Advanced Seminar on Group Theory

Hasse–Weil zeta functions of profinite groups

Profinite groups are topological groups that can be assembled from finite groups. Important examples include Galois groups and arbitrary products of finite groups. They combine properties of finite and infinite groups, making their representation theory over finite fields a natural and interesting subject of study.

The first part of the seminar will focus on the Hasse–Weil zeta functions of profinite groups. These functions encode the numbers of absolutely irreducible representations of a profinite group in different degrees over finite fields. We will give a probabilistic interpretation and investigate their analytic properties, with particular emphasis on their abscissae of convergence.

The initial sequence of talks will introduce the necessary foundations, develop examples and a probabilistic interpretation, and study how these zeta functions behave under open subgroups, products, and split extensions. We will also consider free abelian profinite groups and free pro-p groups. Further topics will be chosen as the seminar develops, and the amount of time devoted to each topic will be determined during the seminar.

More details can be found in the program.

References:

1. G. Corob Cook, S. Kionke, M. Vannacci. Weil zeta functions of group representations over finite fields, Selecta Mathematica 30 (2024), Article 46, 1–57. Springer
2. E. Detomi, A. Lucchini. Probabilistic non-generators in profinite groups, in Ischia Group Theory 2004, Contemporary Mathematics 402, American Mathematical Society, 2006, 149–158. AMS
3. A. Mann. Positively finitely generated groups, Forum Mathematicum 8 (1996), no. 4, 429–460. DOI