MOD 2 AND MOD 5 ICOSAHEDRAL REPRESENTATIONS^{1}

N.I. Shepherd–Barron and R. Taylor^{2}

DPMMS, Cambridge University, 16 Mill Lane, Cambridge CB2 1SB, U.K.

Introduction

We shall call a simple abelian variety A/ℚ modular if it is isogenous (over ℚ) to a factor of the Jacobian of a modular curve. In this paper we shall call a representation \overline{ρ} : G_{ℚ}→GL_{2}(𝔽̄_{l}) modular if there is a modular abelian variety A/ℚ, a number field F/ℚ of degree equal to dim A, an embedding O_{F} ↪ End(A/ℚ) and a homomorphism θ : O_{F} →𝔽̄_{l} such that \overline{ρ} is equivalent to the action of G_{ℚ} on the ker θ torsion points of A. If \overline{ρ} is modular in our sense then there exists a Hecke eigenform f and a homomorphism from the ring generated by its Fourier coefficients to 𝔽̄_{l} such that Tr \overline{ρ}(Frob_{p}) is congruent to the eigenvalue of T_{p} on f modulo ker θ for almost all primes p. (One may further assume either that f is of weight 2 and cuspidal, or that it has level coprime to l.) If \overline{ρ} is irreducible then \overline{ρ} being modular is equivalent to the existence of a Hecke eigenform f related to \overline{ρ} in this way. (This follows from the results of [AS].) However if \overline{ρ} is reducible then our definition appears to be stronger than requiring the existence of such an f, even if we insist that f be cuspidal of weight 2. (This is because the existence of such an f depends only on the semi-simplification of \overline{ρ} not on \overline{ρ} itself.)

Serre [S2] has conjectured that any continuous irreducible homomorphism \overline{ρ} : G_{ℚ} → GL_{2}(𝔽̄_{l}) with odd determinant is modular. Very little is known about this conjecture. It has been proved for representations \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{2}) and \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{3}). In both cases this can be achieved by lifting \overline{ρ} to a continuous odd irreducible representation G_{ℚ} → GL_{2}(O), where O is the ring of integers of some number field, and then using the Langlands-Tunnell theorem which asserts that this latter representation is modular of weight 1 (see [L] and [T]). This makes essential use of the fact that GL_{2}(𝔽_{2}) and GL_{2}(𝔽_{3}) are soluble.

2 N.I. SHEPHERD–BARRON AND R. TAYLOR^{3}

The purpose of this paper is to treat Serre’s conjecture in two further cases where the image of \overline{ρ} is no longer soluble. We show that if \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}) is unramified at 3 and has determinant the cyclotomic character then \overline{ρ} is modular. We also show that if \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{4}) is unramified at 3 and 5 then \overline{ρ} is modular. We do this by proving first that \overline{ρ} (up to twist) can be realized on respectively the 5–division points on an elliptic curve over ℚ and the 2–division points of an Abelian surface with multiplication by O_{ℚ(\sqrt{5})} over ℚ. (In fact we prove this over any given field of characteristic zero. Since SL_{2}(𝔽_{4}) is isomorphic to A_{5}, this can be stated as follows: given any quintic polynomial over a field K of characteristic zero, its splitting field can be obtained by adjoining first the square root of the discriminant d and then the 2-division points of an abelian surface with multiplication by O_{ℚ(\sqrt{5})} defined over K(\sqrt{d}).) We then use a recent result of Wiles (see [W]), as completed in [TW] and extended in [D], to show that these abelian varieties are modular (using respectively their 3-adic and the \sqrt{5}-adic Tate modules). We remark that the ramification conditions arise for technical reasons and do not seem to be essential to the method. On the other hand the restriction on the determinant for representations to GL_{2}(𝔽_{5}) does seem to be essential to the method.

1AMS subject classification : (primary) 11F41, 11G10, 14G05, 14G35.

2Current address: Dept. of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA02138, USA. This author was partially supported by an EPSRC advanced fellowship.

Wiles has asked (private communication) whether in fact any continuous indecomposable homomorphism \overline{ρ} : G_{ℚ} → GL_{2}(𝔽̄_{l}) with odd determinant is modular. He has (unpublished) answered this question affirmatively for representations \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{3}), by exploiting the fact that in this case any odd, reducible but indecomposable representation must have dihedral image. Our methods give an affirmative answer to Wiles’ question for representations \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}) which have determinant the cyclotomic character and which are unramified at 3; as well as to representations \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{4}) which are unramified at 3 and 5.

The question of realizing Galois groups of number fields as the action of Galois on division points of Abelian varieties, especially elliptic curves, has a long history, going back at least to Hermite [H]; see also Klein [K] and for a modern explanation Serre [S1]. Our results seem to be new for the group GL_{2}(𝔽_{4}), while one of us (RT) explained to Wiles in 1992 that the Shimura-Taniyama conjecture implied the result for GL_{2}(𝔽_{5}) (see [W]).

§1. Elliptic curves and mod 5 representations.

If N ≥ 3 is an integer we will let X_{N}/ℚ denote the fine moduli scheme of elliptic curves E with an isomorphism E[N] ≅ μ_{N} × ℤ/Nℤ, where μ_{N} denotes the group scheme of N^{th} roots of unity and where the isomorphism takes the Weil pairing on E[N] to the standard pairing on μ_{N} × ℤ/Nℤ:

(ζ_{1}, a_{1}) × (ζ_{2}, a_{2}) ↦ ζ_{2}^{a_{1}}/ζ_{1}^{a_{2}}.

We will let E → X_{N} denote the universal elliptic curve. It is known that X_{N}/ℚ is smooth, geometrically irreducible and of dimension 1. We will let \overline{X}_{N} denote the unique smooth projective model of its function field. Then X_{N} ⊂ \overline{X}_{N}. (In fact \overline{X}_{N} is the fine moduli space of generalised elliptic curves all whose geometric fibres are either smooth or N-gons together with a level N structure in the above sense (see [DR]).)

If we fix a primitive N^{th} root of unity ζ_{N} then the action of SL_{2}(ℤ/Nℤ) on (μ_{N} × ℤ/Nℤ)/ℚ(ζ_{N}) induces a homomorphism

θ_{N} : SL_{2}(ℤ/Nℤ) ⟶ Aut(E → X_{N})(ℚ(ζ_{N})).

Moreover if χ_{N} : G_{ℚ} → (ℤ/Nℤ)^{×} denotes the cyclotomic character then we have that

σ(θ_{N}(α)) = θ_{N} (( \begin{matrix}χ_{N}(σ) & 0 \\ 0 & 1 \end{matrix} ) α ( \begin{matrix}χ_{N}(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} )) .

Because of the uniqueness of the smooth projective model we see that this action extends uniquely to one on \overline{X}_{N}.

Let K be a field of characteristic zero. Let \overline{ρ} : G_{K} → GL_{2}(ℤ/Nℤ) be a continuous representation with det \overline{ρ} = χ_{N}. We define

ϕ\overline{ρ} : G_{K} ⟶ Aut(E → \overline{X}_{N})(\overline{K})

by

σ ⟼ θ_{N} (\overline{ρ}(σ) ( \begin{matrix}χ_{N}(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} )) .

This is a 1-cocycle; that is, ϕ\overline{ρ}(στ) = ϕ\overline{ρ}(σ)σ(ϕ\overline{ρ}(τ)). We define E\overline{ρ} → \overline{X}\overline{ρ} to be the twist of E → X_{N} by ϕ\overline{ρ}, and we will denote by X\overline{ρ} ⊂ \overline{X}\overline{ρ} the twist of X_{N} ⊂ \overline{X}_{N}. If x ∈ X\overline{ρ}(K) and if E_{x} denotes the fibre of E\overline{ρ} above x then the action of G_{K} on E_{x}[N] is via the representation \overline{ρ}, i.e.

i(σa) = \overline{ρ}(σ)i(a).

