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

Abstract

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 centraliser dimension.Based on this algebro-geometric description, we compute explicit formulae 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 formulae, 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.

The average size of the kernel of a matrix and orbits of linear groups

“…A very similar phenomenon was exploited by O'Brien and Voll [, § 5] in their enumeration of conjugacy classes of certain relatively free ‐groups. Remark We note that the integrals in – are almost of the same shape as those in [, equation (1.4)]. These similarities can be clarified further by rewriting our integrals slightly; see the proof of Theorem .…”

“…For further positive results establishing such functional equations, see, in particular, work of du Sautoy and Lubotzky , Voll , Avni et al . [, § 4] and Stasinski and Voll [, Theorem A]. Using the formalism developed above, we may deduce the following; recall the notation from § 3.3.…”

“…For later applications, in the following, we record ‘projective’ versions of the integrals in Theorem in the spirit of Voll's formalism [, § 2.2], as rewritten by Avni et al . [, § 3.2].…”

“…Using the surjection which sends a matrix to its first column, we rewrite in terms of an integral over ; cf. [, § 4]. In the setting of the explicit formula for derived in § 4.3.5, we may assume that is a matrix of linear forms whence each consists of homogeneous polynomials of degree .…”

Computing topological zeta functions of groups, algebras, and modules, I

“…A much deeper fact [27,Theorem 1] is that each ζ G,p (s) is a rational function in p −s over Q. By considerably advancing the techniques pioneered in [27], du Sautoy and Grunewald [20, Theorem 1.1 (1)] derived a fundamental result on nilpotent groups: the degree of polynomial subgroup growth of G, a priori a positive real number, is in fact rational. Their proof using zeta functions constitutes a crucial step in the development of the theory of zeta functions of groups and rings as a subject of independent interest.…”