Vol. 2, No. 8, 2008

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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
Abstract

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g 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 > 2p(p1). If G2 denotes the second ramification group of G at the unique ramification point of the cover C CG, we display necessary conditions on G2 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 G2 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/