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

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.