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

Avraham Aizenbud, Nir Avni

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 languageEnglish (US)
Pages (from-to)2721-2743
Number of pages23
JournalDuke Mathematical Journal
Volume167
Issue number14
DOIs
StatePublished - Oct 1 2018

ASJC Scopus subject areas

  • Mathematics(all)

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