{
  "id": "2007.10694",
  "version": "v1",
  "published": "2020-07-21T10:10:07.000Z",
  "updated": "2020-07-21T10:10:07.000Z",
  "title": "Rationality of representation zeta functions of compact $p$-adic analytic groups",
  "authors": [
    "Alexander Stasinski",
    "Michele Zordan"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "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 $p$. Moreover, we prove analogous results for twist zeta functions of compact $p$-adic analytic groups, which enumerate irreducible representations up to one-dimensional twists. In the course of the proof, we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-21T10:10:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "representation zeta function",
      "adic analytic group",
      "rationality",
      "twist isoclass",
      "twist zeta functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}