The representation of \({\text{Sp}}_{2g}\) with highest weight \(\lambda\) induces a local system \({\mathbf W}_\lambda\) on \({\mathcal A}_g\), which can then be restricted to interesting loci in \({\mathcal A}_g\). Studying the cohomology of such restrictions is then a natural thing to do, in the light of the Eichler-Shimura correspondence which, in the case \(g=1\), relates such cohomology to modular forms. This paper concentrates on the case \(g=2\) and on the loci \({\mathcal E}_d\) of abelian surfaces with a degree \(d^2\) isogeny to a product of elliptic curves, called \(d\)-elliptic abelian surfaces.
The main algebraic tool is a decomposition formula that shows how the restriction of the representation \(W_{l,m}\) of \({\text{Sp}}_4\) to the wreath product \({\text{Sp}}_2\wr\, {\mathbb S}_2=({\text{Sp}}_2\times {\text{Sp}}_2)\rtimes {\mathbb S}_2\) decomposes into irreducible representations. As this wreath product (with \(\Gamma(d)\) instead of \({\text{Sp}}_2\)) is the modular group that appears when one considers \({\mathcal E}_d\), the Eichler-Shimura theory can then be used to describe the cohomology of the restricted local system. The part coming from Eisenstein series carries no real information and one can concentrate on the rest (the cusp forms) by considering the inner cohomology, which is the image of cohomology with compact supports in ordinary cohomology. The author writes down concise formulae for this.
The case \(d=2\) is slightly special: simpler in some respects but with some additional complications caused by the presence of generic automorphisms (i.e.by the fact that the modular curve of level \(2\) must be considered as a stack). The last part of the paper is devoted to this case, and yields formulae for the compactly supported Euler characteristic, in the Grothendieck group of mixed Hodge structures, of the moduli space of \(n\)-pointed bielliptic genus \(2\) curves.