Weil zeta functions of group representations over finite fields

• Ged Corob Cook ^[1] ; Steffen Kionke ^[2] ; Matteo Vannacci ^[3]
  1. [1] University of Lincoln

     University of Lincoln

     Lincoln District, Reino Unido

     
  2. [2] Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Germany
  3. [3] Matematika Saila, UPV-EHU, Spain

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-57
• Idioma: inglés
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  • 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. ThisWeil 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 ζG(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 ζG is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζG. 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 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.