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 \[ ζ G ( s ) = ∑ λ ∈ Irr ⁡ ( G ) λ ( 1 ) − s = ∑ i = 1 k n i − s f i ( p − s ) . \zeta ^G(s)=\sum _{\lambda \in \operatorname {Irr}(G)} \lambda (1) ^{-s}=\sum _{i=1}^kn_i^{-s}f_i(p^{-s}). \]