Lemma 1.1. If \overline{ρ} : G_{K} → GL_{2}(𝔽_{5}) and det \overline{ρ} = χ_{5} then \overline{X}\overline{ρ} ≅ ℙ_{1}/ℚ.

Proof. It is well known that \overline{X}_{5} ≅ ℙ_{1}/ℚ. (See for instance [RS]. Alternatively note that \overline{X}_{5} is geometrically isomorphic to ℙ_{1} and has a point defined over ℚ. One such is provided by the elliptic curve y^{2} − 10xy − 11y = x^{3} − 11x^{2} another by the generalised elliptic curve with smooth locus 𝔾_{m} × ℤ/5ℤ.) We can think of ϕ\overline{ρ} as giving an element of H^{1}(G_{K}, P SL_{2}(\overline{K})). We must show that this element is trivial. It suffices to show that it comes from an element of H^{1}(G_{K}, SL_{2}(\overline{K})) = (0). For this it suffices to check that θ_{5} : SL_{2}(𝔽_{5}) → P SL_{2}(\overline{K}) lifts to a homomorphism θ̃_{5} : SL_{2}(𝔽_{5}) → SL_{2}(\overline{K}) also satisfying

θ̃_{5} (( \begin{matrix}χ_{5}(σ) & 0 \\ 0 & 1 \end{matrix} ) α ( \begin{matrix}χ_{5}(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} )) = σ(θ̃_{5}(α)),

for all σ ∈ G_{K} and α ∈ SL_{2}(𝔽_{5}).

Note that θ_{5} : P SL_{2}(𝔽_{5}) ↪ P SL_{2}(\overline{K}). As P SL_{2}(𝔽_{5}) has no two dimensional non-trivial representation and a unique non-trivial double cover (see [A] page 2) we see that the preimage of θ_{5}(SL_{2}(𝔽_{5})) in SL_{2}(\overline{K}) is isomorphic to SL_{2}(𝔽_{5}). Thus θ_{5} lifts to a homomorphism θ̃_{5} : SL_{5}(𝔽_{5}) → SL_{2}(\overline{K}), and moreover as SL_{2}(𝔽_{5}) is perfect this lift is unique. Because of this uniqueness we see that

α ⟼ σ(θ̃_{5}) (( \begin{matrix}χ_{5}(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} ) α ( \begin{matrix}χ_{5}(σ) & 0 \\ 0 & 1 \end{matrix} ))

must equal θ̃_{5}, as desired.

Theorem 1.2. Let K be a field of characteristic zero and let \overline{ρ} : G_{K} → GL_{2}(𝔽_{5}) be a homomorphism with det \overline{ρ} = χ_{5}. Then there exists an elliptic curve E/K such that \overline{ρ} is equivalent to the representation of G_{K} on E[5], and if K is a number field we may further suppose that the image of G_{K} in Aut(E[3]) contains SL_{2}(𝔽_{3}).

Proof. For the first part we need only note that X\overline{ρ} has a K-rational point being a non-empty Zariski open subset of ℙ_{1}/K. For the second part let Y\overline{ρ}/X\overline{ρ} denote the quotient of

\{(a, b) ∈ (E\overline{ρ} ×_{X\overline{ρ}} E\overline{ρ})[3]|⟨a, b⟩ ≠ 1\}

by the equivalence relation (a, b) ∼ (±a, b). Then geometrically Y\overline{ρ} is isomorphic to X_{15} and so is geometrically irreducible. Moreover Y\overline{ρ} → X\overline{ρ} is finite of degree 24. If K is a number field we can find (by Hilbert irreducibility) a point x ∈ X\overline{ρ}(K) such that if y is a point of \overline{Y} \overline{ρ} above x then [K(y) : K] = 24. Then if E denotes the fibre of E\overline{ρ} at x we see that the action of G_{K} on E[5] is via \overline{ρ}. Moreover [K(E[3]) : K] ≥ 24, and so, as GL_{2}(𝔽_{3}) has a unique subgroup of index 2, we see that the image of G_{K} in Aut(E[3]) contains SL_{2}(𝔽_{3}).

We remark that lemma 1.1 remains valid with the prime 3 replacing 5. Essentially the same argument applies, since P SL_{2}(𝔽_{3}) has a unique non-trivial double cover and although SL_{2}(𝔽_{3}) is not perfect it has no character of order two. However, there are representations \overline{ρ} : G_{K} → GL_{2}(ℤ/4ℤ) with cyclotomic determinant that cannot be realized on the 4-division points of an elliptic curve. For example, suppose that

K = ℚ, c is complex conjugation and \overline{ρ}(c) = ( \begin{matrix}−1 & 2 \\ 2 & 1 \end{matrix} ). Then \overline{ρ} cannot be lifted

to a λ-adic representation for any λ dividing 2.

§2. Abelian surfaces with trivial level 2 structure.

Let W = 𝔽_{4}^{2} and fix an alternating pairing ⟨ , ⟩ : W × W → 𝔽_{2} by

(a_{1}, a_{2}) × (b_{1}, b_{2}) ↦ tr_{𝔽_{4}/𝔽_{2}}(a_{1}b_{2} − a_{2}b_{1}).

Let η = (1+\sqrt{5})/2, let O = ℤ[η], let τ denote the non-trivial involution of O/ℤ and identify O/2 ≅ 𝔽_{4}. We will denote the group of automorphisms of W as an 𝔽_{2} vector space together with its alternating pairing by Sp_{4}(𝔽_{2}). We will denote by SL_{2}(𝔽_{4}) the subgroup preserving the O structure. We will let S̃L_{2}(𝔽_{4}) denote the subgroup generated by SL_{2}(𝔽_{4}) and the element τ. There is a bijection {i, j} ↦ w_{{i,j}} from cardinality 2 subsets of {1, ..., 6} to W − {0} such that: w_{{i,j}} + w_{{j,k}} = w_{{i,k}} if i, j, k are all distinct; w_{{i,j}} + w_{{k,l}} = w_{{m,n}} if {i, j, k, l, m, n} = {1, ..., 6}; and ⟨w_{S}, w_{T} ⟩ ≡ #S ∩ T mod 2 (see [DO] page 142). Fix such an isomorphism. It gives rise to an isomorphism S_{6} ˜→Sp_{4}(𝔽_{2}), and hence to an embedding j̃: S̃L_{2}(𝔽_{4}) ↪ S_{6}. This embedding arises from the action of S̃L_{2}(𝔽_{4}) on its Sylow 5-subgroups, suitably numbered.

By a principally polarised abelian surface with full level 2 structure we shall mean an abelian surface A together with a principal polarisation λ : A ˜→A^{∨} and an isomorphism β : W ˜→A[2] which takes the Weil pairing onto our standard pairing on W. By an HB principally polarised abelian surface with full level 2 structure we shall mean the same data but in addition a homomorphism i : O → End(A) such that for all x ∈ O we have i(x)^{∨} ∘ λ = λ ∘ i(x) and such that the action of O on A[2] matches (via β) that on W. We will let A_{2}/ℚ denote the coarse moduli scheme of principally polarised abelian surfaces with full level two structure and H_{2}/ℚ the coarse moduli scheme of HB principally polarised abelian surfaces with full level two structure. Note that A_{2} has a natural action of Sp_{4}(𝔽_{2}) and that H_{2} has a natural action of S̃L_{2}(𝔽_{4}) (where τ acts by sending (A, λ, i, β) to (A, λ, i ∘ τ, β ∘ τ)). We have a natural morphism H_{2} → A_{2}, compatible with the inclusion S̃L_{2}(𝔽_{4}) ↪ Sp_{4}(𝔽_{2}).

