s_{q} = 0, by the non-degeneracy of the local Tate pairing.

Let K = Q(A_{5}^{∗})(= Q(A_{5})) be the splitting field of A_{5}^{∗}, i.e., the smallest field through which ρ̄_{0} factors. By lemma 1.4, we have G = Gal(K/Q) ≃ GL_{2}(F_{5}).

Lemma 4.9 The restriction map

H^{1}(Q, A_{5}^{∗}) ⟶ H^{1}(K, A_{5}^{∗})

is injective.

Proof: From the inflation-restriction sequence, the kernel of this map is

H^{1}(G, A_{5}^{∗}) ≃ H^{1}(GL_{2}(F_{5}), Sym^{2}),

where Sym^{2} is the symmetric square of the standard two-dimensional representation of GL_{2}(F_{5}). Let Z ⊂ GL_{2}(F_{5}) denote the group of scalar matrices. The Hochschild-Serre spectral sequence

H^{p}(PGL_{2}(F_{5}), H^{q}(Z, Sym^{2})) ⇒ H^{p+q}(GL_{2}(F_{5}), Sym^{2})

shows that H^{p}(GL_{2}(F_{5}), Sym^{2}) = 0 for all p, since the group Z is of order prime to 5.

We now come to the proof of prop. 4.7. Let s be an element of S_{∅}(Q, A_{5}^{∗}). We will show that s = 0. To show this, it suffices to show that the restriction s̄ of s to H^{1}(K, A_{5}^{∗}) is zero, by lemma 4.9. Let L/K be the smallest extension of K such that s̄ ∈ hom(Gal(K̄/K), A_{5}^{∗}) factors through U = Gal(L/K). If s̄ is non-trivial, then the group U is isomorphic to (Z/5Z)^{3}, and G ≃ GL_{2}(F_{5}) acts on U by conjugation. Furthermore, U is isomorphic as a G-module to the symmetric square of the standard representation of GL_{2}(F_{5}). Let Γ = Gal(L/Q), so that we have an exact sequence of finite groups:

0 ⟶ U ⟶ Γ ⟶ G ⟶ 0.

Now fix a τ ∈ Γ such that the image of τ in G ≃ GL_{2}(F_{5}) is conjugate to

a matrix of the form ( \begin{matrix}2 & 0 \\ 0 & 3 \end{matrix} ), and is of order 4 in Γ. (Such a τ always

exists, since 4 is prime to 5.) Now let h be an arbitrary element of the group U, and choose a prime q of Q whose Frobenius element in Gal(L/Q) = Γ is equal to hτ. Such a prime exists, by the Chebotarev density theorem. Since ρ̄_{0}(Frob_{q}) has eigenvalues 2 and 3, it follows that

a_{q} = 0 (mod 5), q = 1 (mod 5),

so that (q + 1)^{2} − a_{q}^{2} = 4 ≠ 0 (mod 5). Therefore q is a good prime. Hence we can apply lemma 4.8 to conclude that s_{q} = 0, so that, if Q is a prime of K above q,

s̄(Frob_{Q}) = 0.

Since the extension K_{Q}/Q_{q} has residue degree 4, we have

Frob_{Q} = (hτ)^{4} = h(τhτ^{−1})(τ^{2}hτ^{−2})(τ^{3}hτ^{−3}) = h^{+},

where h^{+} is the projection of h ∈ U to the +1-eigenspace, U^{+}, for the action of τ on U. Since h ∈ U was arbitrary, it follows that s̄ annihilates U^{+}. The eigenvalues of τ on U are −1, −1, and +1, and hence U^{+} is non-trivial. Since U is an irreducilbe G-module, and s̄ is G-equivariant, it follows that s̄ = 0, as was to be shown.

4.2.4 Bounding S_{∅}(Q, A_{5})

Let J_{q} ⊂ H^{1}(Q_{q}, A_{5}) be the groups defined in sec. 3.1, but with A_{5} replacing A, and let J_{q}^{∗} ⊂ H^{1}(Q_{q}, A_{5}^{∗}) denote the orhogonal complements under the local Tate pairings, so that

S_{Σ}(Q, A_{5}) = S_{(J_{q})}(Q, A_{5}), S_{Σ}(Q, A_{5}^{∗}) = S_{J_{q}^{∗}}(Q, A_{5}^{∗}).

Applying proposition 4.6 in our situation, with Σ = ∅, we get:

\frac{#S_{∅}(Q, A_{5})}{#S_{∅}(Q, A_{5}^{∗})} = h_{∞}h_{5}h_{17},  (17)

where

h_{∞} = #H^{0}(R, A_{5}^{∗}), h_{q} = #H^{0}(Q_{q}, A_{5}^{∗})/[H^{1}(Q_{q}, A_{5}) : J_{q}].

