Weil zeta functions of group representations over finite fields

Abstract

In this article we define and study a zeta function $$\zeta _G$$ ζ 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 $$\zeta _G(k)^{-1}$$ ζ 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 $$\zeta _G$$ ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of $$\zeta _G$$ ζ 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-$${\mathfrak {C}}$$ C groups, where $${\mathfrak {C}}$$ C is a class of finite groups with prescribed composition factors. We prove that every real number $$a \ge 1$$ a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of $$\zeta _G$$ ζ G are rational functions in $$p^{-s}$$ p - s if G is virtually abelian. For finite groups G we calculate $$\zeta _G$$ ζ G using the rational representation theory of G.