We will also let H_{2\sqrt{5}} denote the fine moduli space of quintuples (A, λ, i, β, α) where (A, λ, i, β) is a principally polarised HB abelian surface with full level two structure and where α : μ_{5} ×ℤ/5ℤ ˜→A[\sqrt{5}] takes the standard pairing on μ_{5} ×ℤ/5ℤ to the Weil pairing on A[\sqrt{5}]. (We remark that one can define a Weil pairing on A[\sqrt{5}] by pulling back the standard pairing between A[\sqrt{5}] and A^{∨}[\sqrt{5}^{∨}] and using the fact that i(\sqrt{5})^{∨} ∘ λ = λ ∘ i(\sqrt{5}).) Equivalently H_{2\sqrt{5}} is the moduli space of sextuples (A, λ, i, β, C, P) where (A, λ, i, β) is a principally polarised HB abelian surface with full level two structure, C ⊂ A[\sqrt{5}] is a subgroup of order 5 and P ∈ A[\sqrt{5}] − C. (Associate (A, λ, i, β, α) to (A, λ, i, β, α(μ_{5}), α(1, 1)). Conversely given (A, λ, i, β, C, P) define α by α(1, 1) = P and α(ζ, 0) = Q_{ζ} where Q_{ζ} ∈ C and the Weil pairing of Q_{ζ} and P equals ζ.)

We have the usual analytic description of H_{2\sqrt{5}} as

Γ_{2\sqrt{5}}\GL_{2}^{+}(ℝ)^{2}/K_{∞},

where Γ_{2\sqrt{5}} is the subgroup of SL_{2}(O) consisting of matrices congruent to 1 modulo 2\sqrt{5}; GL_{2}^{+}(ℝ) denotes the elements of GL_{2}(ℝ) of positive determinant and K_{∞} denotes the commutator in GL_{2}^{+}(ℝ)^{2} of an element I_{0} satisfying I_{0}^{2} = −1 (I_{0} is unique up to conjugation and multiplication by (±1, ±1)). In particular H_{2\sqrt{5}} is geometrically irreducible. (We briefly recall how this analytic description of H_{2\sqrt{5}} is obtained. If we fix a primitive 5^{th} root of unity in ℂ then to give a quintuple (A, λ, i, β, α)/ℂ is the same as giving the following data. A finitely generated torsion free O-module Λ of rank 2; a perfect alternating pairing e : Λ × Λ → ℤ such that e(ax, y) = e(x, ay) if a ∈ O and x, y ∈ Λ; an isomorphism β : W ˜→(1/2Λ)/Λ as 𝔽_{4}-vextor spaces with an alternating pairing; an isomorphism α : 𝔽_{5}^{2} ˜→(1/\sqrt{5}Λ)/Λ taking the standard alternating pairing on 𝔽_{5}^{2} to the one induced by e on (1/\sqrt{5}Λ)/Λ; and an element I ∈ Aut_{O⊗_{ℤ}ℝ}(Λ ⊗_{ℤ} ℝ) such that I^{2} = −1 and (x, y) ↦ e(x, Iy) is a symmetric positive definite bilinear form on Λ ⊗_{ℤ} ℝ. Because O has class number 1 we may take Λ = O^{2}, e to be the pairing (x_{1}, x_{2}) × (y_{1}, y_{2}) ↦ tr_{ℚ(\sqrt{5})/ℚ}(x_{1}y_{2} − x_{2}y_{1})/\sqrt{5}) and α and β to be the obvious maps. The automorphisms of this structure are Γ_{2\sqrt{5}}. It remains to describe the possible choices for I. As Aut_{O⊗_{ℤ}ℝ}(Λ ⊗_{ℤ} ℝ) ≅ GL_{2}(ℝ)^{2}, we need a pair I = (I_{1}, I_{2}) ∈ GL_{2}(ℝ)^{2} such that I^{2} = −1. Then we have automatically that (x, y) ↦ e(x, Iy) is symmetric. There is a unique GL_{2}(ℝ)^{2}-conjugacy class of possible I, which splits into four GL_{2}^{+}(ℝ)^{2}-conjugacy classes. A unique one of these makes (x, y) ↦ e(x, Iy) positive definite.)

We now proceed to describe some concrete models for H_{2} and A_{2}. Our aim is simply to introduce a certain model Y for H_{2} and to prove proposition 2.5.

Let A_{2}^{∗} (resp. H_{2}^{∗}) denote the Satake compactification of A_{2} (resp. H_{2}). Then there is an embedding A_{2}^{∗} ↪ ℙ_{4} given by the modular forms of weight 2 (which are spanned by the 4^{th} powers of the 10 even theta-nulls). Moreover the image is a quartic hypersurface and the action of Sp_{4}(𝔽_{2}) is by linear transformations of ℙ_{4} (see [vG] pages 217-218). Consider the map H_{2}^{∗}− →A_{2}^{∗} ↪ ℙ_{4}. It is given by some space of weight 2 modular forms on H_{2}. This space must be S̃L_{2}(𝔽_{4})-invariant. The full space of weight two modular forms on H_{2} gives an embedding H_{2}^{∗} ↪ ℙ^{4} and the image is the intersection of a quadric and a quartic (see [vG] page 190-191). Moreover the action of S̃L_{2}(𝔽_{4}) ≅ S_{5} is by its permutation representation on ℙ_{4} (loc. cit.). Thus the map H_{2}^{∗}− →A_{2}^{∗} ⊂ ℙ_{4} is given by projection onto an S̃L_{2}(𝔽_{4})-invariant subspace of ℙ^{4}. Using the explicit equations (over ℂ) for the image of H_{2}^{∗} ↪ ℙ_{4} it is not hard to see that none (of the 3) projections to proper invariant subspaces is injective on H_{2}. Thus the weight two modular forms on A_{2} must restrict to the whole space of weight two modular forms on H_{2}. Moreover H_{2}^{∗} ⊂ A_{2}^{∗} ⊂ ℙ_{4}, with A_{2}^{∗} a quartic hypersurface and H_{2}^{∗} the intersection of a quadric and a quartic. Degree considerations show that H_{2}^{∗} is a quadric section of A_{2}^{∗}. The intersection of the singular locus of A_{2}^{∗} (i.e. A_{2}^{∗} − A_{2}) with H_{2}^{∗} is contained in the singular locus of H_{2}^{∗} (i.e. H_{2}^{∗} − H_{2}) (because H_{2} → A_{2}).

Consider the space P_{1}^{n} = P GL_{2}\(ℙ_{1}^{n})^{ss} (see for example [DO]). There is a natural map π : P_{1}^{n+1}→P_{1}^{n}, which is easily seen to be a ℙ_{1}-bundle in the Zariski topology on the stable locus (P_{1}^{n})^{s} (see chapter II of [DO]). It also comes equipped with n-sections s_{1}, ..., s_{n} which are induced by the maps

s_{i} : ℙ_{1}^{n}⟶ℙ_{1}^{n+1}
a_{1} × ... × a_{n} ↦ a_{1} × ... × a_{n} × a_{i}.

The symmetric group S_{n} acts on P_{1}^{n} and P_{1}^{n+1} so that these actions commute with π and so that

σ ∘ s_{i} = s_{σi} ∘ σ.

