Let \(\mathcal{A}_3[2]\) denote the coarse moduli space of principally polarized abelian varieties of dimension \(3\) together with a full level \(2\) structure. There is an action of \(\mathrm{Sp}_2(6)\), the symplectic group of degree \(6\) over the field of two elements, on \(\mathcal{A}_3[2]\), by acting on the level \(2\) structure.
In this paper, the authors compute the weighted Euler characteristic of \(\mathcal{A}_3[2]\), equivariant with respect to the action of \(\mathrm{Sp}_2(6)\). They first stratify \(\mathcal{A}_3[2]\) into three pieces \[ \mathcal{A}_3[2]=\mathcal{A}_3^{\mathrm{in}}[2]\sqcup \mathcal{A}_{2,1}[2] \sqcup \mathcal{A}_{1,1,1}[2] \] where the three strata represents the moduli space of respectively, indecomposable abelian \(3\)-folds, products of an indecomposable abelian \(2\)-fold with an elliptic curve, and products of three elliptic curves.
The weighted Euler characteristic for \(\mathcal{A}_3^{\mathrm{in}}[2]\) was computed by the second author. Their main work is computing the weighted Euler characteristic of the last two pieces \(\mathcal{A}_{2,1}[2]\) and \(\mathcal{A}_{1,1,1}[2]\), in section \(5\) and \(6\). They then combine the results to give the weighted Euler characteristic of \(\mathcal{A}_3[2]\) in section \(4\).