Two problems in the theory of curves over fields of positive characteristic

This thesis consists of two parts. In the first half, we define, so called, generalized Artin-Schreier cover of a scheme X over k. After defining Artin-Schreier group scheme Γ over X, a generalized Artin-Schreier cover is realized as a principal homogeneous space of Γ. We are especially interested in the case when X is P1\{0,1,∞}, a thrice punctured plane. An argument of (generalized) Artin-Schreier field extension and its function field arithmetic follows. The second half is about the coding theory. For a full flag of codes, if it is equivalent to its duals, then it is said to have the isometry-dual property. Introducing characterizations of isometry-dual property for one-point AG codes and its preservation after puncturing at some points, some generalizations in different directions will be given.

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

2019

Author dc:creator

• Hong, Euijin

Contributors dc:contributor

• Haboush, William
• Duursma, Iwan
• Katz, Sheldon
• Dodd, Christoper

• Algebraic Geometry
• Algebraic Geometry Code

Statement dc:rights

• ⃝c 2019 by Euijin Hong. All rights reserved.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/104786

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/104786