Galois theory of Mordell-Weil groups
Let K/k be a finite Galois extension of number fields with Galois group G, and let E be an elliptic curve defined over k. In this thesis we study the problem of trying to determine the Zp[G]-module structure of the p-adic completion E(K)* = E(K) ⊗Z Zp from more easily calculated invariants of K/k an...
| Main Author: | |
|---|---|
| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2019
|
| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/56691/ |
| _version_ | 1848799365776277504 |
|---|---|
| author | Vavasour, Thomas |
| author_facet | Vavasour, Thomas |
| author_sort | Vavasour, Thomas |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Let K/k be a finite Galois extension of number fields with Galois group G, and let E be an elliptic curve defined over k. In this thesis we study the problem of trying to determine the Zp[G]-module structure of the p-adic completion E(K)* = E(K) ⊗Z Zp from more easily calculated invariants of K/k and E/k. In the case where G has cyclic p-Sylow subgroup, a theorem of Yakovlev tells us that the cohomology of E(K)* determines a part of E(K)*, and in the Chapter 3 we study these groups by way of a control theorem describing the cokernel of the natural restriction maps on the p-primary Selmer groups. In Chapter 4 we develop the necessary representation theory of Zp[G]-lattices, that is Zp[G]-modules that are Zp-free, for certain specific groups G whose order is divisible by p precisely once. In particular we calculate their regulator constants which, by a theorem of Torzewski, gives us the necessary ingredient to fully determing E(K)*. In Chapter 5 we combine the results from the previous two chapters to prove various results allowing us to determine the Zp[G]-structure of E(K)* in specific cases. Finally, in Chapter 6 we illustrate these results with concrete examples. |
| first_indexed | 2025-11-14T20:34:31Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-56691 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:34:31Z |
| publishDate | 2019 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-566912025-02-28T14:31:02Z https://eprints.nottingham.ac.uk/56691/ Galois theory of Mordell-Weil groups Vavasour, Thomas Let K/k be a finite Galois extension of number fields with Galois group G, and let E be an elliptic curve defined over k. In this thesis we study the problem of trying to determine the Zp[G]-module structure of the p-adic completion E(K)* = E(K) ⊗Z Zp from more easily calculated invariants of K/k and E/k. In the case where G has cyclic p-Sylow subgroup, a theorem of Yakovlev tells us that the cohomology of E(K)* determines a part of E(K)*, and in the Chapter 3 we study these groups by way of a control theorem describing the cokernel of the natural restriction maps on the p-primary Selmer groups. In Chapter 4 we develop the necessary representation theory of Zp[G]-lattices, that is Zp[G]-modules that are Zp-free, for certain specific groups G whose order is divisible by p precisely once. In particular we calculate their regulator constants which, by a theorem of Torzewski, gives us the necessary ingredient to fully determing E(K)*. In Chapter 5 we combine the results from the previous two chapters to prove various results allowing us to determine the Zp[G]-structure of E(K)* in specific cases. Finally, in Chapter 6 we illustrate these results with concrete examples. 2019-07-18 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/56691/1/PhD.pdf Vavasour, Thomas (2019) Galois theory of Mordell-Weil groups. PhD thesis, University of Nottingham. Galois group Elliptic curves representation theory |
| spellingShingle | Galois group Elliptic curves representation theory Vavasour, Thomas Galois theory of Mordell-Weil groups |
| title | Galois theory of Mordell-Weil groups |
| title_full | Galois theory of Mordell-Weil groups |
| title_fullStr | Galois theory of Mordell-Weil groups |
| title_full_unstemmed | Galois theory of Mordell-Weil groups |
| title_short | Galois theory of Mordell-Weil groups |
| title_sort | galois theory of mordell-weil groups |
| topic | Galois group Elliptic curves representation theory |
| url | https://eprints.nottingham.ac.uk/56691/ |