## Duke Mathematical Journal

### Representation zeta functions of compact $p$-adic analytic groups and arithmetic groups

#### Abstract

We introduce new methods from $\mathfrak{p}$-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 $\mathfrak{p}$-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 $\operatorname {SL}_{3}(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$, and of the groups $\operatorname {SU}_{3}(\mathfrak {O},\mathfrak{o})$, where $\mathfrak {O}$ is an unramified quadratic extension of $\mathfrak{o}$. 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 $A_{2}$. 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 $A_{2}$ defined over number fields.

#### Article information

Source
Duke Math. J., Volume 162, Number 1 (2013), 111-197.

Dates
First available in Project Euclid: 14 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1358172076

Digital Object Identifier
doi:10.1215/00127094-1959198

Mathematical Reviews number (MathSciNet)
MR3011874

Zentralblatt MATH identifier
1281.22005

#### Citation

Avni, Nir; Klopsch, Benjamin; Onn, Uri; Voll, Christopher. Representation zeta functions of compact $p$ -adic analytic groups and arithmetic groups. Duke Math. J. 162 (2013), no. 1, 111--197. doi:10.1215/00127094-1959198. https://projecteuclid.org/euclid.dmj/1358172076

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