On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions
Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/...
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| Format: | Article |
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De Gruyter
2015
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| Online Access: | https://eprints.nottingham.ac.uk/40799/ |
| _version_ | 1848796136308998144 |
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| author | Burns, David Macias Castillo, Daniel Wuthrich, Christian |
| author_facet | Burns, David Macias Castillo, Daniel Wuthrich, Christian |
| author_sort | Burns, David |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/F)(1),Z[Gal(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions. |
| first_indexed | 2025-11-14T19:43:11Z |
| format | Article |
| id | nottingham-40799 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:43:11Z |
| publishDate | 2015 |
| publisher | De Gruyter |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-407992020-05-04T17:08:32Z https://eprints.nottingham.ac.uk/40799/ On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions Burns, David Macias Castillo, Daniel Wuthrich, Christian Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair (h1(A/F)(1),Z[Gal(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse–Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions. De Gruyter 2015-05-13 Article PeerReviewed Burns, David, Macias Castillo, Daniel and Wuthrich, Christian (2015) On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions. Journal für die reine und angewandte Mathematik, 734 . pp. 187-228. ISSN 1435-5345 https://www.degruyter.com/view/j/crelle.2018.2018.issue-734/crelle-2014-0153/crelle-2014-0153.xml |
| spellingShingle | Burns, David Macias Castillo, Daniel Wuthrich, Christian On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title | On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title_full | On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title_fullStr | On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title_full_unstemmed | On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title_short | On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions |
| title_sort | on mordell–weil groups and congruences between derivatives of twisted hasse–weil l-functions |
| url | https://eprints.nottingham.ac.uk/40799/ https://eprints.nottingham.ac.uk/40799/ |