Cohomology of local systems on loci of d-elliptic abelian surfaces

TL;DR

This paper investigates the cohomology of local systems on loci of d-elliptic abelian surfaces, using modular curve quotients to compute mixed Hodge structures and Galois representations, with detailed results for the case d=2.

Contribution

It provides a method to compute cohomology of local systems on d-elliptic loci via modular curve quotients, including explicit calculations for genus 2 bi-elliptic curves.

Findings

Cohomology described as mixed Hodge structures and Galois representations.

Explicit Euler characteristic calculation for bi-elliptic genus 2 curves.

Application of modular quotient descriptions to cohomology computations.

Abstract

We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of d-elliptic surfaces in $A_2$. We show how a description of these loci as quotients of a product of modular curves can be used to calculate cohomology of natural local systems on them, both as mixed Hodge structures and $\ell$-adic Galois representations. We study in particular the case d=2, and compute the Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2 curves in the Grothendieck group of Hodge structures.