Mod 2 homology for GL(4) and Galois representations

TL;DR

This paper computes mod 2 homology for certain congruence subgroups of SL(4,Z), explores Hecke operators' action, and provides evidence linking Hecke eigenclasses to Galois representations, supporting a conjecture.

Contribution

It extends previous computations to degree 1 homology for SL(4,Z) and modifies algorithms to handle Hecke operators without division by 2, offering new computational evidence.

Findings

Found mod 2 homology in degree 1 for Gamma in SL(4,Z)

Identified Hecke eigenclasses with attached Galois representations

Provided computational evidence supporting a conjecture in the field

Abstract

We extend the computations in [AGM4] to find the mod 2 homology in degree 1 of a congruence subgroup Gamma of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is closely related to the cohomology of Gamma with F_2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in [AGM4]. Our method of computation was justified in [AGM5].