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

TL;DR

This paper proves that the representation zeta functions of compact p-adic analytic groups are rational functions in p^{-s}, extending previous results to all primes p without relying on the Kirillov orbit method.

Contribution

It introduces a new proof demonstrating the rationality and meromorphic continuation of these zeta functions for all primes p, generalizing prior work.

Findings

Representation zeta functions are finite sums of rational functions in p^{-s}.

Meromorphic continuation and rationality of the abscissa follow as corollaries.

For pro-p groups, the zeta function is rational in p^{-s}.

Abstract

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 $p$. First part of arXiv:2007.10694, second part uploaded as a separate paper.