Highly reducible Galois representations attached to the homology of $\mathrm {GL}(n,\mathbb {Z})$

Abstract:

Let $n\ge 1$ and $\mathbb {F}$ be an algebraic closure of a finite field of characteristic $p>n+1$. Let $\rho :G_{\mathbb {Q}}\to \mathrm {GL}(n,\mathbb {F})$ be a Galois representation that is isomorphic to a direct sum of a collection of characters and an odd $m$-dimensional representation $\tau$. We assume that $m=2$ or $m$ is odd, and that $\tau$ is attached to a homology class in degree $m(m-1)/2$ of a congruence subgroup of $\mathrm {GL}(m,\mathbb {Z})$ in accordance with the main conjecture of an earlier work of the authors and Pollack. We also assume a certain compatibility of $\tau$ with the parity of the characters and that the Serre conductor of $\rho$ is square-free. We prove that $\rho$ is attached to a Hecke eigenclass in $H_t(\Gamma ,M)$, where $\Gamma$ is a subgroup of finite index in $\rm {SL}$$(n,\mathbb {Z})$, $t=n(n-1)/2$ and $M$ is an $\mathbb {F}\Gamma$-module. The particular $\Gamma$ and $M$ are as predicted by the main conjecture of an earlier work. The method uses modular cosymbols, as in a recent work of the first author.