The variety P_{1}^{6} can be identified with the closed subset of ℙ_{5} defined by ∑_{1}^{6} z_{i} = ∑_{1}^{6} z_{i}^{3} = 0 (see page 16 of [DO]). Note that Out(S_{6}) ≅ C_{2} (see [A] page 4). If we denote the embedding P_{1}^{6} ↪ ℙ_{5} by α then we have that for all z ∈ P_{1}^{6} and all σ ∈ S_{6}:

(α ∘ σ(z))_{σ^{†}(i)} = (α(z))_{i}

where † : S_{6}→S_{6} is some non-inner automorphism. (If it were inner we could assume that (α ∘ σ(z))_{σi} = (α(z))_{i}. Then the −1 eigenspace of the involution σ = (12) on (R_{1}^{6})_{1} (see [DO] page 9) would have dimension 1, but in the notation of [DO] page 15 both t_{0} and t_{2} are −1 eigenvectors, a contradiction.) The stable locus (P_{1}^{6})^{s} coincides with the smooth locus of P_{1}^{6} (see [DO] page 31) which can be identified as P_{1}^{6} less the 15 S_{6}-conjugates of (1 : 1 : 1 : −1 : −1 : −1). (Although [DO] work over ℂ the arguments for these facts work equally well over ℚ.)

Over the open subset U of P_{1}^{6} where the sections s_{1}, ..., s_{6} are distinct, we have a principally polarised abelian surface with full level two structure obtained as the Jacobian of a hyperelliptic curve over U which maps 2 : 1 to P_{1}^{7}|_{U} ramified exactly at s_{1}, ..., s_{6}. This gives a morphism U → A_{2}, which is injective on geometric points by Torelli’s theorem. As P_{1}^{6} and A_{2} are both three dimensional we obtain a birational equivalence P_{1}^{6}− →A_{2}^{∗} which gives an isomorphism between U and the open subset of A_{2} corresponding to Jacobians. Under this equivalence the Sp_{4}(𝔽_{2}) action on A_{2} is compatible with the S_{6} action on P_{1}^{6} under our isomorphism Sp_{4}(𝔽_{2}) ≅ S_{6} (see [DO] page 142).

Lemma 2.1. The birational equivalence between P_{1}^{6} ⊂ ℙ^{4} and A_{2}^{∗} ⊂ ℙ^{4} is given by projective duality (up to an automorphism of ℙ_{4}).

Proof. By Thomae’s formula (see [M] page 120), the fact that A_{2}^{∗} ↪ ℙ_{4} is given by the 4^{th} powers of the 10 even theta-nulls and the formulae in [DO] pages 15-17 we see that it is given by quadratic expressions in the coordinates of P_{1}^{6} ⊂ ℙ_{4}. As a representation of S_{6} these span a space isomorphic to Symm^{2} W_{5} ≅ W_{1} ⊕ W_{5} ⊕ W_{9} where W_{i} is an irreducible representation of dimension i. As the rational map P_{1}^{6}− →A_{2}^{∗} ⊂ ℙ_{4} is equivariant for a linear S_{6} action on ℙ_{4} we see that it is given by the unique irreducible 5-dimensional space of quadratic functions on P_{1}^{6} ⊂ ℙ_{4}. Such a space is provided by the space spanned by the partial derivatives of the defining equation of P_{1}^{6} ⊂ ℙ_{4}. The lemma follows. □

Let Y denote the smooth cubic surface in ℙ_{4} defined by ∑_{1}^{5} y_{i} = ∑_{1}^{5} y_{i}^{3} = 0. Note that the permutation action of S_{5} on ℙ_{4} preserves Y and so we get an action of S_{5} on Y (so that (σ(y))_{σi} = (y)_{i}). Let Y^{1} denote Y less the 10 S_{5}-conjugates of (0 : 0 : 0 : 1 : −1), let Ỹ denote Y blown up in the 10 S_{5}-conjugates of (0 : 0 : 0 : 1 : −1) and let Y^{0} denote Y less the 15 lines which are S_{5} conjugate to (s : −s : t : −t : 0). According to [vG] pages 190-191 we have the following facts. The surface Y is birationally equivalent to H_{2}^{∗} in such a way that the S̃L_{2}(𝔽_{4}) action on H_{2}^{∗} goes over to the S_{5} action on Y under an isomorphism S̃L_{2}(𝔽_{4}) ≅ S_{5} (which is unique up to inner automorphism: this can be easily deduced from Out(A_{5}) ≅ C_{2}, see [A] page 2). We fix such a birational equivalence and the corresponding isomorphism S̃L_{2}(𝔽_{4}) ≅ S_{5}. Moreover Y^{0} becomes isomorphic to an open subset of H_{2}. The birational map Y − →H_{2}^{∗} ⊂ ℙ_{4} is given (up to automorphism of ℙ_{4}) by

(y_{1} : ... : y_{5}) ↦ (1/y_{1} : ... : 1/y_{5}).

It can be factored Y ← Ỹ → H_{2}^{∗}, where Ỹ → H_{2}^{∗} contracts the strict transforms of the five hyperplane sections Y ∩ {y_{i} = 0} to the five singular points of H_{2}^{∗} but is otherwise an isomorphism. (The arguments of [vG] are over ℂ but work equally well over ℚ. To explain this we will use for the rest of this paragraph the notation of pages 190-191 of [vG]. Note that the sheaf of weight two modular forms on X_{Γ}, the set of cusps and the action of S̃L_{2}(𝔽_{4}) are all defined over ℚ. As each cusp is stabilised by a different subgroup of S̃L_{2}(𝔽_{4}) we see that each cusp is individually defined over ℚ. The Eisenstein series E_{i} span the space of weight 2 modular forms. The only ones that vanish at all cusps except cusp 0 are the scalar multiples of E_{0}. Thus replacing E_{0} by some scalar multiple we may assume that it is defined over ℚ. As a subgroup of S̃L_{2}(𝔽_{4}) of index 5 fixes the line generated by E_{0} we see that E_{0} has 5 S̃L_{2}(𝔽_{4})-conjugates all defined over ℚ. Replacing the E_{i} by these new conjugates, it is now clear that the arguments of [vG] work over ℚ.)

We will let E denote the sum of the exceptional divisors of Ỹ → Y and H̃ the pull back of the hyperplane section H of Y. Note that the sum of the exceptional divisors of Ỹ → H_{2}^{∗} is the sum of the strict transforms of the 5 hyperplane sections Y ∩ {y_{i} = 0} of Y. Hence it is linearly equivalent to 5H̃ − 3E.

Lemma 2.2. Y contains 6 disjoint lines each defined over ℚ(\sqrt{5}) which are permuted amongst themselves by the action of A_{5}.

Proof. The A_{5} orbit of the line

{t_{1}(1 : −1 : η : −η : 0) + t_{2}(0 : −η : η : −1 : 1))}

will do. □

Corollary 2.3. The pull back under Ỹ → H_{2}^{∗} ⊂ ℙ_{4} of a hyperplane section is linearly equivalent to 4H̃ − 2E.

Proof. The rational map Y − →H_{2}^{∗} ⊂ ℙ_{4} is given by a system of quartics whose only base points are the 10 S_{5}-conjugates of (0 : 0 : 0 : 1 : −1). Thus the pull back to Ỹ of a hyperplane section is 4H̃ plus a divisor supported on the exceptional divisors of Ỹ → Y. Because the S_{5} action on Ỹ is compatible with a linear action on H_{2}^{∗} ⊂ ℙ_{4} we see that the pull back of a hyperplane section of H_{2}^{∗} defines an S_{5}-invariant class in Pic(Ỹ). Thus it is of the form 4H̃ + mE. On the other hand if E_{1} denotes the preimage of (0 : 0 : 0 : 1 : −1) in Ỹ then the image of E_{1} in H_{2}^{∗} is the conic consisting of points (a : b : c : 0 : 0) such that ab + bc + ca = 0. Thus −m = (E_{1}, 4H̃ + mE) = 2 and the corollary follows. □

The image of H_{2} in A_{2} contains a point corresponding to a Jacobian (e.g. the Jacobian of y^{2} = x(x^{5} − 1)). Thus the composite of rational maps

j : Y − →H_{2}^{∗} ⟶ A_{2}^{∗}− →P_{1}^{6}

makes sense. The actions of S̃L_{2}(𝔽_{4}) ≅ S_{5} and of S_{6} are compatible with j̃ : S̃L_{2}(𝔽_{4}) ↪ S_{6}.

