Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function ζ 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 ζ 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 ζ 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 ζ 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- C C groups, where C C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 a ≥ 1 is the Weil abscissa a ( G ) of some profinite group G . In addition, we show that the Euler factors of ζ G ζ G are rational functions in p⁻ˢ p - s if G is virtually abelian. For finite groups G we calculate ζ G ζ G using the rational representation theory of G .