标题： 辛Steinberg模的表示与$\operatorname{Sp}_{2n}(\mathbb{Z})$的上同调 显示英文标题

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

摘要： Borel-Serre 证明了整辛群$\operatorname{Sp}_{2n}(\mathbb{Z})$是一个虚对偶群，其维度为$n^2$，并且辛 Steinberg 模块$\operatorname{St}^\omega_n(\mathbb{Q})$是它的双重化模。这个模是与$\operatorname{Sp}_{2n}(\mathbb{Q})$相关的 Tits 复形的最高维同调。我们找到了该 Steinberg 模块的一个展示，并用它来证明当 $n \geq 2$, $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) \cong 0$ 时，$\operatorname{Sp}_{2n}(\mathbb{Z})$的余维数-1 有理上同调消失。 等价地，主极化阿贝尔簇的模叠积 $\mathcal{A}_n$ 在维数为 $2n$ 时的有理上同调在相同的次数下消失。 我们的研究结果表明高维上同调在次数 $n^2-i$ 存在一个消失模式，这与 Church-Farb-Putman 对特殊线性群所猜测的模式相似。 显示英文摘要

摘要： 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.