Finding Galois representations corresponding to certain Hecke eigenclasses

TL;DR

This paper presents new computational methods to identify Galois representations attached to specific Hecke eigenclasses, providing evidence for their uniqueness and advancing understanding in arithmetic cohomology.

Contribution

It introduces a combination of Hunter searches, class field theory, and elliptic curves to find Galois representations linked to previously unresolved Hecke eigenclasses.

Findings

Successfully identified Galois representations for several eigenclasses

Provided strong evidence for the uniqueness of these Galois representations

Extended computational techniques for connecting Galois representations to cohomology classes

Abstract

In 1992, Avner Ash and Mark McConnell presented computational evidence of a connection between three-dimensional Galois representations and certain arithmetic cohomology classes. For some examples they were unable to determine the attached representation. For several Hecke eigenclasses (including one for which Ash and McConnell did not find the Galois representation), we find a Galois representation which appears to be attached and show strong evidence for the uniqueness of this representation. The techniques that we use to find defining polynomials for the Galois representations include a targeted Hunter search, class field theory, and elliptic curves.