Repository logo
 

Counting Artin-Schreier curves over finite fields

Date

2015

Authors

Ho, Anne M., author
Pries, Rachel, advisor
Achter, Jeff, committee member
Lee, Myung Hee, committee member
Penttila, Tim, committee member

Journal Title

Journal ISSN

Volume Title

Abstract

Several authors have considered the weighted sum of various types of curves of a certain genus g over a finite field k := Fq of characteristic p where p is a prime and q = pm for some positive integer m. These include elliptic curves (Howe), hyperelliptic curves (Brock and Granville), supersingular curves when p = 2 and g = 2 (Van der Geer and Van der Vlught), and hyperelliptic curves of low genus when p = 2 (Cardona, Nart, Pujolàs, Sadornil). We denote this weighted sum ∑[C] 1/|Autk(C)|' where the sum is over k-isomorphism classes of the curves and Autk(C) is the automorphism group of C over k. Many of these curves mentioned above are Artin-Schreier curves, so we focus on these in this dissertation. We consider Artin-Schreier curves C of genus g = d(p - 1)/2 for 1 ≤ d ≤ 5 over finite fields k of any characteristic p. We also determine a weighted sum for an arbitrary genus g in one-, two-, three-, and four-branch point cases. In our cases, we must consider a related weighted sum ∑/[C] 1/|CentAutk(C)‹t›|' where CentAutk(C) ‹t› is the centralizer of ‹t› in Autk(C). We discuss our methods of counting, our results, applications, as well as geometric connections to the moduli space of Artin-Schreier covers.

Description

Rights Access

Subject

arithmetic geometry
Artin-Schreier
finite fields
number theory

Citation

Associated Publications