An elementary abelian $p$-cover of the Hermitian curve with many automorphisms

TL;DR

This paper determines the automorphism group of a specific elementary abelian p-cover of the Hermitian curve, revealing its large Sylow p-subgroups and exploring related geometric properties in characteristic p.

Contribution

It explicitly computes the automorphism group of the cover and analyzes its structure, including Sylow p-subgroups, in relation to Nakajima's bound, and investigates various geometric features.

Findings

Automorphism group determined explicitly

Sylow p-subgroups are near Nakajima's bound

Properties like Weierstrass points and Galois points are studied

Abstract

The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.