On degree 2 Galois representations over 𝔽₄

Abstract

We discuss proofs of some new special cases of Serre’s conjecture on odd, degree 2 representations of G_ℚ.

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. If A/ℚ is a modular abelian variety then F = End⁰(A/ℚ) is a number field of degree dim A. Replacing A by an isogenous (over ℚ) abelian variety we may assume that End(A/ℚ) = 𝒪_F. If λ is a prime of 𝒪_F with residue characteristic l, then G_ℚ acts on A[λ] ⊗ 𝔽̄_l, so that there is a continuous representation ρ_A,λ: G_ℚ → GL₂(𝔽̄_l). We shall call a representation arising in this way modular. If c denotes complex conjugation then det ρ_A,λ(c) = −1, i.e., ρ_A,λ is odd.

The following two conjectures have been extremely influential. The first is a generalization of the Shimura–Taniyama conjecture, the second is due to Serre (1).

Conjecture 1: If A/ℚ is a simple abelian variety and End⁰(A/ℚ) is a number field of degree dim A then A is modular.

Conjecture 2: If ρ: G_ℚ → GL₂(𝔽̄_l) is odd and irreducible then ρ is modular.

Very little is known about Serre’s conjecture, but we do have the following deep result of Langlands (2) and Tunnell (3).

Theorem 1: If ρ: G_ℚ → GL₂(𝔽₂) or GL₂(𝔽₃) is odd and absolutely irreducible then ρ is modular.

Recent work of Wiles (4) completed by Taylor and Wiles (5) and extended by Diamond (6) proves the following theorem.

Theorem 2: Suppose A/ℚ is a simple abelian variety and that End(A/ℚ) is the ring of integers in a number field, F, of degree dim A. Suppose also that there is a prime λ of 𝒪_F with residue characteristic l ≠ 2 such that A has semi-stable reduction at l, ρ_A,λ restricted to G_ℚ() is absolutely irreducible and ρ_A,λ is modular. Then A is modular.

In ref. 7 we obtain a few new cases of Serre’s conjecture. In fact we prove the following theorem.

Theorem 3: 1. If ρ: G_ℚ → GL₂(𝔽₅) has determinant the cyclotomic character and if #ρ(I₃)|10 then ρ is modular.

2. If ρ: G_ℚ → GL₂(𝔽₄) is unramified at 3 and 5 then ρ is modular.

This is an easy consequence of the two theorems cited above and the following algebro-geometric result. By a abelian surface we shall mean a triple (A, λ, i) where A is an abelian surface, λ: A ⥲ A^∨ is a principal polarization and i: ℤ[(1 + )/2] ↪ End(A), which has image fixed by the Rosati involution coming from λ (that is, λ is ℤ[(1 + )/2]-linear).

Theorem 4: 1. If ρ: G_ℚ → GL₂(𝔽₅) has determinant the cyclotomic character then there exists an elliptic curve E/ℚ such that ρ ≅ ρ_E,5 and ρ_E,3: G_ℚ → GL₂(𝔽₃) is surjective.

2. If ρ: G_ℚ → SL₂(𝔽₄) then there is a abelian surface (A, λ, i)/ℚ such that ρ ≅ ρ_A,2and ρ_A,: G_ℚ → GL₂(𝔽₅) is surjective.

Part 1 of this theorem is a slight generalization of an old result of Hermite (8); see also refs. 9 and 10. [We remark that the analogous statement for representations G_ℚ → GL₂(ℤ/4ℤ) is false.] In this form (except for the surjectivity of ρ_E,3) one of us (R.T.) pointed it out to Wiles in 1992 and explained how it could be used to deduce part 1 of Theorem 3 from the Shimura–Taniyama conjecture (see ref. 4). Part 2 seems to be new. The same argument also gives the following result [recall that SL₂(𝔽₄) ≅ A₅].

Proposition 1: Let K be a field of characteristic zero, f ∈ K[X] a quintic polynomial with discriminant d and L/K the splitting field for f. Then there is a abelian surface A/K() such that L = K()(A[2]).

We will now sketch the proof of part 2 of Theorem 4 (see ref. 7 for the details). Let Y denote the cubic surface

It has an obvious action of S₅. The 27 lines on Y divide into 3 orbits of length 15, 6, and 6 under the action of A₅. The lines in the orbit of length 15 are all defined over ℚ. We will let Y⁰ denote their complement. The other 12 lines are each defined over ℚ(). The lines in each orbit of length 6 are disjoint.

Y⁰ is the open subspace of the coarse moduli space of abelian surfaces with full level 2 structure which parametrizes abelian surfaces which are not the product of two elliptic curves. [Over ℂ this was discovered by Hirzebruch (see for example ref. 11).] We can twist Y and Y⁰ by ρ: G_ℚ → SL₂(𝔽₄) ≅ A₅ to obtain Y_ρ and Y_ρ⁰. Then Y_ρ is still a cubic surface because the action of A₅ extends to one on the ambient ℙ₃ which itself lifts to a homomorphism A₅ → GL₄. Y_ρ also contains 6 disjoint lines collectively defined over ℚ() and blowing them down we obtain ℙ₂/ℚ() (again because the action of A₅ lifts to a representation A₅ → GL₃). If X_ρ denotes the restriction of scalars from ℚ() to ℚ of Y_ρ then we deduce that X_ρ/ℚ is a rational 4-fold. There is also a dominant rational map θ: X_ρ → Y_ρ which on geometric points sends a pair (y₁, y₂) to the third point of intersection of the line y₁y₂ with Y_ρ. We deduce that Y_ρ⁰ contains many rational points.

Unfortunately, a rational point y ∈ Y_ρ⁰ does not necessarily give rise to a abelian surface A which is defined over ℚ. However if it does then ρ ≅ ρ_A,2. Over Y⁰ there is no universal abelian surface. However there is a canonical ℙ₁-bundle C/Y⁰ (for the Zariski topology) and six sections s₁, … , s₆, such that if y ∈ Y⁰(ℚ̄) then the abelian surface parametrized by y is the Jacobian of the double cover of C_y ramified exactly at s₁(y), … , s₆(y). The action of A₅ extends to C, where it permutes s₁, … , s₆ transitively, so that we get a ℙ₁-bundle C_ρ/Y_ρ⁰ (now for the étale topology). A point of Y_ρ⁰(ℚ) gives rise to a abelian surface if and only if it is in the image of C_ρ(ℚ). Although C_ρ/Y_ρ⁰ is not split, one can show that its pull back to X_ρ is split. Thus points in θ(X_ρ(ℚ)) do correspond to abelian surfaces defined over ℚ. This is sufficient to prove part 2 of Theorem 4. [To show that the pull back of C_ρ to X_ρ splits we first show that it extends outside codimension two and hence is equivalent to a constant bundle (as X_ρ is rational). Then we find one rational point on it above the boundary of X_ρ.]