Bell, James Joseph;
(2025)
Isogenies and Selmer Groups of Abelian Varieties.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The rank of an abelian variety is its most important invariant, determining the structure of its rational points, however there is no known algorithm to compute it. A procedure to find it, which works for some abelian varieties, is by descent, which involves computing a Selmer group. The Selmer groups of the variety give an upper bound, and the difference from the correct rank can be explained by the Tate–Shafarevich group, which measures the failure of a local-to-global principle. This thesis deduces results about Selmer and Tate–Shafarevich groups from the existence of certain isogenies. We give results about the size of the Tate–Shafarevich group in the cases of abelian varieties with complex multiplication, and elliptic curves over dihedral extensions. We also show that the Selmer groups and certain other invariants do not determine the isomorphism class of an abelian variety.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Isogenies and Selmer Groups of Abelian Varieties |
| 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/10215608 |
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