Vol. 13, No. 1, 2019

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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

Let 𝒳 be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus 𝔤(𝒳) 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| 34(𝔤(𝒳) + 1)32. There are known curves attaining this bound up to the constant 34. For p odd, our result improves the classical Nakajima bound |G| 84(𝔤(𝒳) 1)𝔤(𝒳) and, for solvable groups G, the Gunby–Smith–Yuan bound |G| 6(𝔤(𝒳)2 + 1221𝔤(𝒳)32) where 𝔤(𝒳) > cp2 for some positive constant c.

algebraic curves, algebraic function fields, positive characteristic, automorphism groups
Mathematical Subject Classification 2010
Primary: 14H37
Secondary: 14H05
Received: 25 October 2016
Revised: 18 October 2018
Accepted: 20 November 2018
Published: 13 February 2019
Gábor Korchmáros
Dipartimento di Matematica, Informatica ed Economia
Università degli Studi della Basilicata
Maria Montanucci
Dipartimento di Tecnica e Gestione dei Sistemi Industriali
Università degli Studi di Padova