Icosahedral Galois representations

In memory of Olga Taussky-Todd

Richard Taylor Department of Mathematics,

Harvard University, Cambridge, MA 02138,

U.S.A.

August 24, 1999

Introduction

Let K be a number field. We will let G_{K} denote its absolute Galois group; and for each finite prime ℘ of K we will let G_{℘} ⊂ G_{K} denote a decomposition group for ℘, I_{℘} ⊲ G_{℘} the inertia subgroup and Frob_{℘} ∈ G_{℘}/I_{℘} the arithmetic Frobenius element. These are all uniquely defined up to conjugation in G_{K}. We will also let N℘ denote the cardinality of the residue field of ℘. If ρ : G_{K} → GL_{d}(ℂ) is a continuous representation then it has finite image. Following Artin [A2] we introduce the L-function

L(ρ, s) = ∏_{℘} det(1 − ρ^{I_{℘}}(Frob_{℘})(N℘)^{−s})^{−1},

where the product is over all finite primes of K and where ρ^{I_{℘}} denotes the representation of G_{℘}/I_{℘} on the I_{℘} invariants of ρ. This definition is easilly seen to independent of the choices of G_{℘}, I_{℘} and Frob_{℘}. The product converges for the real part of s greater than 1. Brauer [B] showed that L(ρ, s) has meromorphic continuation to the whole complex plane and satisfies a certain functional equation relating the values at s and 1 − s. Artin [A2] conjectured that L(ρ, s) is holomorphic except for a possible pole at s = 1 when the trivial representation is a constituent of ρ. Because any such representation is semi-simple and because L(ρ_{1} ⊕ ρ_{2}, s) = L(ρ_{1}, s)L(ρ_{2}, s) we see that it suffices to treat the case where ρ is irreducible. It is now generally expected (the “strong Artin conjecture”) that in the case where ρ is irreducible there should be a cuspidal automorphic representation π(ρ) of GL_{n}(𝔸_{K}) such that L(π(ρ), s) = L(ρ, s). This implies Artin’s original conjecture, but appears to be strictly stronger. When Artin made his conjecture he was already aware that in the case dim ρ = 1 the existence of π(ρ) follows from class field theory ([A1]) and that the holomorphy of L(π(ρ), s) had been established even earlier by Hecke ([He]). Artin also showed that if ρ is induced from a one dimensional representation χ of the Galois group G_{L} of a finite extension L/K then L(ρ, s) = L(χ, s) and hence the Artin conjecture is also true for ρ. However even in this case the strong Artin conjecture is still not known in general (but it is known if L/K is normal and soluble (see [AC])). The Artin conjecture itself is still open for d-dimensional representations for any d ≥ 2, even if K = ℚ.

For the rest of this article we will restrict attention to two dimensional continuous irreducible representations ρ : G_{K} → GL_{2}(ℂ). Such representations can be classified according to the image of the associated projective representation proj(ρ) : G_{K} → P GL_{2}(ℂ). It is known that the image of proj(ρ) is either the dihedral group D_{2n} of order 2n for some n ≥ 2, the alternating group A_{4}, the symmetric group S_{4} or the alternating group A_{5}. In the case that the image of proj(ρ) is dihedral then Artin proved that L(ρ, s) is holomorphic and the strong Artin conjecture is also known to be true. (In this case ρ is induced from a one dimensional representation of the Galois group of a quadratic extension.)

Langlands [L], using his theory of base change, succeeded in treating the strong Artin conjecture when the image of proj(ρ) is A_{4}. He also treated some cases where K = ℚ and the image of proj(ρ) is S_{4}. Tunnell [T] combining Langlands techniques with the existence of non-normal cubic base change for GL_{2} of Jacquet, Piatetski-Shapiro and Shalika (see [JPSS]), treated the general case where the image is S_{4}. These methods relied essentially on the solubility of A_{4} and S_{4} and seem to offer little insight into the so called icosahedral case where the image of proj(ρ) is A_{5}.

