Large automorphism groups of ordinary curves in characteristic $2$

TL;DR

This paper establishes an upper bound on the size of solvable automorphism groups of ordinary algebraic curves over fields of characteristic 2, relating group order to the genus of the curve.

Contribution

It provides a new bound on automorphism group size for ordinary curves in characteristic 2, specifically when the group is solvable and the genus is even.

Findings

Automorphism group order is less than 35(g+1)^{3/2} for the specified curves.

The bound applies to ordinary curves with even genus in characteristic 2.

The result constrains the symmetry groups of such algebraic curves.

Abstract

Let $\mathcal{X}$ be a (projective, non-singular, irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is ordinary. In this paper, we deal with large automorphism groups $G$ of ordinary curves. Under the hypotheses that $p = 2$, $g(\mathcal{X})$ is even and $G$ is solvable, we prove that $|G| < 35(g(\mathcal{X}) +1)^{3/2}$.