Counting Artin-Schreier curves over finite fields
| dc.contributor.author | Ho, Anne M., author | |
| dc.contributor.author | Pries, Rachel, advisor | |
| dc.contributor.author | Achter, Jeff, committee member | |
| dc.contributor.author | Lee, Myung Hee, committee member | |
| dc.contributor.author | Penttila, Tim, committee member | |
| dc.date.accessioned | 2016-01-11T15:13:33Z | |
| dc.date.available | 2016-01-11T15:13:33Z | |
| dc.date.issued | 2015 | |
| dc.description.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. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Ho_colostate_0053A_13247.pdf | |
| dc.identifier.uri | http://hdl.handle.net/10217/170280 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.subject | arithmetic geometry | |
| dc.subject | Artin-Schreier | |
| dc.subject | finite fields | |
| dc.subject | number theory | |
| dc.title | Counting Artin-Schreier curves over finite fields | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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