Ordinary algebraic curves with many automorphisms in positive characteristic

Korchmáros, Gábor; Montanucci, Maria

doi:10.2140/ant.2019.13.1

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Ordinary algebraic curves with many automorphisms in positive characteristic

Gábor Korchmáros and Maria Montanucci

Vol. 13 (2019), No. 1, 1–18

DOI: 10.2140/ant.2019.13.1

Abstract

Let $𝒳$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $𝔤 (𝒳) \geq 2$ defined over an algebraically closed field $𝕂$ of odd characteristic $p$ . Let $Aut (𝒳)$ be the group of all automorphisms of $𝒳$ which fix $𝕂$ elementwise. For any solvable subgroup $G$ of $Aut (𝒳)$ we prove that $| G | \leq 34 {(𝔤 (𝒳) + 1)}^{3 ∕ 2}$ . There are known curves attaining this bound up to the constant $34$ . For $p$ odd, our result improves the classical Nakajima bound $| G | \leq 84 (𝔤 (𝒳) - 1) 𝔤 (𝒳)$ and, for solvable groups $G$ , the Gunby–Smith–Yuan bound $| G | \leq 6 (𝔤 {(𝒳)}^{2} + 12 \sqrt{21} 𝔤 {(𝒳)}^{3 ∕ 2})$ where $𝔤 (𝒳) > c p^{2}$ for some positive constant $c$ .

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Keywords

algebraic curves, algebraic function fields, positive characteristic, automorphism groups

Mathematical Subject Classification 2010

Primary: 14H37

Secondary: 14H05

Milestones

Received: 25 October 2016

Revised: 18 October 2018

Accepted: 20 November 2018

Published: 13 February 2019

Authors

Gábor Korchmáros
Dipartimento di Matematica, Informatica ed Economia Università degli Studi della Basilicata Potenza Italy

Maria Montanucci
Dipartimento di Tecnica e Gestione dei Sistemi Industriali Università degli Studi di Padova Vicenza Italy

Vol. 13, No. 1, 2019