On the enumeration of orbits of unipotent groups over finite fields

2022/08/09 by Tobias Rossmann, Rossmann, Tobias
Computer Science · Mathematics · #05A15 #11G25 #20D15 #20E45 #Algebraic Geometry (math.AG) #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2208.04646

openalex publication_date 2022/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the enumeration of linear orbits and conjugacy classes of Z-defined unipotent groups over finite fields is "wild" in the following sense: given an arbitrary scheme Y of finite type over Z and integer n\geqslant 1, the numbers # Y(F_q) \bmod qⁿ can be expressed, uniformly in q, in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many Z-defined unipotent groups over F_q and finitely many Laurent polynomials in q.

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