Algebra & Number Theory

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

Michel Matignon and Magali Rocher

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Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g2. 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 CCG, 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.

Article information

Algebra Number Theory, Volume 2, Number 8 (2008), 887-926.

Received: 1 February 2008
Revised: 14 August 2008
Accepted: 17 September 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H37: Automorphisms
Secondary: 11R37: Class field theory 11G20: Curves over finite and local fields [See also 14H25] 14H10: Families, moduli (algebraic)

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


Matignon, Michel; Rocher, Magali. Smooth curves having a large automorphism $p$-group in characteristic $p\gt 0$. Algebra Number Theory 2 (2008), no. 8, 887--926. doi:10.2140/ant.2008.2.887.

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