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

TL;DR

This paper links the singularities of schemes over finite rings to point counting asymptotics and uses this to establish polynomial bounds on the number of irreducible representations of certain arithmetic lattices.

Contribution

It provides a partial answer to a question of Mustata by relating scheme singularities to point counts and establishes polynomial bounds on representation counts of high-rank arithmetic lattices.

Findings

Established a relation between scheme singularities and point counting over finite rings.

Proved that the number of irreducible n-dimensional representations of certain arithmetic lattices grows at most polynomially.

Provided explicit polynomial bound C=746 for the growth of representation counts.

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.