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15 January 2013 Representation zeta functions of compact p-adic analytic groups and arithmetic groups
Nir Avni, Benjamin Klopsch, Uri Onn, Christopher Voll
Duke Math. J. 162(1): 111-197 (15 January 2013). DOI: 10.1215/00127094-1959198

Abstract

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.

Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

Citation

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Nir Avni. Benjamin Klopsch. Uri Onn. Christopher Voll. "Representation zeta functions of compact p-adic analytic groups and arithmetic groups." Duke Math. J. 162 (1) 111 - 197, 15 January 2013. https://doi.org/10.1215/00127094-1959198

Information

Published: 15 January 2013
First available in Project Euclid: 14 January 2013

zbMATH: 1281.22005
MathSciNet: MR3011874
Digital Object Identifier: 10.1215/00127094-1959198

Subjects:
Primary: 20F69, 22E50, 22E55
Secondary: 11M41, 20C15, 20G25, 22E40

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 1 • 15 January 2013
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