Lemma 2.4. The rational map j is in fact a morphism and, up to conjugation by an element of S_{6} is given by

j : Y ⟶ P_{1}^{6} ≅ Z
(y) ↦ (4y_{1}^{2} − s : 4y_{2}^{2} − s : 4y_{3}^{2} − s : 4y_{4}^{2} − s : 4y_{5}^{2} − s : s),

where s = y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2} + y_{5}^{2}.

Proof. The action of S_{5} on Ỹ is compatible with a linear action on P_{1}^{6} ⊂ ℙ_{4} and so the pull back under j of a hyperplane section of P_{1}^{6} defines an S_{5} invariant class in Pic(Ỹ). The map H_{2}^{∗} ↪ A_{2}^{∗}− →P_{1}^{6} ⊂ ℙ_{4} is given by a linear system of cubics whose only base points are the intersection of H_{2}^{∗} with the singular locus of A_{2}^{∗} and so are contained in the singular locus of H_{2}^{∗}. Thus the pull back of a hyperplane section of P_{1}^{6} to Ỹ is of the form 3(4H̃ − 2E) plus a divisor supported on the exceptional divisors of Ỹ → H_{2}^{∗}. Using the S_{5} invariance we see that it must in fact be

3(4H̃ − 2E) − r(5H̃ − 3E) = (12 − 5r)H̃ + 3(r − 2)E,

where r ∈ ℤ_{≥0}.

We will show that under Ỹ − →P_{1}^{6} an irreducible component E_{i} of E gets contracted to a point. According to [vG] pages 190-191 the image of E_{i} in H^{∗} parametrises principally polarised abelian surfaces which split as a product of two elliptic curves. Hence by [vG] page 217 we see that the image of E_{i} in A^{∗} is contained in the locus of vanishing of some theta-null. However it is not difficult to see that the hyperplane defined by the vanishing of a theta-null is tangent to A at every point on its intersection with A. It follows that E_{i} is contracted to a point in P_{1}^{6}. (To verify the fact that the hyperplane defined by the vanishing of a theta-null is tangent to A at every point of contact with A we use the model of A^{∗} ⊂ ℙ^{5} given in the notation of page 217 of [vG] by s_{1} = s_{2}^{2} − 4s_{4} = 0. We see that the singular locus of A^{∗} consists of the S_{6} conjugates of the line (r : r : s : s : t : t) where r + s + t = 0. Finding the six sets of five disjoint lines of singularity we see that with a suitable numbering the triple {λ_{12}, λ_{13}, λ_{23}} (see [vG] page 216) becomes the set of lines {(r : s : t : r : s : t), (r : s : t : s : t : r), (r : s : t : t : r : s)}. The unique hyperplane containing these three lines is x_{1} + x_{2} + x_{3} = x_{4} + x_{5} + x_{6} = 0 and so the 10 S_{6}-conjugates of this hyperplane are the 10-hyperplanes defined by the vanishing of an even theta-null. These hyperplanes each meet A^{∗} in a quadric taken with multiplicity 2.

Hence either we have (E, (12−5r)H̃ +3(r−2)E) = 0 or Ỹ − →P_{1}^{6} has base points on E. We deduce that either r = 2 or that all points in the intersection of E with an exceptional divisor of Ỹ − →H_{2}^{∗} are in the base locus of Ỹ − →P_{1}^{6} (because the base locus is S_{5} invariant). In the latter case ((12−5r)H̃ +3(r−2)E, 5H̃ −3E) ≥ 30 and so r ≥ 2. On the other hand (12 − 5r)H̃ + 3(r − 2)E must be effective so that r ≤ 2. We deduce that r = 2 and that the pull back under j of a hyperplane section is linearly equivalent to 2H on Y.

Let V ⊂ |2H| be the linear system defining j. As a representation of S_{5} we have V ⊂ Symm^{2} V_{4} ≅ V_{1} ⊕ V_{4} ⊕ V_{5}, where V_{1} is the trivial representation, V_{1} ⊕ V_{4} is the 5-dimensional permutation representation and V_{5} is an irreducible 5-dimensional representation. Thus V = V_{1} ⊕ V_{4} and is spanned by y_{1}^{2}, ..., y_{5}^{2}. Using the S_{5}- symmetry and letting s = y_{1}^{2} + ... + y_{5}^{2} we see that up to composition with an element of S_{6}

j : (y_{1} : ... : y_{5}) ⟼ (Ay_{1}^{2} + Bs : ... : Ay_{5}^{2} + Bs : −(A + 5B)s).

In order that (A + 5B)^{3}s^{3} − ∑_{i=1}^{5}(Ay_{i}^{2} + Bs)^{3} vanish on Y it is easy to check that we must have A/B = −4. Thus j has the form described in the lemma as is easily checked to be everywhere defined. □

We remark that the map Y ↪ P_{1}^{6} described in [vG] pages 191, 218 and 220 is not the one induced by their moduli space interpretation . The correct map is the one given in the above lemma.

We see that j(Y^{1}) ⊂ (P_{1}^{6})^{s}. We will let C denote the pull back to Y^{1} of P_{1}^{7}/P_{1}^{6}. We will now summarise what we will need from the proceeding calculations.

Proposition 2.5. The surface H_{2} is birationally equivalent to Y. Under this equivalence Y^{0} embeds as an open subset of H_{2} and the action of S_{5} on Y matches with the action of S̃L_{2}(𝔽_{4}) on H_{2} under an isomorphism S_{5} ≅ S̃L_{2}(𝔽_{4}) (which is unique up to inner automorphism). Moreover there is a ℙ_{1}-bundle for the Zariski topology C/Y^{1} which has an action of S̃L_{2}(𝔽_{4}) compatible with the action on Y^{1} and six sections s_{1}, ..., s_{6} such that if σ ∈ S̃L_{2}(𝔽_{4}) then

σ ∘ s_{i} = s_{j̃(σ)i} ∘ σ.

If y is a geometric point of Y^{0} then the principally polarised abelian surface which is parametrised by y can be realised as the Jacobian of the genus 2 curve which maps 2 : 1 to C_{y} ramified exactly at s_{1}(y), ..., s_{6}(y). Over the S̃L_{2}(𝔽_{4})-conjugates of (1 : −1 : 1 : −1 : 0) the six sections s_{1}, ..., s_{6} coalesce in pairs.

Proof. The only part which needs further explanation is the final sentence. Note that j(1 : −1 : 1 : −1 : 0) = (0 : 0 : 0 : 0 : −1 : 1). Also from the formulae in [DO] pages 15-17 we see that (0 : 0 : 0 : 0 : −1 : 1) parametrises 6-tuples (P, Q, R, R, Q, P) with P, Q, R distinct points in ℙ_{1}. □

As H_{2} is a coarse moduli space we can not hope for a universal family of abelian surfaces over H. However one might hope for a universal family of Kummer surfaces, or what comes to much the same thing a universal family of conics with six marked points. Our family C is probably such a family. At least it has a natural action of S̃L_{2}(𝔽_{4}) and from a geometric fibre of this family we can reconstruct the abelian surface associated to that point in Y^{0}. We can not use C to construct a universal family of abelian surfaces as there is no canonical choice of genus 2 curve mapping 2 : 1 to C ramified at the 6 marked points. Also note that our calculations show that C extends to a larger open set (namely Y^{1}) than one might have expected. This will be crucial in the sequel.

