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  • Research Article
  • Open Access

Irreducible canonical representations in positive characteristic

Research in Number Theory20151:3

https://doi.org/10.1007/s40993-015-0004-8

  • Received: 26 September 2014
  • Accepted: 19 November 2014
  • Published:

Abstract

For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H 0 (X,Ω X) is irreducible. Any curve with an irreducible canonical representation must either be superspecial or ordinary. Having a small automorphism group is an obstruction to having irreducible canonical representation; with this motivation, the bulk of the paper is spent bounding the size of automorphism groups of superspecial and ordinary curves. After proving that all automorphisms of an \(\mathbb {F}_{q^{2}}\)‐maximal curve are defined over \(\mathbb {F}_{q^{2}}\), we find all superspecial curves with g>8 2 having an irreducible representation. In the ordinary case, we provide a bound on the size of the automorphism group of an ordinary curve that improves on a result of Nakajima.

Keywords

  • Automorphism Group
  • Rational Point
  • Maximal Curve
  • Canonical Representation
  • Hyperelliptic Curve

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