On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Burns, David; Macias Castillo, Daniel; Wuthrich, Christian

Abstract

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.

Citation

Burns, D., Macias Castillo, D., & Wuthrich, C. (2018). On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions. Journal für die reine und angewandte Mathematik, 2018(734), 187-228. https://doi.org/10.1515/crelle-2014-0153

• Journal Article Type: Article
• Acceptance Date: Feb 5, 2015
• Online Publication Date: May 13, 2015
• Publication Date: 2018-01
• Deposit Date: Feb 23, 2017
• Publicly Available Date: Feb 23, 2017
• Journal: Journal für die reine und angewandte Mathematik
• Print ISSN: 0075-4102
• Electronic ISSN: 1435-5345
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 2018
• Issue: 734
• Pages: 187-228
• DOI: https://doi.org/10.1515/crelle-2014-0153
• Public URL: https://nottingham-repository.worktribe.com/output/752101
• Publisher URL: https://www.degruyter.com/view/journals/crll/2018/734/article-p187.xml