Abstract
We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of GL(4,ℤ), with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of GL(n,ℚ) has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution of ℤ by GL(n,ℚ)-modules consisting of sums of Steinberg modules for all subspaces of ℚ n .
Highlights
Serre’s conjecture [23] gives a connection between odd irreducible Galois representations ρ : GQ → GL(2, Fp) and modular forms that are simultaneous eigenvectors of all the Hecke operators
Via the Eichler–Shimura isomorphism [24], it can be interpreted as giving a connection between such Galois representations and elements of a cohomology group H1(Γ0(N), V) for an appropriate coefficient module V
The conjecture is known for twodimensional Galois representations
Summary
Serre’s conjecture [23] ( a theorem of Khare, Wintenberger, and Kisin [18, 19, 20]) gives a connection between odd irreducible Galois representations ρ : GQ → GL(2, Fp) and modular forms that are simultaneous eigenvectors of all the Hecke operators. In [3], odd Galois representations that are sums of characters are shown to correspond to cohomology eigenclasses. We prove that Galois representations of the form ρ1 ⊕ ρ2 of squarefree level N with ρ1 and ρ2 two-dimensional irreducible and odd are attached to Hecke eigenclasses in the cohomology of Γ0(4, N). For any odd Galois representation ρ : GQ → GL(n, F), we may find an integer N (called the level), an irreducible GL(n, Fp)-module M (called the weight), and a Dirichlet character (called the nebentype), such that ρ fits Hk (Γ0(N), M ). For i = 1, 2, let ρi : GQ → GL(2, F) be odd, irreducible Galois representations with squarefree relatively prime levels. Ρ1 ⊕ ρ2 fits at least one of H6(Γ0(n, N), M ) or H2(Γ0(n, N), M ), with N, M, and as predicted by [8]
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