Weil zeta functions of group representations over finite fields

TL;DR

This paper introduces a zeta function for profinite groups that counts irreducible representations over finite fields, explores its properties, and computes it for various classes of groups, revealing its well-behaved nature and connections to group generation probabilities.

Contribution

It defines and analyzes a new zeta function for profinite groups, providing explicit formulas, convergence properties, and bounds for various classes of groups, including free and virtually abelian groups.

Findings

The zeta function converges on a half-plane for all UBERG groups and has an Euler product.

The abscissa of convergence can be explicitly calculated or bounded for many classes of groups.

For virtually abelian groups, the Euler factors are rational functions in p^{-s}.

Abstract

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value $\zeta_G(k)^{-1}$ at a positive integer $k$ coincides with the probability that $k$ random elements generate the completed group ring of $G$. The explicit formulas obtained so far suggest that $\zeta_G$ is rather well-behaved. A central object of this article is the abscissa of convergence $a(G)$ of $\zeta_G$. We calculate the abscissae for free abelian, free abelian pro-$p$, free pro-$p$, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the abscissae of free pro-$\mathfrak{C}$ groups, where $\mathfrak{C}$ is a class of finite groups with prescribed composition factors. We prove that every real number $a \geq 1$ is the abscissa $a(G)$ of some profinite group $G$. In addition, we show that the Euler factors of $\zeta_G$ are rational functions in $p^{-s}$ if $G$ is virtually abelian. For finite groups $G$ we calculate $\zeta_G$ using the rational representation theory of $G$.