Sums of Galois representations and arithmetic homology

Abstract:

Let $\Gamma _0(n,N)$ denote the usual congruence subgroup of type $\Gamma _0$ and level $N$ in $\text {SL}(n,{\mathbb Z})$. Suppose for $i=1,2$ that we have an irreducible odd $n$-dimensional Galois representation $\rho _i$ attached to a homology Hecke eigenclass in $H_*(\Gamma _0(n,N_i),M_i)$, where the level $N_i$ and the weight and nebentype making up $M_i$ are as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that $n$ is odd, that $N_1N_2$ is squarefree, and that $\rho _1\oplus \rho _2$ is odd. We prove two theorems that assert that $\rho _1\oplus \rho _2$ is attached to a homology Hecke eigenclass in $H_*(\Gamma _0(2n,N),M)$, where $N$ and $M$ are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of $M_1$ and $M_2$ 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 $M_1$ and $M_2$. This conjecture is known to be true for $n=3$, so we obtain unconditional results for $\text {GL}(6)$. A similar result for $\text {GL}(4)$ appeared in an earlier paper.