On nilpotent automorphism groups of function fields

TL;DR

This paper investigates the bounds on the size of nilpotent automorphism groups of function fields with genus at least 2 over algebraically closed fields of positive characteristic, establishing a sharp upper limit and characterizing cases of equality.

Contribution

It establishes a new upper bound for the order of nilpotent automorphism groups of function fields and characterizes the cases where this bound is attained.

Findings

Bound of 16(g-1) for non-p-group nilpotent automorphism groups

If the bound is attained, g-1 is a power of 2

Existence of an infinite family of function fields reaching the bound

Abstract

We study the automorphisms of a function field of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. More precisely, we show that the order of a nilpotent subgroup $G$ of its automorphism group is bounded by $16 (g-1)$ when G is not a $p$-group. We show that if $|G|=16(g-1) $, then $g-1$ is a power of $2$. Furthermore, we provide an infinite family of function fields attaining the bound.