Grossi, Giada;
(2020)
Euler systems and their applications.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The main theme of this thesis is the theory of Euler and Kolyvagin systems. Such systems are norm compatible classes in the Galois cohomology of p-adic representations. We focus on two aspects of this theory: how to prove these norm compatibilities in the case of the Asai representation attached to a quadratic Hilbert modular form on one hand and how to use norm compatible classes to bound Selmer groups in the case of elliptic curves with a rational p-isogeny on the other. More precisely, in the first part of this thesis we study certain classes in motivic cohomology of Hilbert modular surfaces, first constructed by Lei-Loeffler-Zerbes. We prove norm relations for the Euler system built from these classes for the Asai representation attached to a Hilbert modular form over a quadratic real field F. Under a strong condition on the underlying real quadratic field, we give a proof of the norm relations for primes that split in F, using the technique introduced by the authors. We then redefine the classes in the language used by Loeffler-Skinner-Zerbes in the GSp(4) case and prove norm relations using local representation theory. With this technique we are able to remove the above-mentioned assumption and prove tame norm relations for all inert and split primes. In the second part, we present part of a joint work with F. Castella, J. Lee and C. Skinner in which we use the Heegner point Kolyvagin system to prove a bound on the Selmer group attached to a rational elliptic curve with a rational p-isogeny, extending a result by Howard. This result is crucial in the proof of the anticyclotomic Iwasawa main conjecture, which is used in the above-mentioned work to prove new cases of the p-part of the Birch and Swinnerton-Dyer conjecture.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Euler systems and their applications |
| Event: | UCL (University College London) |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/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 |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10117229 |
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