In this article we will describe an approach to the strong Artin conjecture for odd, icosahedral representations ρ : G_{ℚ} → GL_{2}(ℂ). By odd we mean that det ρ(c) = −1, with c denoting complex conjugation. I first outlined this approach to Wiles in 1992 when I learnt of his progress on the Shimura-Taniyama conjecture. Since then some progress has been made on this approach and it is perhaps time to describe the overall strategy in print. In the first section we will recall some background material. In the next section we will sketch the basic strategy, and in the last two sections we will fill out this sketch somewhat.

1 Notation and background

In this section we shall establish some notation and recall some basic facts about modular forms. We will let H denote the upper half complex plane. By a modular form of weight k and level N we shall mean a holomorphic function f : H → ℂ such that

1. f((az + b)/(cz + d)) = (cz + d)^{k}f(z) for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N),

2. for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) and for the real part of z in any bounded

interval, f((az + b)/(cz + d)) → 0 as the imaginary part of z tends to infinity.

Here Γ_{1}(N) denotes the set of ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) such that c ≡ d − 1 ≡

0 mod N. We will let S_{k}(N) denote the finite dimensional complex vector space of modular forms of weight k and level N. This space has a natural action of (ℤ/Nℤ)^{×} by \overline{d} ↦ ⟨\overline{d}⟩, where f|⟨\overline{d}⟩(z) = (cz + d)^{−k}f((az + b)/(cz + d)) for

any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) with c ≡ 0 mod N and d ≡ \overline{d} mod N. Any f ∈ S_{k}(N)

can be expressed

∑_{n=1}^{∞} c_{n}(f)q^{n},

where q = e^{2πiz}.

If p |̸N is a prime number we define an operator T_{p} on S_{k}(N) by

f|T_{p}(z) = ∑_{n=1}^{∞} c_{np}(f)q^{n} + ∑_{n=1}^{∞} c_{n}(f|⟨p⟩)q^{np}.

If p|N is a prime then we define an operator U_{p} on S_{k}(N) by

f|U_{p}(z) = ∑_{n=1}^{∞} c_{np}(f)q^{n}.

For any n ∈ ℤ_{≥1} we define an operator T (n) on S_{k}(N) by the formulae

1. T (mn) = T (m)T (n) if (m, n) = 1,

2. T (p^{r}) = U_{p}^{r} if p|N,

3. ∑_{r=0}^{∞} T (p^{r})X^{r} = (1 − T_{p}X + p^{k−1}⟨p⟩X^{2})^{−1} if p |̸N.

We have the useful formula c_{1}(f|T (n)) = c_{n}(f). Finally we define S_{p} = p^{k−2}⟨p⟩ = (T (p)^{2} − T (p^{2}))/p if p |̸N.

We let 𝕋_{k}(N) denote the ℤ-algebra generated by ⟨\overline{d}⟩ for \overline{d} ∈ (ℤ/Nℤ)^{×}, by T_{p} for p |̸N and by U_{p} for p|N all acting on S_{k}(N). It is also the ℤ-algebra generated by the T (n) for all n ∈ ℤ_{≥1} acting on S_{k}(N). It is commutative and finitely generated as an abelian group. The pairing

\begin{matrix}S_{k}(N) & × & (𝕋_{k}(N) ⊗_{ℤ} ℂ) & ⟶ & ℂ \\ (f & , & T ⊗ λ) & ⟼ & λc_{1}(f|T). \end{matrix}

is perfect. For any ring R we define S_{k}(N, R) = Hom_{ℤ}(𝕋_{k}(N), R). If R ⊂ ℂ then S_{k}(N, R) is simply the space of f ∈ S_{k}(N) with c_{n}(f) ∈ R for all n. An element f ∈ S_{k}(N, R) is called an eigenform if the map f : 𝕋_{k}(N) → R is a ring homomorphism. For R ⊂ ℂ, f is an eigenform if and only if f (as an element of S_{k}(N)) is a common eigenvector for all the elements of 𝕋_{k}(N).

