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

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

Authors

David Burns

Daniel Macias Castillo



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.

Journal Article Type Article
Publication Date Jan 1, 2018
Journal Journal für die reine und angewandte Mathematik
Print ISSN 0075-4102
Electronic ISSN 1435-5345
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 734
Pages 187-228
APA6 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, 734, (187-228). doi:10.1515/ crelle-2014-0153. ISSN 0075-4102
DOI https://doi.org/10.1515/+crelle-2014-0153
Publisher URL https://www.degruyter.com/view/j/crelle.2018.2018.issue-734/crelle-2014-0153/crelle-2014-0153.xml
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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