The surface Y is not rational over ℚ. It will be useful to replace it by a related variety X which is rational. In fact X will be a certain blow up of the restriction of scalars from ℚ(\sqrt{5}) to ℚ of Y ×_{ℚ} ℚ(\sqrt{5}). More precisely consider the blow up X′ of Y × Y along its diagonal. It has an involution t which switches the two factors and an action of S̃L_{2}(𝔽_{4}) which commutes with t and is compatible with the two projections X′→Y. We also have a rational map

θ : X′− →Y

which takes a pair (y^{(1)}, y^{(2)}) to the third point of intersection of the line through y^{(1)} and y^{(2)} with Y. The rational map θ is well defined outside the strict transforms of the subvarieties L×L ⊂ Y ×Y as L runs over lines lying in Y. Also θ is compatible with the S̃L_{2}(𝔽_{4}) actions on X′ and Y. We will let X denote the twist of X′ by the homomorphism

G_{ℚ}→Gal(ℚ(\sqrt{5})/ℚ) ˜→\{1, t\} ⊂ Aut(X′/ℚ).

Then X inherits an action of S̃L_{2}(𝔽_{4}) and a rational map θ : X′− →Y compatible with the S̃L_{2}(𝔽_{4}) actions, all still defined over ℚ. We note that X is birationally equivalent over ℚ to the restriction of scalars from ℚ(\sqrt{5}) to ℚ of Y ×_{ℚ} ℚ(\sqrt{5}). We let X^{1} ⊂ X denote the locus where θ is defined and maps to Y^{1}. We note that the complement of X^{1} in X has codimension at least 2. We will also let D denote the pull back via θ of C to X^{1}. It inherits an action of S̃L_{2}(𝔽_{4}) and six sections s_{1}, ..., s_{6}.

Over some non-empty open subset X^{0} ⊂ X, s_{1}, ..., s_{6} are distinct and we can form an abelian surface A/X^{0} as the Jacobian of a genus 2 curve over X^{0} which maps 2 : 1 to D ramified exactly at s_{1}, ..., s_{6}. Then A comes with a principal polarisation λ : A→A^{∨} and an isomorphism β : A[2] ˜→W which takes the Weil pairing to the standard pairing on W. Note however that the construction of A is not canonical and that A does not have an action of SL_{2}(𝔽_{4}). Let x (resp. \overline{x}) denote the generic (resp. geometric generic) point of X. From the interpretation of Y^{0} as a coarse moduli space we see that there is an embedding i : O ↪ End(A\overline{x}) which is compatible via θ with the moduli theoretic interpretation of Y and such that the action of O (via i) on A\overline{x}[2] matches that on W. In fact there is a unique such map i. For if i′ were a second such map then i′ = f ∘ i ∘ f^{−1} for some automorphism f of A\overline{x} which is the identity on A\overline{x}[2]. We see that the representation T_{2}(f) on the 2-adic Tate module of A can only have eigenvalues ±1. Suppose that f ≠ ±1. Then (1+f)A must be an elliptic curve, A must be isogenous to (1+f)A⊕(1−f)A and we have a non-zero isogeny

(1 + f)A ⟶ A ⟶ (1 − f)A,

where the first map is multiplication by i(\sqrt{5}) and the second (1 − f). This will all be defined over a non-empty open subset U of some finite cover of H_{2}. Then we can map U to the j-line by taking z to the j-invariant of (1 + f)A_{z}. It is easy to see that this morphism has countable fibres and so must in fact have finite fibres, a contradiction. Thus f = ±1 and i′ = i. In particular we see that i is stable under the action of Gal(k(\overline{x})/k(x)) and so defined (non-canonically) over k(x). Thus replacing X^{0} by a smaller non-empty open subset we see that there is a homomorphism i : O ↪ End(A) such that i(a)^{∨} ∘ λ = λ ∘ i(a) for all a ∈ O and such that the actions of O on A[2] and W match under β.

We will let

X̃^{0} = \{(a, b) ∈ (A ×_{X^{0}} A)[\sqrt{5}]|⟨a, b⟩ ≠ 1\}/ ∼

where (a, b) ∼ (a′, b′) if and only if (a, b) = ±(μa′, b′) for some μ ∈ 𝔽_{5}^{×}. (Although the construction of A is non-canonical, A\overline{x} is canonical and the only automorphisms of (A\overline{x}, λ, i, β) are ±1. Thus at least generically the construction of X̃^{0} is canonical (i.e. if we start with a different hyperelliptic curve mapping 2 : 1 to F we get in some neighbourhood of the generic point of X^{0} the same variety X̃^{0}→X^{0}).)

Lemma 2.7. After perhaps again shrinking X^{0}, X̃^{0} is geometrically irreducible.

Proof. We shall work over ℂ. We can perform the same construction over some non-empty open subset Y^{00} ⊂ Y^{0}, except that we will use the equivalence relation (a, b) ∼ (a′, b′) if and only if (a, b) = (μa′, b′) for some μ ∈ 𝔽_{5}^{×}. We obtain a variety Z/Y^{00} which has an action of {±1} which associates (y, [(a, b)]) with (y, [(−a, −b)]). We will denote by Ỹ^{0} the quotient Z/{±1}. Using the fact that the construction of X̃^{0} is generically canonical we see that (after perhaps shrinking X^{0}) X̃^{0} = Ỹ^{0} ×_{Y^{00}} X^{0}. After perhaps again shrinking X^{0} we may suppose that X̃^{0} dominates X^{0}. Assume for now that Ỹ^{0} is irreducible. The general fibres of X^{0} → Y^{00} and hence of X̃^{0} → Ỹ^{0} are irreducible (in fact birationally equivalent to Y). If y is the generic point of Y^{00} then Ỹ_{y}^{0} is a field and X̃_{y}^{0} is irreducible. Thus the generic fibre of X̃^{0} → X^{0} is irreducible and hence X̃^{0} is irreducible.

It remains to show that Ỹ^{0} is irreducible. We have a family Ã/Z of principally polarised HB abelian surfaces with full level 2-structure and with a subgroup C ⊂ A[\sqrt{5}] of order 5 and a point P ∈ Ã[\sqrt{5}] − C. (Let Ã = Z ×_{Y^{00}} A and over a point (y, [(a, b)]) let C be the subgroup generated by a and let P = b.) Thus we get a map Z → H_{2\sqrt{5}} over Y^{00} → H_{2}. Looking at points one sees that Z → H_{2\sqrt{5}} factors through a map Ỹ^{0} → H_{2\sqrt{5}}. Because [Ỹ^{0} : Y^{00}] ≤ 60 and [H_{2\sqrt{5}} : H_{2}] = 60 we see that Ỹ^{0} is birationally equivalent to H_{2\sqrt{5}} and hence irreducible. □

§3. Abelian surfaces with twisted level 2 structure.

We will now fix both a field K of characteristic zero and a homomorphism \overline{ρ} : G_{K}→SL_{2}(𝔽_{4}). If V/K is a variety with an action of SL_{2}(𝔽_{4}) defined over K we will let V\overline{ρ} denote the twist of V by \overline{ρ} : G_{K}→SL_{2}(𝔽_{4})→Aut(V)(K). Thus we have

D\overline{ρ} ⟶ X1 ; \overline{ρ} ⊂ X\overline{ρ}
↓  ↓
C\overline{ρ} ⟶ Y 1 ; \overline{ρ} ⊂ Y\overline{ρ}.