Suppose l|N. Then Hida introduced the idempotent e = lim_{n→∞} U_{l}^{n!} ∈ 𝕋_{k}(N) ⊗ ℤ_{l}. It has the property that U_{l} is a unit in e(𝕋_{k}(N) ⊗ ℤ_{l}) and is topologically nilpotent in (1 − e)(𝕋_{k}(N) ⊗ ℤ_{l}). Now suppose l |̸N. Then Hida showed that there exists a finite torsion free Λ = ℤ_{l}[[X]]-algebra 𝕋^{0}(N) with distinguished elements T (n) for all n ∈ ℤ_{≥1} such that for any k ≥ 1 and any r ≥ 0 there is a map

𝕋^{0}(N)/((1 + X)^{l^{r}} − (1 + p)^{l^{r}(k−1)})𝕋^{0}(N) → → e(𝕋_{k}(Nl^{1+r}) ⊗ ℤ_{l})

which takes T (n) to T (n) for all n and which is an isomorphism after tensoring with ℚ_{l} if k ≥ 2. (See [H2]. We remark that there are various normalisations one can use here. In our normalisation if p is a prime ≡ 1 mod Nl and if p = (1 + l)^{s} in ℤ_{l} then pS_{p} = (1 + X)^{s} in 𝕋^{0}(N).) If l ≥ 5 then it is known that 𝕋^{0}(N) is free over Λ and that for k ≥ 2 the above maps are isomorphisms before tensoring with ℚ_{l} (see [H1]). If k = 1 then it is known that in general the above surjection is not an isomorphism even after tensoring with ℚ_{l} (see [MW]).

The quotient Y_{1}(N)^{an} = Γ_{1}(N)\H can be algebraised to a smooth irreducible curve Y_{1}(N)/ℂ with smooth compactification X_{1}(N). We let cusps denote the reduced divisor X_{1}(N) − Y_{1}(N). If N ≥ 5 then Y_{1}(N) is the (fine) moduli space for elliptic curves with a point of exact order N. In this case the pull back along the zero section of of the relative differentials of the universal elliptic curve over Y_{1}(N) gives a line bundle ω/Y_{1}(N). This line bundle has a natural extension to X_{1}(N) which we shall also denote ω. Then S_{k}(N) = H^{0}(X_{1}(N), ω^{⊗k}(−cusps)). All these objects have natural models over ℤ, except that the extension of ω to X_{1}(N) may only be defined over ℤ[1/N] (see [KM]).

Suppose that l |̸N and let \overline{SS} ⊂ Y_{1}(N) ⊗ \overline{𝔽}_{l} denote the finite set of points which parametrise supersingular elliptic curves. For each \overline{x} ∈ \overline{SS} choose T\overline{x} ∈ O_{Y_{1}(N)⊗W (\overline{𝔽}_{l}),\overline{x}} such that O_{Y_{1}(N)⊗W (\overline{𝔽}_{l}),\overline{x}}^{∧} ≅ W (\overline{𝔽}_{l})[[T\overline{x}]]. If r ∈ l^{ℚ} ∩ [1, 1/l) we let SS_{<r} denote the union over \overline{x} ∈ \overline{SS} of the lifts x of \overline{x} with |T\overline{x}(x)|_{l} < r. If r ∈ l^{ℚ} ∩ (1, 1/l] we let X_{1}(N)_{>r} denote the \overline{ℚ}_{l} points x of X_{1}(N) which either do not reduce to a point of \overline{SS} or reduce to \overline{x} ∈ \overline{SS} but |T\overline{x}(x)|_{l} > r. Both SS_{>r} and X_{1}(N)_{<r} are admissible open subspaces of the rigid space attached to X_{1}(N)⊗K for a suitable finite extension K/ℚ_{l} (depending on r). If r_{1} < r_{2} then X_{1}(N) = X_{1}(N)_{>r_{1}} ∪ SS_{<r_{2}} is an admissible cover. These definitions do not depend on the choice of the parameters T\overline{x}.

