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

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 $\mathrm{\Gamma }$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\mathrm{\Gamma })$ be the number of irreducible $n$-dimensional representations of $\mathrm{\Gamma }$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\mathrm{\Gamma })=O(n^C)$ for every such $\mathrm{\Gamma }$. This answers a question of Larsen and Lubotzky.