Smooth curves having a large automorphism p-group in characteristic p > 0

Matignon, Michel; Rocher, Magali

doi:10.2140/ant.2008.2.887

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Smooth curves having a large automorphism $p$-group in characteristic $p\gt 0$

Michel Matignon and Magali Rocher

Vol. 2 (2008), No. 8, 887–926

DOI: 10.2140/ant.2008.2.887

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \geq 2$ . This paper continues our study of big actions, that is, pairs $(C, G)$ where $G$ is a $p$ -subgroup of the $k$ -automorphism group of $C$ such that $| G | ∕ g > 2 p ∕ (p - 1)$ . If $G_{2}$ denotes the second ramification group of $G$ at the unique ramification point of the cover $C \to C ∕ G$ , we display necessary conditions on $G_{2}$ for $(C, G)$ to be a big action, which allows us to pursue the classification of big actions.

Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J.-P. Serre and continued by Lauter and by Auer. In particular, we obtain explicit examples of big actions with $G_{2}$ abelian of large exponent.

Keywords

automorphisms, curves, $p$-groups, ray class fields, Artin–Schreier–Witt theory

Mathematical Subject Classification 2000

Primary: 14H37

Secondary: 11R37, 11G20, 14H10

Milestones

Received: 1 February 2008

Revised: 14 August 2008

Accepted: 17 September 2008

Published: 23 November 2008

Authors

Michel Matignon
Institut de Mathématiques de Bordeaux Université de Bordeaux 1 351 cours de la Libération 33405 Talence Cedex France
http://www.math.u-bordeaux1.fr/~matignon/
Magali Rocher
Institut de Mathématiques de Bordeaux Université de Bordeaux 1 351 cours de la Libération 33405 Talence Cedex France
http://www.math.u-bordeaux.fr/~mrocher/

Vol. 2, No. 8, 2008