Date of Completion

2018

Document Type

Honors College Thesis

Department

Mathematics and Statistics

Type of Thesis

Honors College, College of Arts and Science Honors

First Advisor

Professor Christelle Vincent

Keywords

hyperelliptic curves, genus 2, isomorphism class, isogeny class

Abstract

In this thesis, we investigate curves over finite fields. More precisely, fixing a base field F_q and a genus g, we aim to enumerate a representative from each isogeny and isomorphism class of curves defined over that field and of that genus. As a step towards this goal, in this work we provide code that, given a finite field of any characteristic, generates a list of models of hyperelliptic curves of genus 2 which we can guarantee contains one representative from each isomorphism class of curves defined over that field. Furthermore, our code allows us to sort these models into isogeny classes. Finally, if the field is of odd characteristic, we can further sort the models into isomorphism classes.

As an application of our software, we obtain representatives for every isogeny class of hyperelliptic curves of genus 2 defined over the finite field F_2. We also give a model for each isogeny and isomorphism class of hyperelliptic curves of genus 2 defined over F_3. In these investigations, we discovered that Theorem 5 of Isomorphism Classes of Genus-2 Hyperelliptic Curves Over Finite Fields by Encinas, Menezes, and Masqué may be more accurately stated as giving the number of isomorphism classes of pointed hyperelliptic curves rather than isomorphism classes of hyperelliptic curves.

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

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