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

Abstract

Borel-Serre proved that the integral symplectic group ${\mathrm{Sp}}_{2n}(\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module ${\mathrm{St}}_n^{\omega }(\mathbb{Q})$ is its dualising module. This module is the top-dimensional homology of the Tits building associated to ${\mathrm{Sp}}_{2n}(\mathbb{Q})$. We find a presentation of this Steinberg module and use it to show that the codimension-1 rational cohomology of ${\mathrm{Sp}}_{2n}(\mathbb{Z})$ vanishes for $n\ge 2$, $H^{n^2-1}({\mathrm{Sp}}_{2n}(\mathbb{Z});\mathbb{Q})\cong 0$. 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.

98 pages, 34 figures. Comments welcome!