On the classification problem for the genera of quotients of the\n Hermitian curve

2018/05/23 by Francesca Dalla Volta, Volta, Francesca Dalla, Maria Montanucci +3 · 1 citation
Mathematics · #11G20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1805.09118

openalex publication_date 2018/05/23 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

In this paper we characterize the genera of those quotient curves\n\H_q/G of the mathbbF_q²-maximal Hermitian curve\n\H_q for which G is contained in the maximal subgroup\n\M₁ of rm Aut(\H_q) fixing a self-polar triangle, or\nq is even and G is contained in the maximal subgroup \M₂ of\n rm Aut(\H_q) fixing a pole-polar pair (P,\ℓ) with respect to\nthe unitary polarity associated to \H_q( mathbbF_q²). In this\nway several new values for the genus of a maximal curve over a finite field are\nobtained. Together with what is known in the literature, our results leave just\ntwo open cases to provide the complete list of genera of Galois subcovers of\nthe Hermitian curve; namely, the open cases in [Bassa-Ma-Xing-Yeo, J. Combin.\nTheory Ser. A, 2013] when G fixes a point P \∈\n\H_q( mathbbF_q²) and q is even, and the open cases in\n[Montanucci-Zini, Comm. Algebra, 2018] when G\≤\M₂ and q is\nodd.\n

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• The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes

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