University of Illinois Urbana-Champaign

Covers and invariants of Deligne-Lusztig curves

Skabelund, Dane C.

Content Files
SKABELUND-DISSERTATION-2018.pdf

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

Description

Title
Covers and invariants of Deligne-Lusztig curves
Author(s)
Skabelund, Dane C.
Issue Date
2018-04-20
Director of Research (if dissertation) or Advisor (if thesis)
Duursma, Iwan
Doctoral Committee Chair(s)
Ahlgren, Scott
Committee Member(s)
  • Nevins, Thomas
  • Allen, Patrick
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2018-09-04T20:27:22Z
Keyword(s)
  • Deligne-Lusztig curves
  • Ree curve
  • maximal curves
  • supersingular curves
  • Stohr-Voloch theory
  • Weierstrass points
  • p-torsion
  • de Rham cohomology
Abstract
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.
Graduation Semester
2018-05
Type of Resource
text
Permalink
http://hdl.handle.net/2142/101037
Copyright and License Information
Copyright 2018 Dane Skabelund

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