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

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We consider the loci of d-elliptic curves in M₂, and corresponding loci of d-elliptic surfaces in A₂. 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 ℓ-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.