Large automorphism groups of ordinary curves of even genus in odd\n characteristic

2019/08/11 by Maria Montanucci, Montanucci, Maria, Pietro Speziali +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1908.04684

openalex publication_date 2019/08/11 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Let \X be a (projective, non-singular, geometrically irreducible)\ncurve of even genus g(\X) \≥ 2 defined over an algebraically\nclosed field K of odd characteristic p. If the p-rank\n\γ(\X) equals g(\X), then \X is\n\ordinary. In this paper, we deal with \large automorphism groups\nG of ordinary curves of even genus. We prove that |G| <\n821.37g(\X)^7/4. The proof of our result is based on the\nclassification of automorphism groups of curves of even genus in positive\ncharacteristic, see citegiulietti-korchmaros-2017. According to this\nclassification, for the exceptional cases rm Aut(\X) \≅ rm\nAlt₇ and rm Aut(\X) \≅ rmM₁₁ we show that the\nclassical Hurwitz bound | rm Aut(\X)| < 84(g(\X)-1)\nholds, unless p=3, g(\X)=26 and rm Aut(\X) \≅\n rmM₁₁; an example for the latter case being given by the modular curve\nX(11) in characteristic 3.\n

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