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1 October 2018 Counting points of schemes over finite rings and counting representations of arithmetic lattices
Avraham Aizenbud, Nir Avni
Duke Math. J. 167(14): 2721-2743 (1 October 2018). DOI: 10.1215/00127094-2018-0021

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.

Citation

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Avraham Aizenbud. Nir Avni. "Counting points of schemes over finite rings and counting representations of arithmetic lattices." Duke Math. J. 167 (14) 2721 - 2743, 1 October 2018. https://doi.org/10.1215/00127094-2018-0021

Information

Received: 5 May 2016; Revised: 13 February 2018; Published: 1 October 2018
First available in Project Euclid: 28 September 2018

zbMATH: 06982205
MathSciNet: MR3859363
Digital Object Identifier: 10.1215/00127094-2018-0021

Subjects:
Primary: 14G05
Secondary: 14B05, 14G10, 20F69, 20G05, 20G30

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 14 • 1 October 2018
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