Counting points of schemes over finite rings and counting representations of arithmetic lattices

Abstract

We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than $1$, then let $r_n(\Gamma )$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (in fact, any $C>40$ suffices) such that $r_n(\Gamma )=O\left(n^C\right)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky.