Konstantinou, Alexandros;
(2025)
Galois covers of curves and the Birch and Swinnerton-Dyer conjecture.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The calculation of the rank of an abelian variety over a number field is a central, yet notoriously difficult, problem in number theory, intimately tied to the Birch and Swinnerton-Dyer conjecture. In this work, we study two main aspects of this conjecture: the parity conjecture and the Tate–Shafarevich group. For the former, we investigate parities of ranks by using Galois covers of curves, isogenies and certain relations between permutation representations; for the latter, we use Galois extensions and isogenies to give a positive answer to a conjecture by Stein regarding the possible orders of Tate–Shafarevich groups of abelian varieties.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Galois covers of curves and the Birch and Swinnerton-Dyer conjecture |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. 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.ucl.ac.uk/id/eprint/10204028 |
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