In this paragraph suppose again that N ≥ 5 and that l |̸N. Also suppose that r ∈ l^{ℚ} ∩ (1, l^{−l/(1+l)}]. If l |̸n then there is a natural Hecke operator T (n) on H^{0}(X_{1}(N)_{>r}, ω^{⊗k}) with the usual effect on q-expansions. The theory of the canonical subgroup (see [K]) allows us to define an action of U_{l} on H^{0}(X_{1}(N)_{>r}, ω^{⊗k}) and hence of the Hecke operators T (n) for all n. It also gives a map

S_{k}(Nl, K)^{(ℤ/lℤ)^{×}} ↪ H^{0}(X_{1}(N)_{>r}, ω^{⊗k})

which is equivariant for all the Hecke operators T (n). (Here (ℤ/lℤ)^{×} acts via (ℤ/lℤ)^{×} ↪ (ℤ/Mlℤ)^{×}.) At least if l ≥ 5 then e is naturally defined on H^{0}(X_{1}(N)_{>r}, ω^{⊗k}) and

eH^{0}(X_{1}(N)_{>r}, ω^{⊗k}) ≅ Hom_{Λ}(𝕋^{0}(N), K)^{(ℤ/lℤ)^{×}}.

Here Λ acts on K by letting X act by multiplication by (1 + l)^{k−1} − 1 and \overline{d} ∈ (ℤ/lℤ)^{×} sends f to p^{2−k}f ∘ S_{p} where p is a prime with p ≡ 1 mod N and where p ≡ \overline{d} mod l. (See [Go]. These results remain true for k = 1, as seems to be well known to experts, but see [BT] for a proof.)

We now turn our attention to Galois representations. Let 𝔪 be a maximal ideal of 𝕋_{k}(N) (resp. 𝕋^{0}(N)) with residue field k(𝔪) of characteristic l. There is a unique continuous semi-simple representation \overline{ρ}_{𝔪} : G_{ℚ} → GL_{2}(k(𝔪)) which is unramified at all primes p |̸Nl and satisfies tr \overline{ρ}_{𝔪}(Frob_{p}) = T_{p} and det \overline{ρ}_{𝔪}(Frob_{p}) = pS_{p} at these primes. If \overline{ρ}_{𝔪} is absolutely reducible we call 𝔪 Eisenstein. If 𝔪 is not Eisenstein then there is a unique continuous representation ρ_{𝔪} : G_{ℚ} → GL_{2}(𝕋_{k}(N)_{𝔪}) (resp. GL_{2}(𝕋^{0}(N)_{𝔪})) which is unramified at all primes p |̸Nl and satisfies tr ρ_{𝔪}(Frob_{p}) = T_{p} and det ρ_{𝔪}(Frob_{p}) = pS_{p} at these primes (see [C]). If θ : 𝕋_{k}(N)_{𝔪} → R (resp. θ : 𝕋_{k}(N)_{𝔪} → R) is a local map to a noetherian complete local ring R then we will let ρ_{θ} : G_{ℚ} → GL_{2}(R) denote θ ∘ ρ_{𝔪}. If θ : 𝕋_{k}(N) → \overline{ℚ}_{l} is a ring homomorphism then there is a unique continuous irreducible representation ρ_{θ} : G_{ℚ} → GL_{2}(\overline{ℚ}_{l}) which is unramified at all primes p |̸Nl and satisfies tr ρ_{θ}(Frob_{p}) = T_{p} and det ρ_{θ}(Frob_{p}) = pS_{p} at these primes. All the representations described in this paragraph are called modular.

Let us make some remarks on these Galois representations. Firstly if ρ : G_{ℚ} → GL_{2}(ℂ) is a continuous representation then the strong Artin conjecture holds for ρ if and only if for some isomorphism i : ℂ →^{∼} \overline{ℚ}_{l} the composite i ∘ ρ is modular.

Secondly we remark that Serre (see [S]) has conjectured that any odd, irreducible, two dimensional representation of G_{ℚ} over a finite field is modular.