Thus to compute #S_{∅}(Q, A_{5}) we are reduced to a series of local computations. We recall briefly the facts of local Galois cohomology that we will use. (For proofs of these, see [Sr2].)

1. If A is any finite module over D_{q} = G_{Q_{q}}, let h^{i}(A) denote #H^{i}(Q_{q}, A). Then h^{i}(A) = 1 if i > 2, and the Euler characteristic of A is defined by

χ(A) = \frac{h^{0}(A)h^{2}(A)}{h^{1}(A)}.

The Poitou-Tate formula for the local Euler characteristic is

χ(A) = |#A|_{q},

where |x|_{q} = q^{−ord_{q}(x)} is the usual q-adic valuation.

2. It is a direct consequence of the local Tate duality that h^{i}(A) = h^{2−i}(A^{∗}).

Lemma 4.10 h_{∞} = 25.

Proof: Since ρ̄_{0} is odd, complex conjugation acts on A_{5} with eigenvalues 1, −1, and − 1, and hence with eigenvalues −1, 1, 1 on A_{5}^{∗}. The result follows.

Lemma 4.11 h_{17} = 1.

Proof: Let A_{5}^{0} be the module (A_{(17)}^{0})_{5}, so that J_{1}7 is defined by the exact sequence

0 ⟶ J_{17} ⟶ H^{1}(Q_{17}, A_{5}) ⟶ H^{1}(Q_{17}, A_{5}/A_{5}^{0}).

The group G_{Q_{17}} acts on the line A_{5}^{0} by the cyclotomic character χ. More generally, there is a filtration

A_{5}^{0} ⊂ A_{5}^{1} ⊂ A_{5}

with one dimensional quotients, and the action of G_{Q_{17}} on the sucessive quotients is by χ, 1, and χ^{−1}. Using this explicit description, one verifies that h^{0}(A_{5}) = h^{0}(A_{5}^{∗}) = 1, that h^{0}(A_{5}^{0}) = 1, h^{0}(A_{5}^{0∗}) = 5, etc.

From the long exact cohomology sequence associated to

0 ⟶ A_{5}^{0} ⟶ A_{5} ⟶ A_{5}/A_{5}^{0} ⟶ 0,

we obtain

[H^{1}(Q_{17}, A_{5}) : J_{17}] = \frac{h^{1}(A_{5}/A_{5}^{0})h^{2}(A_{5})}{h^{2}(A_{5}^{0})h^{2}(A_{5}/A_{5}^{0})} = \frac{h^{0}(A_{5}/A_{5}^{0})}{χ(A_{5}/A_{5}^{0})} · \frac{h^{0}(A_{5}^{∗})}{h^{0}(A_{5}^{0∗})}
= \frac{5}{1} × \frac{1}{5} = 1.

Since h^{0}(A_{5}^{∗}) = 1, one obtains h_{17} = 1, which proves the lemma.

Lemma 4.12 h_{5} ≤ 1/25.

Proof: Denote now by A_{5}^{0} the module (A_{(5)}^{0})_{5}, so that J_{5} is defined by the exact sequence

0 ⟶ J_{5} ⟶ H^{1}(Q_{5}, A_{5}) ⟶ H^{1}(I_{5}, A_{5}/A_{5}^{0}).

The inertia group I_{5} acts on the line A_{5}^{0} by the cyclotomic character χ. More generally, there is a filtration

A_{5}^{0} ⊂ A_{5}^{1} ⊂ A_{5}

with one dimensional quotients, and the action of the decomposition group G_{Q_{5}} on the sucessive quotients is by χψ^{−2}, 1, and χ^{−1}ψ^{2}, where ψ is an unramified character of order 4. This follows from part 2 of lemma 1.4.

Using this explicit description, one verifies that h^{0}(A_{5}) = h^{0}(A_{5}^{∗}) = 1, that h^{0}(A_{5}^{0}) = h^{0}(A_{5}^{0}) = 1, etc.

Now, consider the composite map

ϕ : H^{1}(Q_{5}, A_{5}) ⟶^{ϕ_{1}} H^{1}(Q_{5}, A_{5}/A_{5}^{0}) ⟶^{ϕ_{2}} H^{1}(I_{5}, A_{5}/A_{5}^{0}).

Then we have

[H^{1}(Q_{5}, A_{5}) : J_{5}] = #Im(ϕ) ≥ #(Imϕ_{1})/#(ker ϕ_{2}).  (18)

By the inflation restriction sequence,

ker ϕ_{2} = H^{1}(D_{5}/I_{5}, (A_{5}/A_{5}^{0})^{I_{5}}) = ((A_{5}/A_{5}^{0})^{I_{5}})_{D_{5}/I_{5}},

so that

# ker ϕ_{2} = h^{0}(A_{5}/A_{5}^{0}) = 5.  (19)

On the other hand, by the same argument as in the proof of lemma 4.11, we find:

#Imϕ_{1} = \frac{h^{0}(A_{5}/A_{5}^{0})}{χ(A_{5}/A_{5}^{0})} · \frac{h^{0}(A_{5}^{∗})}{h^{0}(A_{5}^{0∗})} = \frac{5}{25^{−1}} × \frac{1}{1} = 125.

