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Enumerating Curves of Genus 2 Over Finite Fields
Steinberg, Rose K
Steinberg, Rose K
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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.
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Date
2018-01-01