Thirdly we remark that if \overline{ρ} : G_{ℚ} → GL_{2}(k) is absolutely irreducible and modular (k a finite field) then \overline{ρ} = \overline{ρ}_{𝔪} for 𝔪 a maximal ideal of 𝕋_{2}(Nl^{2}) where l |̸N. If moreover \overline{ρ}_{I_{l}} is not trivial we may take 𝔪 to be a maximal ideal of 𝕋_{2}(Nl) such that U_{l} mod 𝔪 is an eigenvalue of \overline{ρ}_{I_{l}}(Frob_{l}) (see [E]). At the possible cost of increasing N we may also assume that U_{p} ∈ 𝔪 for all p|N (elementary to verify).

Fourthly we remark that we have the following criterion for modularity which is rather easy to prove. Suppose A/ℚ is an abelian variety of dimension d and that there exists an embedding i : O_{K} ↪ End (A/ℚ), where K is a number field of degree d. Suppose that for one prime λ the representation of G_{ℚ} on the Tate module T_{λ}A is modular. Then for all primes λ′ of K the representations of G_{ℚ} on both T_{λ′}A and on A[λ′] are both modular.

We finish this section by summarising some recent lifting theorems of Wiles [W], Taylor-Wiles [TW] and Diamond [D]. We will summarise three different theorems. Suppose that l |̸N and l ≠ 2. Let 𝔪 be a maximal ideal of 𝕋_{2}(N) in case 1, of 𝕋_{2}(Nl) in case 2 and of 𝕋^{0}(N) in case 3. Suppose that 𝔪 contains U_{p} for all p|N. Suppose also that \overline{ρ}_{𝔪}|_{ℚ(\sqrt{(−1)^{(l−1)/2}l})} is absolutely irreducible and that \overline{ρ}_{𝔪}|_{G_{l}} does not act by scalars. Let R be a complete noetherian local W (k(𝔪))-algebra with finite residue field and let ρ : 𝔾_{ℚ} → GL_{2}(R) be a continuous representation lifting \overline{ρ} and unramified outside Nl. Suppose finally that:

1. in the first case ,det ρ|_{I_{l}} is the cyclotomic character and for each open ideal 𝔞 of R there is a finite flat group scheme G/ℤ_{l} such that ρ mod 𝔞 is equivalent to the action of G_{l} on G(\overline{ℚ}_{l});

2. in the second case, ρ|_{G_{l}} ∼ ( \begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix} ) with χ_{1} tamely ramified, χ_{2}

unramified and U_{l} − (χ_{2} mod 𝔪_{R}) ∈ 𝔪;

3. in the third case, ρ|_{G_{l}} ∼ ( \begin{matrix}∗ & ∗ \\ 0 & χ_{2} \end{matrix} ) with χ_{2} unramified and U_{l} −(χ_{2} mod

𝔪_{R}) ∈ 𝔪;

Then there exists a unique homomorphism θ : 𝕋_{2}(N)_{𝔪} → R in the first case, 𝕋_{2}(Nl) → R in the second case, 𝕋^{0}(N) → R in the third case, such that ρ ∼ ρ_{θ}, θ(U_{p}) = 0 if p|N and in the second and third cases θ(U_{l}) = χ_{2}(Frob_{l}).

2 The basic strategy

Let ρ : G_{ℚ} → GL_{2}(ℂ) be an odd, icosahedral representation. Because ρ has finite image we may suppose (possibly after conjugation), that the image of ρ is contained in GL_{2}(E) for some number field E. Let λ be a prime of E with residue characteristic l. Then we may suppose (possibly after another conjugation) that ρ : G_{ℚ} → GL_{2}(O_{E,(λ)}) (where O_{E,(λ)} denotes the localisation of the ring of integers of E at λ). Let \overline{ρ} denote the reduction of ρ modulo λ. In [BT] Kevin Buzzard and I prove the following theorem.

