A presentation of symplectic Steinberg modules and cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$

Abstract

Borel-Serre proved that the integral symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module $\operatorname{St}^\omega_n(\mathbb{Q})$ is its dualising module. This module is the top-dimensional homology of the Tits building associated to $\operatorname{Sp}_{2n}(\mathbb{Q})$. We find a presentation of this Steinberg module and use it to show that the codimension-1 rational cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes for $n \geq 2$, $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) \cong 0$. Equivalently, the rational cohomology of the moduli stack $\mathcal{A}_n$ of principally polarised abelian varieties of dimension $2n$ vanishes in the same degree. Our findings suggest a vanishing pattern for high-dimensional cohomology in degree $n^2-i$, similar to the one conjectured by Church-Farb-Putman for special linear groups.