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

Stasinski, Alexander; Zordan, Michele

doi:10.1090/tran/9210

Rationality of twist representation zeta functions of compact $p$-adic analytic groups
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by Alexander Stasinski and Michele Zordan;

Trans. Amer. Math. Soc. 377 (2024), 7601-7631

DOI: https://doi.org/10.1090/tran/9210

Published electronically: August 30, 2024

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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.

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