Theorem 1 Suppose that l ≥ 5, that ρ is unramified at l and that the order of \overline{ρ}(G_{l}) is divisible by a prime other than l. Suppose also that \overline{ρ} is modular. Then the strong Artin conjecture holds for ρ.

It may be useful to discuss the various assumptions in this theorem. The assumptions that ρ be unramified at l can be weakened (see some forthcoming work of Buzzard). Whether it is reasonable to hope it can be removed entirely is not clear to me. Perhaps it can be avoided by using base change. The assumption that the order of \overline{ρ}(G_{l}) is divisible by a prime other than l seems to be of a technical nature. The assumption that l ≥ 5 also seems to be of a technical nature. This restriction comes not from the paper [BT], but from the papers cited therein. My expectation is that this restriction is only serious if l = 2 and then it is only serious for the invoking of the lifting theorems summarised at the end of the last section. In these the assumption l ≠ 2 is being used in an important way in at least two places. The lifting theorems are proved by first reducing the level to a certain minimal case (following Ribet [R] and others), lifting in the minimal case (using the method of Taylor-Wiles [TW]) and then extending to the more general case again (using the method of Wiles [W]). Both the first two steps use l ≠ 2 in a significant way. The first uses an auxiliary prime p ≢ 1 or 0 mod l. The second requires a numerical coincidence in Galois cohomology for the method to work, which becomes delicate if l = 2. The first use of l ≠ 2 is currently being investigated by Ribet and his student David Jones. In this case it seems likely that the difficulty can be avoided by imposing an additional local assumption that 2 divides the order of \overline{ρ}(I_{p}) for some p ≠ 2. The second use of l ≠ 2 is currently being investigated by Mark Dickinson for his Harvard PhD. Let me formulate as a conjecture a generalisation of this theorem which I am hopeful will be proven in the near future.

Conjecture 1 Suppose that ρ is unramified at l, that the order of \overline{ρ}(G_{l}) is divisible by a prime other than l and that if l = 2 then 2 divides the order of \overline{ρ}(I_{p}) for some p ≠ l. Suppose also that \overline{ρ} is modular. Then the strong Artin conjecture holds for ρ.

The final assumption in the theorem and conjecture, that \overline{ρ} is modular, is a special case of a conjecture Serre’s conjecture [S]. In general it is probably very deep. However in the special case l = 2, Shepherd-Barron and I have the following partial result ([ST]).

Theorem 2 Suppose l = 2 and \overline{ρ} is unramified at 3 and 5, then \overline{ρ} is modular.

Corollary 1 Suppose that ρ is unramified at 2, 3 and 5. Suppose moreover that proj(ρ)(G_{2}) has odd order and that proj(ρ)(I_{p}) has even order for some p ≠ 2. If conjecture 1 is true then the strong Artin conjecture is true for ρ.

It may be helpful to comment on the ramification assumptions in theorem 2. They result from restrictions in the available lifting results for modular Galois representations (see [W], [TW], [D]). One can expect them to be weakened in the near future, but it is less clear how soon it will be possible to remove them entirely. In fact using the results of [CDT] one may already reduce the assumption at 3 from being unramified to being tamely ramified.

In the rest of this paper we will comment briefly on the proofs of theorems 1 and 2. Both rely in an essential way on the work of Wiles [W] as completed by Wiles and the author [TW].

3 Mod 2 icosahedral representations.

We have isomorphisms \overline{𝔽}_{2}^{×} × SL_{2}(\overline{𝔽}_{2}) →^{∼} GL_{2}(\overline{𝔽}_{2}) and SL_{2}(\overline{𝔽}_{2}) →^{∼} P SL_{2}(\overline{𝔽}_{2}). The only subgroups of GL_{2}(\overline{𝔽}_{2}) isomorphic to A_{5} are those conjugate to SL_{2}(𝔽_{4}). (This can be seen by examining the mod 2 modular characters of A_{5}.) Thus to prove theorem 2 we must show that any continuous representation \overline{ρ} : G_{ℚ} → SL_{2}(𝔽_{4}) which is unramified at 3 and 5 is modular.

