On the enumeration of orbits of unipotent groups over finite fields

TL;DR

This paper demonstrates that counting linear orbits and conjugacy classes of unipotent groups over finite fields is extremely complex, capable of encoding arbitrary finite scheme point counts modulo powers of the field size.

Contribution

It establishes the 'wild' complexity of enumerating orbits and conjugacy classes of unipotent groups over finite fields, linking it to arbitrary scheme point counts.

Findings

Enumeration can encode arbitrary scheme point counts modulo q^n.

Counting problems are as complex as arbitrary finite scheme enumeration.

Results show the enumeration problem is 'wild' in complexity.

Abstract

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