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

TL;DR

This paper presents a detailed study of symplectic Steinberg modules, providing a presentation and demonstrating the vanishing of certain high-dimensional rational cohomology groups of symplectic groups and related moduli stacks.

Contribution

It introduces a presentation of the symplectic Steinberg module and proves the vanishing of codimension-1 rational cohomology for symplectic groups, extending understanding of their cohomological properties.

Findings

The symplectic Steinberg module has a specific presentation.

Codimension-1 rational cohomology of Sp_{2n}(Z) vanishes for n ≥ 2.

High-dimensional cohomology vanishes in degree n^2 - 1.

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.