Cohomology with twisted one-dimensional coefficients for congruence subgroups of SL(4,Z) and Galois representations

TL;DR

This paper computes the cohomology of congruence subgroups of SL(4,Z) with twisted coefficients, identifies attached Galois representations, and advances algorithms for such complex computations, providing evidence for theoretical conjectures.

Contribution

It extends previous cohomology computations to twisted coefficients in degree five for SL(4,Z) subgroups and improves algorithms to identify unique Galois representations.

Findings

Confirmed the existence of unique Galois representations attached to cohomology classes.

Extended computational methods to handle twisted coefficients in high-degree cohomology.

Supported theoretical predictions about Galois representations and the Borel-Serre boundary.

Abstract

We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a field K, twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In practice 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 a Galois representation that appears to be attached to it. Our computations show that in every case this Galois representation is the only one that could be attached to it. The existence of the attached Galois representations agrees with a theorem of Scholze and sheds light on the Borel-Serre boundary for Gamma. The computations require serious modifications to our previous algorithms to accommodate the twisted coefficients. Nontrivial coefficients add a layer of complication to our data structures, and new possibilites arise that must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder so that it reports when the attached Galois representation is uniquely determined by our data.