Zeta function of representations of compact 𝑝-adic analytic groups

Abstract

Let G G be an FAb compact p p -adic analytic group and suppose that p > 2 p>2 or p = 2 p=2 and G G is uniform. We prove that there are natural numbers n 1 , … , n k n_1, \ldots , n_k and functions f 1 ( p − s ) , … , f k ( p − s ) f_1(p^{-s}),\ldots , f_k(p^{-s}) rational in p − s p^{-s} such that \[ ζ G ( s ) = ∑ λ ∈ Irr ⁡ ( G ) λ ( 1 ) − s = ∑ i = 1 k n i − s f i ( p − s ) . \zeta ^G(s)=\sum _{\lambda \in \operatorname {Irr}(G)} \lambda (1) ^{-s}=\sum _{i=1}^kn_i^{-s}f_i(p^{-s}). \]

Arithmetic groups have rational representation growth

“…where f i (p, x) are rational functions in x (this is proved in [12]). A sequence of numbers, k(p), indexed by the primes of O Σ , is called geometric if there is a variety K , defined over O Σ , such that for every p, k(p) is equal to the number of points of the variety K over the finite field O Σ /p (this terminology is taken from [13]).…”

“…This will enable us to connect the g 1 p 's for different primes p. The analytic definition is useful in order to treat other pro-p subgroups of G p ; we shall promptly do this. The Lazard definition is used in [12], to which we shall refer.…”

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

“…Remark The introduction of the additional variable to express a generating function as an integral in Lemma mimics similar formulae of Jaikin‐Zapirain [, § 4] and Voll [, § 2.2].…”

Zeta functions of groups and rings

“…The formalism of zeta functions has been applied successfully in [7] to solve conjecture P, which had appeared in connection with periodicity in trees connected with the classification problem for finite p-groups in terms of coclass. (6) Thinking of Hilbert's basis theorem we might expect a connection between the ideal counting zeta function of a ring R and that of the polynomial ring R[x] over R. This expectation is confirmed by a beautiful formula of D. Segal [45] which holds for Dedekind rings R. (7) The formalism of zeta functions has been used to count representations of arithmetic and p-adic analytic groups in the papers [37] of B. Martin and A. Lubotzky, [30] of A. Jaikin-Zapirain and [33] of M. Larsen and A. Lubotzky.…”