On the enumeration of orbits of unipotent groups over finite fields

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 $q$.