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

TL;DR

This paper proves that the twist representation zeta functions of certain compact p-adic analytic groups are rational functions, with implications for their meromorphic continuation and rationality of the abscissa, especially for pro-p groups.

Contribution

It establishes the rationality and meromorphic properties of twist representation zeta functions for compact p-adic groups, introducing a new Clifford theory and cohomological invariant.

Findings

Zeta functions are finite sums of rational functions in p^{-s}

Meromorphic continuation and rationality of the abscissa are proven

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

Abstract

We prove that for any twist rigid compact $p$-adic analytic group $G$, its twist 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 twist representation zeta function is rational in $p^{-s}$. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. Second part of arXiv:2007.10694.