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

2015/02/28 by Avraham Aizenbud, Nir Avni
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Arithmetic #Coding theory and cryptography #Combinatorics #Discrete mathematics #Isomorphism (crystallography) #Lattice (music) #Mathematics #Rank (graph theory) #math.AG #math.GR #math.NT #math.RT #msc:14B05 #msc:14G05 #msc:14G10 #msc:20F69 #msc:20G05 #msc:20G25

paper · pdf · doi:10.1215/00127094-2018-0021

published as Duke Math. J. 167, no. 14 (2018), 2721-2743 · version 2: The prove of the main theorem was simplified and a result on arithmetic algebraic geometry was added. 29 pages

arxiv created 2015/08/14 · openalex created_date 2016/06/24 · openalex publication_date 2018/09/14 · arxiv updated 2018/11/14 · openalex updated_date 2026/08/05

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.

Citations

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