## Duke Mathematical Journal

### 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 $C\gt 40$ suffices) such that $r_{n}(\Gamma)=O(n^{C})$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky.

#### Article information

Source
Duke Math. J., Volume 167, Number 14 (2018), 2721-2743.

Dates
Revised: 13 February 2018
First available in Project Euclid: 28 September 2018

https://projecteuclid.org/euclid.dmj/1538121919

Digital Object Identifier
doi:10.1215/00127094-2018-0021

Mathematical Reviews number (MathSciNet)
MR3859363

Zentralblatt MATH identifier
06982205

#### Citation

Aizenbud, Avraham; Avni, Nir. Counting points of schemes over finite rings and counting representations of arithmetic lattices. Duke Math. J. 167 (2018), no. 14, 2721--2743. doi:10.1215/00127094-2018-0021. https://projecteuclid.org/euclid.dmj/1538121919

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