Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function 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 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 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. 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 groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G.

In addition, we show that the Euler factors of ζ_G are rational functions in p^−s if G is virtually abelian. For finite groups G we calculate ζ_G using the rational representation theory of G.