The key construction in [ST] is the proof of the existence of a principally polarised abelian surface (A, ψ)/ℚ, together with an embedding i : ℤ[(1 + \sqrt{5})/2] ↪ End ((A, ψ)/ℚ) such that the representation of G_{ℚ} on A[2] is equivalent to \overline{ρ}. (In fact we can ensure that the representation of G_{ℚ} on A[\sqrt{5}] has image the whole of GL_{2}(𝔽_{5}) and (by twisting) that A has semistable reduction at 3 and 5.) To make this construction one examines the appropriate twisted coarse moduli space. This space is birationally a cubic surface Y\overline{ρ} in ℙ^{3}. More specificly Y\overline{ρ} is (birationally) the twist of the subvariety Y ⊂ ℙ^{4} defined by y_{1} + y_{2} + y_{3} + y_{4} + y_{5} = y_{1}^{3} + y_{2}^{3} + y_{3}^{3} + y_{4}^{3} + y_{5}^{3} = 0, by \overline{ρ} : G_{ℚ} → SL_{2}(𝔽_{4}) ≅ A_{5} ⊂ Aut (Y). (Here A_{5} acts on Y by permuting the coordinates.) The variety Y\overline{ρ} is itself a cubic surface in ℙ^{3} which is in fact unirational: if X\overline{ρ} denotes the restriction of scalars from ℚ(\sqrt{5}) to ℚ of Y\overline{ρ} then we show that X\overline{ρ} is rational and because Y\overline{ρ} is a cubic surface collinearity gives a dominant rational map θ from X\overline{ρ} to Y\overline{ρ}. As Y\overline{ρ} is not a fine moduli space, rational points on Y\overline{ρ} do not necessarily correspond to rationally defined abelian surfaces. However by a rather explicit calculation we show that this is the case for the rational points θ(X\overline{ρ}(ℚ)). (We obtain the assertion about the action on A[\sqrt{5}] by combining the above argument with a Hilbert irreducibility argument (as Wiles does in [W]).)

Once we have constructed the desired triple (A, λ, i) we show that T\sqrt{5}A is modular and deduce that A[\sqrt{2}] is also. By the results of Wiles [W], Taylor-Wiles [TW] and Diamond [D], to show that T\sqrt{5}A is modular it suffices to show that A[\sqrt{5}] is modular.

In fact one can show that if \overline{ρ}′ : G_{ℚ} → GL_{2}(𝔽_{5}) is any representation with determinant the cyclotomic character and which is tamely ramified at 3 then \overline{ρ}′ is modular. The argument is similar. One finds an elliptic curve E/ℚ which realises \overline{ρ}′ on its 5 division points. (This seems to be implicit in work going back to Hermite, but we could not find it explicitly in the literature.) Using Hilbert irreducibility we may ensure (following Wiles) that the representation of G_{ℚ} on E[3] has image GL_{2}(𝔽_{3}). Then, by [D], T_{3}E is modular and so E[5] is also. (In 1992 I explained to Wiles how this argument could be used to show that the Shimura-Taniyama conjecture implied Serre’s conjecture for representations G_{ℚ} → GL_{2}(𝔽_{5}) with determinant the cyclotomic character. He later combined the argument with Hilbert irreducibility as part of his attack on the Shimura-Taniyama conjecture.)

4 l-adic representations unramified at l

We will now briefly discuss the proof of theorem 1. In fact it is no more difficult to prove a stronger theorem. We will put ourselves in the following situation. Let O denote the ring of integers of a finite extension of ℚ_{l}, λ its maximal ideal and k its residue field. Let ρ : 𝔾_{ℚ} → GL_{2}(O) be a continuous representation which is ramified only at a finite set of primes. Let \overline{ρ} denote the reduction of ρ modulo λ. In [BT] we prove the following theorem, which is stronger than theorem 1.

