Reducible Galois representations and the homology of GL(3,Z)

TL;DR

This paper proves that certain reducible Galois representations of dimension three over finite fields are associated with Hecke eigenclasses in the homology of arithmetic groups, confirming predictions of a conjecture for specific coefficient modules.

Contribution

It establishes a link between reducible Galois representations and homology classes in GL(3,Z), confirming a conjecture about the coefficient modules involved.

Findings

Proves the association of reducible Galois representations with homology classes.

Confirms the predicted coefficient modules match the conjecture.

Shows the result holds for squarefree conductors.

Abstract

We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $\rho$ is squarefree, we prove that $\rho$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $\Gamma$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, Pollack, and Sinnott.