Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alexander Stasinski, Michele Zordan

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 $p$. First part of arXiv:2007.10694, second part uploaded as a separate paper.

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

Abstract

We prove that for any FAb compact -adic analytic group , its representation zeta function is a finite sum of terms , where are natural numbers and are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If is moreover a pro- group, we prove that its representation zeta function is rational in . These results were proved by Jaikin-Zapirain for or for uniform and pro-, respectively. We give a new proof which avoids the Kirillov orbit method and works for all . First part of arXiv:2007.10694, second part uploaded as a separate paper.

Paper Structure

This paper contains 18 sections, 21 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Main result and consequences
  3. Outline of the paper and the proofs
  4. Remarks on the positive characteristic case
  5. Basics from model theory
  6. The language of $p$-adic analytic groups
  7. The analytic language of the $p$-adic numbers
  8. Rationality and definable enumerations in $\mathcal{M}_{\mathrm{an}}$
  9. Preliminaries on projective representations
  10. Projective representations and Clifford theory
  11. Reduction to the partial zeta series
  12. Cohomology classes and degree one characters
  13. Rationality of the partial zeta series
  14. Bases for $p$-adic analytic groups
  15. The fibres of $\mathcal{C}$ in terms of degree one characters.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G$ be a FAb compact $p$-adic analytic group. Then $\zeta_{G}(s)$ is virtually rational in $p^{-s}$. If in addition $G$ is pro-$p$, then $\zeta_{G}(s)$ is rational in $p^{-s}$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2: duSautoy-rationality
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: hrumar2015definable
  • ...and 41 more