On the enumeration of orbits of unipotent groups over finite fields

We show that the enumeration of linear orbits and conjugacy classes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -defined unipotent groups over finite fields is “wild” in the following sense: given an arbitrary scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite type over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-slanted-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:mo> ⩾ </mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\!\geqslant \! 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="number-sign upper Y left-parenthesis bold upper F Subscript q Baseline right-parenthesis mod q Superscript n"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> # </mml:mi> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\#Y(\mathbf {F}_q) \bmod q^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be expressed, uniformly in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -defined unipotent groups over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and finitely many Laurent polynomials in 𝑞.