Top-dimensional rational cohomology of the congruence subgroup ${\Gamma }_{0,n}^{+}(p)$

Abstract

Let ${\Gamma }_{0,n}^{+}(p)\subset {\mathrm{SL}}_n(\mathbb{Z})$ be the congruence subgroup of level-$p$ whose first column is of the form $(\ast ,0,\dots ,0{)}^t\mathrm{mod}p$. We prove that the top-dimensional cohomology group $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}({\Gamma }_{0,n}^{+}(p);\mathbb{Q})$ vanishes for $p\in \{2,3,5,7,13\}$ if $n\ge 3$, as well as for $p\le 6n-14$. Additionally, we prove a non-vanishing result, showing that this cohomology group is nonzero for $n=2$ for every prime $p$, and for $n=3$ for all primes $p\notin \{2,3,5,7,13\}$.

95 pages