Weil zeta functions of group representations over finite fields

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 $G$.