Sur la cohomologie des syst\`emes locaux sur les espaces des modules des courbes de genre 2 et des surfaces ab\'eliennes

TL;DR

This paper investigates the cohomology of local systems on moduli spaces of genus 2 curves and abelian surfaces, providing explicit formulas for Eisenstein cohomology and insights into Siegel modular forms through finite field curve counting.

Contribution

It offers an explicit formula for Eisenstein cohomology and a conjectural expression for endoscopic contributions in the context of moduli spaces of genus 2 curves and abelian surfaces.

Findings

Explicit formula for Eisenstein cohomology

Conjectural formula for endoscopic contributions

Connection between curve counting over finite fields and Siegel modular forms

Abstract

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms.