Large odd prime power order automorphism groups of algebraic curves in any characteristic

TL;DR

This paper extends bounds on automorphism groups of algebraic curves with prime power order in any characteristic, focusing on extremal cases for the prime 3, and classifies their structures.

Contribution

It generalizes Zomorrodian's bound for automorphism groups to algebraic curves over any algebraically closed field, and classifies extremal 3-group cases.

Findings

Proves Zomorrodian's bound for any algebraically closed field.

Classifies the structure of extremal 3-Zomorrodian curves.

Constructs infinite families of extremal curves.

Abstract

Let $\mathcal{X}$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p\ge 0$, and let $\rm{Aut}(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any a subgroup $G$ of $\rm{Aut}(\mathcal{X})$ whose order is a power of an odd prime $d$ other than $p$, the bound proven by Zomorrodian for Riemann surfaces is $|G|\leq 9(g-1)$ where the extremal case can only be obtained for $d=3$. We prove Zomorrodian's result for any $\mathbb{K}$. The essential part of our paper is devoted to extremal $3$-Zomorrodian curves $\mathcal{X}$. Two cases are distinguished according as the quotient curve $\mathcal{X}/Z$ for a central subgroup $Z$ of $\rm{Aut}(\mathcal{X})$ of order $3$ is either elliptic, or not. For elliptic type extremal $3$-Zomorrodian curves $\mathcal{X}$, we completely determine the two possibilities for the abstract structure of $G$ using deeper results on finite $3$-groups. We also show infinite families of extremal $3$-Zomorrodian curves for both types, elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.