Covers and invariants of Deligne-Lusztig curves

This thesis is comprised of three parts, each dealing with one or more of the Hermitian, Suzuki, and Ree curves, which are three families of algebraic curves over finite fields which have pronounced arithmetic and geometric properties. In the first part, we use a ray class field construction to produce covers of each of these three families of curves which meet the Hasse-Weil bound over suitable base fields. In the Hermitian case, the family of covers constructed coincide with the family of Giulietti-Korchmáros curves. In the second part, we study a certain linear series D on the Ree curve which gives an embedding in P^13. We compute the orders of vanishing of sections of D, and use this to determine the set of Weierstrass points of D. The third part is a computational project studying the structure of the 3-torsion group scheme of the Jacobian of the smallest Ree curve, which has genus 3627. This is accomplished by computing the action of the Frobenius and Verschiebung operators on the de Rham cohomology of the curve. As a result, we determine the Ekedahl-Oort type and a decomposition for the Dieudonné module of this curve.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Skabelund, Dane C.

Contributors dc:contributor

• Duursma, Iwan
• Ahlgren, Scott
• Nevins, Thomas
• Allen, Patrick

• Deligne-Lusztig curves
• Ree curve
• maximal curves
• supersingular curves
• Stohr-Voloch theory
• Weierstrass points
• p-torsion
• de Rham cohomology

Statement dc:rights

• Copyright 2018 Dane Skabelund

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101037

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101037