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 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.

version 2: The prove of the main theorem was simplified and a result on arithmetic algebraic geometry was added. 29 pages

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• When does the zero fiber of the moment map have rational singularities?
• A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^k\mathbb{Z}$-points
• Rational singularities for moment maps of totally negative quivers