D\overline{ρ}/X1 ; \overline{ρ} is now a ℙ_{1}-bundle in the étale topology, but not necessarily in the Zariski topology. We have sections s_{1}, ..., s_{6} : X1 ; \overline{ρ} × \overline{K}→D\overline{ρ} × \overline{K} such that for all σ ∈ G_{K}

σ(s_{i}) = s_{j̃∘\overline{ρ}(σ)i}.

In particular the collection {s_{1}, ..., s_{6}} is defined over K.

Lemma 3.1. The surface Y\overline{ρ} is birationally equivalent to ℙ_{2} over K(\sqrt{5}). X\overline{ρ} is birationally equivalent to ℙ_{4} over K.

Proof. By lemma 2.2 there is a system of six disjoint lines on Y\overline{ρ} collectively defined over K(\sqrt{5}). Blowing them down we obtain a variety V which is geometrically isomorphic to ℙ_{2} and has a zero cycle of degree 10 rationally defined over K(\sqrt{5}), namely the SL_{2}(𝔽_{4}) orbit of (1 : −1 : 0 : 0 : 0) ∈ Y (\overline{K}). As 10 is coprime to 3 we see that V ˜→ℙ_{2} over K(\sqrt{5}). (Because on V^{∨} the line bundle O(1) ≅ O(10) ⊗ (Ω_{V^{∨}}^{2})^{⊗3} is defined over K(\sqrt{5}).) For the second part note that X\overline{ρ} is birationally equivalent over K to the restriction of scalars from K(\sqrt{5}) to K of Y\overline{ρ} ⊗ K(\sqrt{5}). □

Proposition 3.2. D\overline{ρ}/X1 ; \overline{ρ} is a ℙ_{1} bundle in the Zariski topology.

Proof. We will use Br_{2}(V) to denote H_{et}^{2}(V, 𝔾_{m})[2], the 2-torsion in the cohomological Brauer group of V. We must check that [D\overline{ρ}] ∈ Br_{2}(X1 ; \overline{ρ}) is trivial. If [L : K] is odd then Br_{2}(X1 ; \overline{ρ}) ↪ Br_{2}(X1 ; \overline{ρ} ⊗ L), and so we may reduce to the case that the image of \overline{ρ} is contained in a Sylow 2-subgroup of SL_{2}(𝔽_{4}). (We thank Bogomolov for pointing this out to us.)

The natural inclusion

Br_{2}(K) ↪ Br_{2}(X1 ; \overline{ρ})

is an isomorphism because

Br_{2}(X_{ρ}^{1}) ≅ Br_{2}(X_{ρ}) ≅ Br_{2}(ℙ_{4}/K) ≅ Br_{2}(K)

(using respectively: purity for the cohomological Brauer group ([G3] corollary 6.2), birational invariance of the cohomological Brauer group ([G3] theorem 7.4) and the calculation of the cohomological Brauer group of projective space ). Thus if x is any K-rational point of X_{ρ}^{1} then the map Br_{2}(X1 ; \overline{ρ})→Br_{2}(K) obtained by taking the fibre at x is an isomorphism. We see that it suffices to find a K-rational point on D\overline{ρ}.

Suppose as we may that im \overline{ρ} acts on Y so as to fix the first coordinate. Then (0 : 1 : 1 : −1 : −1), (0 : 1 : −1 : 1 : −1) and (0 : 1 : −1 : −1 : 1) ∈ Y (\overline{K}) all define K-rational points on Y 1 ; \overline{ρ}. Above these points the fibre of C\overline{ρ} has a rationally defined cycle of degree 3 (coming from the sections s_{1}, ..., s_{6}, see corollary 2.6) and so splits. We can find a Severi-Brauer variety C′ ; \overline{ρ}→Y\overline{ρ} such that C′ ; \overline{ρ}|_{Y 1 ; \overline{ρ}} ∼ C\overline{ρ} (purity of the cohomological Brauer group: [G3] corollary 6.2, and the cohomological description of the Brauer group of a regular surface: [G2] theorem 2.1). The fibre of C′ ; \overline{ρ} at these three points will also be split and so the restriction of C′ ; \overline{ρ} to each of the three copies of ℙ_{1} that make up (Y ∩ \{y_{1} = 0\})\overline{ρ} is split (as Br_{2}(K) ˜→Br_{2}(ℙ_{1}/K)). We deduce that above any K-rational point of (Y^{1} ∩ \{y_{1} = 0\})\overline{ρ} there is a K-rational point of C\overline{ρ}. Thus it suffices to find a K-rational point of X1 ; \overline{ρ} which maps under θ to a point of (Y ∩ \{y_{1} = 0\})\overline{ρ}.

Let L denote the line \{(0 : s : −s : t : −t)\} ⊂ Y. Let y ∈ L\overline{ρ}(K) and suppose that, considered as a point of Y (\overline{K}), y ≠ (0 : 1 : −1 : 1 : −1), (0 : 1 : −1 : −1 : 1), (0 : 1 : −1 : 0 : 0) or (0 : 0 : 0 : 1 : −1). Consider the blow up of Y\overline{ρ} at y less the strict transform of L\overline{ρ}, which we will denote by V. It has an involution w which maps a point y′ to the third point of intersection of the line yy′ with Y. Let V′ denote the twist of V by the homomorphism

G_{K} ⟶ Gal(K(\sqrt{5})/K) ↪ \{1, w\} ⊂ Aut(V/K).

Then V′ is isomorphic over K to the fibre of θ at y. On the other hand V′ has a K-rational point, namely the intersection of the exceptional divisor with the strict transform of the conic (Y ∩ T_{y}Y − L)\overline{ρ}. This concludes the proof of the proposition. □

The same arguments used before lemma 2.7 allow one to deduce the following corollary.

Corollary 3.3. There is a non-empty Zariski open subset X^{0}(\overline{ρ}) ⊂ X1 ; \overline{ρ}, a principally polarised HB abelian surface (A(\overline{ρ}), λ, i)/X^{0}(\overline{ρ}) and an isomorphism β : (A(\overline{ρ}) × \overline{K})[2] ˜→W such that for all σ ∈ G_{K}

β ∘ σ = \overline{ρ}(σ) ∘ β.

Note that as SL_{2}(𝔽_{4}) does not act on A/X^{0} we can not define A\overline{ρ} directly. The abelian surface A(\overline{ρ}) may be thought of as a substitute.

We will let

X̃^{0}(\overline{ρ}) = \{(a, b) ∈ (A(\overline{ρ}) ×_{X^{0}(\overline{ρ})} A(\overline{ρ}))[\sqrt{5}]|⟨a, b⟩ ≠ 1\}/ ∼

where (a, b) ∼ (a′, b′) if and only if (a, b) = ±(μa′, b′) for some μ ∈ 𝔽_{5}^{×}. Again, although the construction of A(\overline{ρ}) is non-canonical, generically the construction of X̃^{0}(\overline{ρ}) is canonical. Thus over \overline{K} we have that X̃^{0}(\overline{ρ}) is birationally equivalent to X̃^{0} and in particular X̃^{0}(\overline{ρ}) is absolutely irreducible (by lemma 2.7).

