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. This Weil representation 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 sufficiently large 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 Weil abscissa, i.e., the abscissa of convergence a(G) of \(\zeta _G\). We calculate the Weil 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 Weil 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 \(a \ge 1\) is the Weil 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.