Irreducible Canonical Representations in Positive Characteristic

2014/08/17 by Benjamin Gunby, Alexander Smith, Gunby, Benjamin +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Finite Group Theory Research #math.AG #msc:14H37

paper · pdf · doi:10.48550/arxiv.1408.3830

29 pages

arxiv created 2014/08/17 · arxiv updated 2014/08/19

Abstract

For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H⁰(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 \mathbbF_q²-maximal curve are defined over \mathbbF_q², we find all superspecial curves with g > 82 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.

Cited by

• Non-isomorphic subfields of the BM and GGS maximal function fields

Related

• Large automorphism groups of ordinary curves in characteristic 2
• Algebraic curves with a large cyclic automorphism group
• Representations of field automorphism groups
• Constraints on the automorphism group of a curve
• Large automorphism groups of ordinary curves of even genus in odd\n characteristic