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

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.

Author's Accepted Manuscript version. To appear in Trans. Amer. Math. Soc. Second part of arXiv:2007.10694