  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.
