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

Alexander Stasinski, Michele Zordan

Published 2020-07-21Version 1

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ 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 one-dimensional 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.