@article{dc1eb53f2d3d4caa95f696857d9be3fc,

title = "Counting points of schemes over finite rings and counting representations of arithmetic lattices",

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.",

author = "Avraham Aizenbud and Nir Avni",

note = "Funding Information: Aizenbud{\textquoteright}s work was partially supported by Israel Science Foundation grant 687/13, National Science Foundation (NSF) grant DMS-1100943, United States– Israel Binational Science Foundation (BSF) grant 2012247, and a Minerva foundation grant. Avni{\textquoteright}s work was partially supported by NSF grants DMS-0901638 and DMS-1303205 and by BSF grant 2012247. Funding Information: The current proof of TheoremAwas suggested by Johannes Nicaise and is simpler than the proof we originally had. We thank him for showing us his proof. We also thank Vladimir Hinich for useful discussions. Part of this article was written during the program Multiplicity Problems in Harmonic Analysis held at the Hausdorff Institute (2012-2014), and we thank that institution for its hospitality. Aizenbud's work was partially supported by Israel Science Foundation grant 687/13, National Science Foundation (NSF) grant DMS-1100943, United States-Israel Binational Science Foundation (BSF) grant 2012247, and a Minerva foundation grant. Avni's work was partially supported by NSF grants DMS-0901638 and DMS-1303205 and by BSF grant 2012247. Publisher Copyright: {\textcopyright} 2018.",

year = "2018",

month = oct,

day = "1",

doi = "10.1215/00127094-2018-0021",

language = "English (US)",

volume = "167",

pages = "2721--2743",

journal = "Duke Mathematical Journal",

issn = "0012-7094",

publisher = "Duke University Press",

number = "14",

}