Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p²

2023/07/27 by Barbara Gatti, Gatti, Barbara, Gábor Korchmáros +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.15192

openalex publication_date 2023/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field \mathbbF_q² is maximal if the number N_q² of its \mathbbF_q²-rational points attains the Hasse-Weil upper bound, that is N_q²=q²+2\mathfrakgq+1 where \mathfrakg is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p² where p is the characteristic of \mathbbF_q². Doing so we also determine the \mathbbF_q²-isomorphism classes of such curves and describe their full \mathbbF_q²-automorphism groups.

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