On the enumeration of orbits of unipotent groups over finite fields
Tobias Rossmann · Proceedings of the American Mathematical Society · 2024
We show that the enumeration of linear orbits and conjugacy classes of Z \mathbf {Z} -defined unipotent groups over finite fields is “wild” in the following sense: given an arbitrary scheme Y Y of finite type over Z \mathbf {Z} and integer n ⩾ 1 n\!\geqslant \! 1 , the numbers # Y ( F q ) mod q n \#Y(\mathbf {F}_q) \bmod q^n can be expressed, uniformly in q q , in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many Z \mathbf {Z} -defined unipotent groups over F q \mathbf {F}_q and finitely many Laurent polynomials in 𝑞.