Duke Mathematical Journal

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

Avraham Aizenbud and Nir Avni

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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.

Article information

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

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

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Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 20F69: Asymptotic properties of groups 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture) 20G05: Representation theory 20G30: Linear algebraic groups over global fields and their integers

representation growth Igusa zeta function


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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