Rationality of representation zeta functions of compact $p$-adic analytic groups

Abstract

We prove that for any FAb compact p-adic analytic group G, its representation zeta function is a finite sum of terms n −s i f i (p −s ), where n i are natural numbers and f i (t) ∈ Q(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G is moreover a pro-p group, we prove that its representation zeta function is rational in p −s . These results were proved by Jaikin-Zapirain for p > 2 or for G uniform and pro-2, respectively. We give a new proof which avoids the Kirillov orbit method and works for all p.Moreover, we prove analogous results for twist zeta functions of compact p-adic analytic groups, which enumerate irreducible representations up to onedimensional twists. In the course of the proof, we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.