Theorem 3.4. Suppose that K is a field of characteristic 0 and suppose that ρ : G_{K}→SL_{2}(𝔽_{4}) is a continuous homomorphism. Then there is an abelian surface A/K together with a principal polarisation λ : A→A^{∨} and an embedding i : O ↪ End(A) (both defined over K) such that

(1) λ ∘ i(a) = i(a)^{∨} ∘ λ for all a ∈ O;

(2) the action of G_{K} on A[2] ≅ 𝔽_{4}^{2} is equivalent to \overline{ρ};

(3) and, if K is a number field, the action of G_{K} on A[\sqrt{5}] ≅ 𝔽_{5}^{2} is via a homomorphism G_{K} ↠ GL_{2}(𝔽_{5}) whose image contains SL_{2}(𝔽_{5}).

Proof. Choose a point x ∈ X^{0}(\overline{ρ})(K). If K is a number field we may, by Hilbert irreducibility, assume that if x̃ ∈ X̃^{0}(\overline{ρ}) is a point above x then [K(x̃) : K] = 60. Let A = A(\overline{ρ})_{x}. The only part of the theorem that is not obvious is part 3. So suppose that K is a number field. If G denotes the image of G_{K} in GL_{2}(𝔽_{5}) ≅ Aut(A[\sqrt{5}]) then we have that

#G /G ∩ \{( \begin{matrix}μ & 0 \\ 0 & ν \end{matrix} )| ν = ±1, μ ∈ 𝔽_{5}^{×}\} = 60.

An easy bit of group theory now shows that G ⊃ SL_{2}(𝔽_{5}). □

Corollary 3.5. Suppose K is a field of characteristic zero, f ∈ K[X] is a quintic polynomial with discriminant d and L/K is the splitting field for f. Then there is a principally polarised HB abelian surface A/K(\sqrt{d}) such that L = K(\sqrt{d})(A[2]).

Hermite [H] showed that in the situation of this corollary there is a solvable extension K_{1}/K and an elliptic curve E/K_{1} such that K_{1}(E[5]) ⊃ L. However we do not in general have equality, nor even an equality between K_{1}(E[5]) and L(ζ_{5}). Apparently Kronecker found this lack of equality unsatisfactory (see [K]).

We remark, in the context of Theorem 3.4, that if F is a real quadratic field in which 2 is inert such that every ρ : G_{K} → SL_{2}(𝔽_{4}) can be realized on A[2] for some abelian surface A/K with multiplication by O_{F}, then necessarily F = ℚ(\sqrt{5}). The reason is that then we could, for any prime p ≡ 1 mod 5, find an abelian surface with multiplication by F such that 5 divides the image of inertia at p in Aut(T_{2}A) and hence in Aut(T_{λ}A) for all λ not dividing p. Thus [F_{λ}(ζ_{5}) : F_{λ}] ≤ 2 for all but finitely many primes λ and so [F (ζ_{5}) : F] ≤ 2. Thus F = ℚ(\sqrt{5}).

§4. Applications.

Theorem 4.1. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}) has determinant the cyclotomic character and that #\overline{ρ}(I_{3})|10; then \overline{ρ} is modular.

Proof. After twisting by a character we may replace the third condition by the assumption that \overline{ρ}(I_{3}) is unipotent. Then by Theorem 1.2 we can find an elliptic curve E/ℚ such that \overline{ρ}_{E,5} ≅ ρ and \overline{ρ}_{E,3} is surjective onto GL_{2}(𝔽_{3}). Then we see by considering the action of I_{3} on the 5-adic Tate module of E that E is semi-stable at 3. Hence by [D], E is modular and so \overline{ρ} is modular. □

Theorem 4.2. Suppose that A/ℚ is an Abelian surface, λ : A→A^{∨} is a principal polarisation, and i : O→ End(A) is an embedding all defined over ℚ and such that λ ∘ i(a) = i(a)^{∨} ∘ λ for all a ∈ O. Suppose moreover that A has semi-stable reduction at 3 and 5 and that the representation of G_{ℚ(\sqrt{5})} on the \sqrt{5} division of A is irreducible. Then A is modular (i.e. a factor of the Jacobian of a modular curve).

Proof. Let \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{5}) denote the representation of G_{ℚ} on A[\sqrt{5}]. The Weil pairing on A[\sqrt{5}] shows that det \overline{ρ} is the cyclotomic character. Also \overline{ρ}(I_{3}) has order 1 or 5 because A has semi-stable reduction at 3. Thus by theorem 4.1 \overline{ρ} is modular.

Let ρ : G_{ℚ}→GL_{2}(O\sqrt{5}) denote the representation of G_{ℚ} on the \sqrt{5}-adic Tate module of A. Then ρ is either flat or Selmer (in the sense of Wiles [W]), det ρ is the cyclotomic character and the reduction of ρ modulo \sqrt{5} is modular and irreducible when restricted to G_{ℚ(\sqrt{5})}. By [D], ρ is modular and hence by Faltings isogeny theorem A is modular. □

We remark that we have in fact proved the following more general result, which it may be convenient to record here.

Theorem 4.2´ . Let F be a totally real number field of degree d and let μ be a prime of O_{F} with norm 5 (resp. 3). Let A/ℚ be an abelian d-fold with semistable reduction at 3 and 5 (resp. at 3). Suppose that there is an isomorphism i : F ˜→ End(A) ⊗_{ℤ} ℚ. This gives rise to a representation (well defined up to semisimplification) \overline{ρ}_{A,μ} : G_{ℚ}→GL_{2}(O_{F} /μ). We further suppose that this representation is absolutely irreducible when restricted to ℚ(\sqrt{5}) (resp. ℚ(\sqrt{−3})). Then A is modular.

We should first explain what we mean by \overline{ρ}_{A,μ}. Choose an abelian variety A′ isogenous to A with an embedding O_{F} ↪ End(A′). Then let \overline{ρ}_{A,μ} denote the action of G_{ℚ} on A′[μ]. If for one choice of A′ this is absolutely irreducible it is for all choices of A′ and is independent of the choice of A′.

Thus to prove the proposition we may assume that R = O_{F}. We may choose a polarisation λ on A and it is known that the Rosati involution it induces on i(O_{F}) must be positive and hence the identity. We see that the action of G_{ℚ} on T_{μ}A ≅ O_{F,μ}^{2} has determinant the cyclotomic character and so det \overline{ρ}_{A,μ} is the cyclotomic character. The result now follows exactly as in the previous proposition. We remark that the case μ has norm 3 is due to Wiles et al (see [W], [TW], [D]).

Theorem 4.3. Suppose that \overline{ρ} : G_{ℚ}→GL_{2}(𝔽_{4}) is a representation which is unramified at 3 and 5. Then \overline{ρ} is modular.

Proof. Twisting, we may replace GL_{2}(𝔽_{4}) by SL_{2}(𝔽_{4}). By theorem 3.6 we can find an abelian surface A/ℚ, a principal polarisation λ : A→A^{∨} and an embedding i : O ↪ G_{ℚ} all defined over ℚ and such that

(1) λ ∘ i(a) = i(a)^{∨} ∘ λ for all a ∈ O;

(2) \overline{ρ} is equivalent to the representation of G_{ℚ} on A[2];

(3) the representation of G_{ℚ(\sqrt{5})} on A[\sqrt{5}] is irreducible. It is not hard to see that the action of the inertia groups I_{3} and I_{5} on the 2-adic Tate module of A can be conjugated (separately) to lie in the group of matrices of the form

± ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) .

Thus, possibly after replacing A by a quadratic twist, we may also assume that A is semi-stable at 3 and 5. Thus, by theorem 4.2, A is modular; and hence \overline{ρ} is modular.

Acknowledgements We are grateful to Fedya Bogomolov, for explaining to us how to improve our original argument for proposition 3.2 so that it would work over any field of characteristic 0, rather than just over ℚ. NIS-B is grateful to Columbia University for its hospitality, and especially to Bob Friedman and Troels Jorgenson. We are both grateful to John Coates for his support.

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