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

Avraham Aizenbud, Nir Avni

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 Γ is an arithmetic lattice whose Q-rank is greater than 1, then let rn.(Γ)be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that rn(Γ) = O(nC) for every such Γ. This answers a question of Larsen and Lubotzky.

Original language English (US) 2721-2743 23 Duke Mathematical Journal 167 14 https://doi.org/10.1215/00127094-2018-0021 Published - Oct 1 2018

Finite Rings
Counting
n-dimensional
Isomorphism
Count
Singularity
Partial

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

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In: Duke Mathematical Journal, Vol. 167, No. 14, 01.10.2018, p. 2721-2743.

Research output: Contribution to journalArticle

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AU - Avni, Nir

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AB - 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 Γ is an arithmetic lattice whose Q-rank is greater than 1, then let rn.(Γ)be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that rn(Γ) = O(nC) for every such Γ. This answers a question of Larsen and Lubotzky.

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