Theorem 3 Suppose that l ≥ 5 and that

1. ρ is unramified at l;

2. \overline{ρ} is modular;

3. \overline{ρ}|_{G_{ℚ(\sqrt{(−1)^{(l−1)/2}})}} is absolutely irreducible;

4. \overline{ρ}(Frob_{l}) has two distinct eigenvalues.

Then ρ has a finite image and the strong Artin conjecture is true for ρ.

This theorem also provides some evidence for a conjecture of Fontaine and Mazur (see [FM]) that any continuous l-adic representation of G_{ℚ} which is ramified at only finitely many primes and is finitely ramified at l (i.e. the image of I_{l} is a finite group) has finite image. As far as we are aware the only previous evidence for this conjecture was in the case of one dimensional representations. It also gives a hint about how to prove theorem 1: one should try to only exploit the finiteness of ρ(I_{l}), not the finiteness of ρ(G_{ℚ}).

Let α and β be the two eigenvalues of ρ(Frob_{l}). We may suppose that α and β lie in O. Because ρ is modular we can find, perhaps after interchanging α and β, a homomorphism \overline{f}_{α} : 𝕋_{2}(Nl) → k such that

1. N is not divisible by l;

2. \overline{f}_{α}(T_{p}) = tr \overline{ρ}(Frob_{p}) for p |̸Nl;

3. p\overline{f}_{α}(S_{p}) = det \overline{ρ}(Frob_{p}) for p |̸Nl;

4. \overline{f}_{α}(U_{l}) = α;

5. \overline{f}_{α}(U_{p}) = 0 for p|N.

We can then produce a second homomorphism \overline{f}_{β} : 𝕋_{2}(Nl) → k with the same properties except that \overline{f}_{β}(U_{l}) = β. The construction of \overline{f}_{β} is a deep theorem. To stress how surprising the existence of \overline{f}_{β} is, it may be worth remarking that any lifts f_{α} and f_{β} of \overline{f}_{α} and \overline{f}_{β} to homomorphisms 𝕋_{2}(Nl) → \overline{ℚ}_{l} will differ on most Hecke operators T_{p}. Gross [Gr] gives a construction for all l, but dependent on some unproven functorialities in some rigid cohomology. Coleman and Volloch [CV] then gave an unconditional construction but one which excludes the prime l = 2.

We can now apply the lifting results of Wiles, Taylor-Wiles and Diamond summarised at the end of section 1 to produce homomorphisms F_{α} and F_{β} : 𝕋^{0}(N) → O such that

1. F_{α}(T_{p}) = tr ρ(Frob_{p}) for p |̸Nl;

2. pF_{α}(S_{p}) = det ρ(Frob_{p}) for p |̸Nl;

3. F_{α}(U_{l}) = α;

4. F_{α}(U_{p}) = 0 for p|N.

and similarly for F_{β}. Here again we are using l ≠ 2, this time in what may be a more serious way (see section 2).

Because det ρ is unramified at l we see that F_{α} and F_{β} factor through 𝕋^{0}(N)/X𝕋^{0}(N). Thus, at least if l ≥ 5, they define sections of ω over the rigid space X_{1}(N)_{>l^{−l/(1+l)}}. We introduce F = (αF_{α} − βF_{β})/(α − β) and F′ = (F_{α} − F_{β})/(α − β). Then F = ∑_{n=1}^{∞} a_{n}q^{n} is formally a weight one level N normalised eigenform, while F′ = ∑_{n=1}^{∞} a_{n}q^{ln}. We have that

1. a_{p} = tr ρ(Frob_{p}) for p |̸N (including p = l);

2. a_{p} = 0 for p|N.

In [BT] we fabricate from F′ a rigid section of ω over SS_{<l^{−1/(1+l)}} which we show matches F on SS_{<l^{−1/(1+l)}} ∩ X_{1}(N)_{>l^{−l/(1+l)}}. Gluing and applying rigid GAGA we see that F extends to a classical weight one modular form such that ρ_{f} ∼ ρ. Thus ρ has finite image and the strong Artin conjecture holds for ρ.

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