2022/12/07 by Ged Corob Cook, Cook, Ged Corob, Steffen Kionke +3
Chemistry · Mathematics · #11M41 #20C20 #20E18 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Number Theory (math.NT) #Synthesis and Reactivity of Heterocycles
paper · pdf · doi:10.48550/arxiv.2212.03748
openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we define and study a zeta function ζG - 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 ζG(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 ζG is rather well-behaved. A central object of this article is the abscissa of convergence a(G) of ζG. 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-\mathfrakC groups, where \mathfrakC is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζG are rational functions in p-s if G is virtually abelian. For finite groups G we calculate ζG using the rational representation theory of G.