Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

TL;DR

This paper computes torsion in various homology theories of congruence subgroups of SL(4,Z), providing evidence for a conjecture that Hecke eigenclasses have attached Galois representations, especially at small torsion primes.

Contribution

It presents the first detailed computations of torsion in these homology theories for SL(4,Z) and supports a conjecture linking Hecke eigenclasses to Galois representations.

Findings

Identified torsion classes at primes 2, 3, 5 in levels up to 31.

Provided evidence for the conjecture in 15 cases of odd torsion.

Analyzed the structure of Hecke modules in the computed homology.

Abstract

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.