Rationality of representation zeta functions of compact $p$-adic analytic groups
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arXiv
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| Format: | Preprint |
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2020
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| author | Stasinski, Alexander Zordan, Michele |
| author_facet | Stasinski, Alexander Zordan, Michele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_10694 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rationality of 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 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. |
| title | Rationality of representation zeta functions of compact $p$-adic analytic groups |
| topic | Group Theory Primary 20E18, Secondary 20C15, 22E35, 20J06 |
| url | https://arxiv.org/abs/2007.10694 |