Explicit computation of a Galois representation attached to an eigenform over SL3 from the H2 \'etale of a surface

TL;DR

This paper presents a method for explicitly computing mod l Galois representations from the H2 étale cohomology of surfaces, applied to an eigenform over SL(3) to produce a polynomial with a specific Galois group.

Contribution

It introduces a novel computational approach for Galois representations in H2 étale cohomology and applies it to a case over SL(3), producing explicit polynomials with desired Galois groups.

Findings

Computed a polynomial with Galois group PSU(3,9)

Representation ramified only at 2 and 3

Demonstrated the method's effectiveness for SL(3) eigenforms

Abstract

We sketch a method to compute mod $\ell$ Galois representations contained in the H2 \'etale of surfaces. We apply this method to the case of a representation with values in GL(3,9) attached to an eigenform over a congruence subgroup of SL(3). We obtain in particular a polynomial with Galois group isomorphic to the simple group PSU(3,9) and ramified at 2 and 3 only.