(Now, of course, h^{i}(M) denotes #H^{i}(Q_{5}, M) instead of #H^{i}(Q_{17}, M)!) Hence, by eqn. (18) and (19),

[H^{1}(Q_{5}, A_{5}) : J_{5}] ≥ 25.

Since h^{0}(A_{5}^{∗}) = 1, one obtains h_{5} ≤ 1/25, which proves the lemma.

Corollary 4.13 #S_{∅}(Q, A_{5}) = #S_{∅}(Q, A_{5}^{∗}) = 1.

Proof: Combining lemmas 4.10, 4.11, and 4.12 yields the inequality

h_{∞}h_{5}h_{17} ≤ 1.

The corollary now follows from eqn. (17). This finishes the proof of inequality (10), and hence of thm. 1.2.

5 Wiles’ general strategy

We conclude by briefly mentionning Wiles’ strategy for proving the Shimura-Taniyama conjecture for a large class of elliptic curves, containing all of the semi-stable ones. This can also be found in other surveys, eg. [RS], [Gou].

1. If E is an arbitrary (semistable) elliptic curve over Q, to get the machinery of Wiles going one needs to know that E_{p} is modular for some prime p. It was conjectured by Serre [Sr3] that any odd representation with values in GL_{2}(F_{p}) arises from a modular form with apropriate weight, character and level, but this conjecture seems quite difficult and is very much an open problem, even for p = 5. For p = 3, however, something very fortunate occurs: the image of the mod 3 representation is isomorphic to a double cover of S_{4} which is a finite solvable subgroup of GL_{2}(C). A result of Langlands [La] and Tunnell [Tu] then guarantees that ρ_{3} arises from a modular form (mod 3) of weight 1. By multiplying by an appropriate Eisenstein series, one gets a form of weight 2 mod 3 which gives the corresponding representation. Finally, the work of Ribet and others allows one to then show that there is a form of “minimal” level given by the recipe explained in the conjectures of [Sr3]. (See for example [Di].) Using this minimal level, Wiles constructs the Hecke ring T_{∅} and the base point map T_{∅} ⟶ O, where O is a local ring with residue field F_{3}. He also gets rings T_{Σ} for sets of primes Σ by increasing the level.

2. Wiles then needs to construct a deformation ring R_{Σ} which “captures” the 3-adic representation associated to E. This leads him to consider a variety of deformation problems, corresponding to the cases where E is good ordinary, multiplicative, or supersingular at 3. In the ordinary and multiplicative cases, the deformation given by T a_{3}(E) is ordinary, and the existence of the universal rings R_{Σ} is shown by Mazur and Tilouine [MT]. When E is supersingular at 3 one needs to consider flat deformations; here one makes essential use of the work of Ramakrishna [Ra], where the existence and key properties of the universal deformation ring are established.

3. Wiles then needs to show that the map R_{Σ} ⟶ T_{Σ} is an isomorphism for all sets Σ. Using the ideas explained in this report, he succeeds in reducing this isomorphism to the inequality

length(S_{∅}(Q, A)) ≤ length(O/η_{T_{∅}}O),  (20)

for the ring T_{∅} that arises in the minimal deformation problem. At the time of writing, this inequality has not yet been established in sufficient generality, although it can be established in many specific instances thanks to the ideas of Wiles and Flach, as we have explained.

4. The inequality (20) (for p = 3) is sufficient to show that any semi-stable elliptic curve such that E_{3} is absolutely irreducible is modular. (N. Elkies has shown that this fact alone is enough to prove that all of the Frey curves, whose modularity implies Fermat’s Last Theorem, satisfy the Shimura Taniyama conjecture.) By an elementary and ingenious argument, Wiles shows that all semistable curves are modular, assuming (20), as follows: If the mod 3 representation ρ_{3} arising from E is reducible, Wiles constructs an auxiliary curve E′ such that E3^{′} is absolutely irreducible as a Galois module, and E_{5} is isomorphic to E5^{′} as Galois modules. The key point here is that the modular curve X(5) is of genus 0, so that there is a plentiful supply of curves E′ satisfying E5^{′} ≃ E_{5}. Now, applying his argument to the irreducible representation attached to E3^{′}, Wiles shows that E′ is modular. Hence, so is E5^{′}, and therefore E_{5}. By an analysis of the rational points on X_{0}(15), one knows that E_{5} must be irreducible; hence, Wiles can apply his analysis, again assuming equation (20) with p = 5, to conclude that E is modular.

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