%0 Journal Article
%T Counting points of schemes over finite rings and counting representations of arithmetic lattices
%A Avraham Aizenbud
%A Nir Avni
%J Mathematics
%D 2015
%I arXiv
%X 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 one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky.
%U http://arxiv.org/abs/1502.07004v2