Sums of Galois representations and arithmetic homology

Abstract

Let Γ 0 ( n , N ) \Gamma _0(n,N) denote the usual congruence subgroup of type Γ 0 \Gamma _0 and level N N in SL ( n , Z ) \text {SL}(n,{\mathbb Z}) . Suppose for i = 1 , 2 i=1,2 that we have an irreducible odd n n -dimensional Galois representation ρ i \rho _i attached to a homology Hecke eigenclass in H ∗ ( Γ 0 ( n , N i ) , M i ) H_*(\Gamma _0(n,N_i),M_i) , where the level N i N_i and the weight and nebentype making up M i M_i are as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that n n is odd, that N 1 N 2 N_1N_2 is squarefree, and that ρ 1 ⊕ ρ 2 \rho _1\oplus \rho _2 is odd. We prove two theorems that assert that ρ 1 ⊕ ρ 2 \rho _1\oplus \rho _2 is attached to a homology Hecke eigenclass in H ∗ ( Γ 0 ( 2 n , N ) , M ) H_*(\Gamma _0(2n,N),M) , where N N and M M are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of M 1 M_1 and M 2 M_2 are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of M 1 M_1 and M 2 M_2 . This conjecture is known to be true for n = 3 n=3 , so we obtain unconditional results for GL ( 6 ) \text {GL}(6) . A similar result for GL ( 4 ) \text {GL}(4) appeared in an earlier paper.