Docking, Jordan;
(2023)
Arithmetic of Genus Three Curves and Their Jacobians.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The Birch–Swinnerton-Dyer Conjecture predicts that, given an abelian variety A over a number field K, its rank, rk(A/K), is equal to the order of vanishing of its L-function L(A/K, s) at s = 1. A consequence of this is the Parity Conjecture; rk(A/K) and the order of vanishing at s=1 of L(A/K, s) are expected to have the same parity. The parity of the latter is given by the root number w(A/K), and so the Parity Conjecture states that (−1)^rk(A/K) = w(A/K). This thesis investigates what can be said about the Parity Conjecture when A is the Jacobian of a curve of genus 3. Part of this requires developing the local theory of non-hyperelliptic genus 3 curves. We introduce a combinatorial object called an octad diagram, which we conjecture to recover the essential data of stable models.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Arithmetic of Genus Three Curves and Their Jacobians |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2023. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10167930 |
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