Mod 2 and Mod 5 Icosahedral Representations

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 ρ̄ : GQ→GL2(F̄l) modular if there is a modular abelian variety A/Q, a number field F/Q of degree equal to dimA, an embedding OF ↪→ End(A/Q) and a homomorphism θ : OF→F̄l such that ρ̄ is equivalent to the action of GQ on the ker θ torsion points of A. If ρ̄ is modular in our sense, then there exists a Hecke eigenform f and a homomorphism from the ring generated by its Fourier coefficients to F̄l such that Tr ρ̄(Frobp) is congruent to the eigenvalue of Tp 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. This follows from the results of [AS].) If ρ̄ is irreducible, then ρ̄ being modular is equivalent to the existence of a Hecke eigenform f related to ρ̄ in this way. However if ρ̄ 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 ρ̄ and not on ρ̄ itself.) Serre [S2] has conjectured that any continuous irreducible homomorphism ρ̄ : GQ → GL2(F̄l) with odd determinant is modular. Very little is known about this conjecture. It has been proved for representations ρ̄ : GQ → GL2(F2) and ρ̄ : GQ → GL2(F3). In both cases this can be achieved by lifting ρ̄ to a continuous odd irreducible representation GQ → GL2(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 GL2(F2) and GL2(F3) are soluble. The purpose of this paper is to treat Serre’s conjecture in two further cases where the image of ρ̄ is no longer soluble. We show that if ρ̄ : GQ → GL2(F5) is unramified at 3 and has determinant the cyclotomic character, then ρ̄ is modular. We also show that if ρ̄ : GQ → GL2(F4) is unramified at 3 and 5, then ρ̄ is modular. We do this by proving first that ρ̄ (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 OQ(√5) over Q. (In fact we prove this over any given field of characteristic zero. Since SL2(F4) is isomorphic to A5, 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