Codimensions one and two cohomology of Hecke congruence subgroups

Abstract

For $n\ge 1$ and $p$ a prime, the Hecke congruence subgroup ${\Gamma }_{0,n}(p)\le {\mathrm{SL}}_n(\mathbb{Z})$ is the subgroup of matrices whose first column is of the form $(\ast ,0,\dots ,0{)}^t\mathrm{mod}p$. Borel--Serre showed that ${\Gamma }_{0,n}(p)$ has virtual cohomological dimension $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. The first author proved that the rational cohomology in this top degree $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ vanishes for $n$ sufficiently large compared to $p$. We prove analogous results in codimension $1$ and $2$.

14 pages, 1 figure