Equivariant Cohomology of the Moduli Space of Genus Three Curves with Symplectic Level Two Structure via Point Counts

TL;DR

This paper computes the cohomology of the moduli space of genus three curves with symplectic level two structure by counting points over finite fields, revealing detailed representation-theoretic information.

Contribution

It provides explicit cohomological calculations for the moduli space of genus three curves with symplectic level two structure using point counting methods.

Findings

Cohomology groups of the quartic locus are determined as symmetric group representations.

Cohomological data related to genus three curves with level structures are explicitly computed.

The approach links point counts over finite fields to cohomological invariants.

Abstract

We make cohomological computations related to the moduli space of genus three curves with symplectic level two structure by means of counting points over finite fields. In particular, we determine the cohomology groups of the quartic locus as representations of the symmetric group on seven elements.