Rationality of twist representation zeta functions of compact $p$-adic analytic groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913337154994176 |
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| author | Stasinski, Alexander Zordan, Michele |
| author_facet | Stasinski, Alexander Zordan, Michele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05825 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rationality of twist representation zeta functions of compact $p$-adic analytic groups Stasinski, Alexander Zordan, Michele Group Theory Primary 20E18, Secondary 20C15, 22E35, 20J06 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. |
| title | Rationality of twist representation zeta functions of compact $p$-adic analytic groups |
| topic | Group Theory Primary 20E18, Secondary 20C15, 22E35, 20J06 |
| url | https://arxiv.org/abs/2212.05825 |