\mathbbF_p²-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

2017/08/13 by Daniele Bartoli, Bartoli, Daniele, Maria Montanucci +3
Computer Science · Mathematics · #11G #14G #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1708.03933

openalex publication_date 2017/08/13 · openalex created_date 2017/08/31 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF be the finite field of order q², q=p^h with p prime. It is commonly atribute to J.P. Serre the fact that any curve \mathbbF-covered by the Hermitian curve H_q+1: y^q+1=x^q+x is also \mathbbF-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that q>8 and h≡ 0\pmod3. In this paper, we show that if an \mathbbF-maximal curve X of genus g≥ 2 where q=p is such that |Aut(X)|>84(g-1) then X is Galois-covered by H_p+1. Also, we show that the hypothesis on the order of Aut(X) is sharp, since there exists an \mathbbF-maximal curve X for q=71 of genus g=7 with |Aut(X)|=84(7-1) which is not Galois-covered by the Hermitian curve H₇₂.

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