Cohomology with Sym^g coefficients for congruence subgroups of SL_4(Z) and Galois representations

TL;DR

This paper extends cohomology computations for congruence subgroups of SL_4(Z) with Sym^g coefficients over finite fields, identifying Galois representations attached to Hecke eigenclasses in the top cuspidal degree.

Contribution

It advances computational methods to determine cohomology with higher-dimensional coefficients and attaches Galois representations to Hecke eigenclasses in degree five.

Findings

Computed cohomology in degree five for SL_4(Z) with Sym^g coefficients.

Identified Galois representations associated with Hecke eigenclasses.

Modified algorithms to handle non-one-dimensional coefficient modules.

Abstract

We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In this paper we take K to be a finite field of large characteristic, as a proxy for the complex numbers. For each Hecke eigenclass found, we produce the unique Galois representation that appears to be attached to it. The computations require modifications to our previous algorithms to accommodate the fact that the coefficients are not one-dimensional.