On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil LfunctionsTools Burns, David, Macias Castillo, Daniel and Wuthrich, Christian (2015) On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil Lfunctions. Journal für die reine und angewandte Mathematik, 734 . pp. 187228. ISSN 14355345 Full text not available from this repository.
Official URL: https://www.degruyter.com/view/j/crelle.2018.2018.issue734/crelle20140153/crelle20140153.xml
AbstractLet 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 nottoostringent conditions on A and F we compute explicitly the algebraic part of the pcomponent 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 pcomponent 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 nonabelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural padic congruences between the values at s = 1 of derivatives of the Hasse–Weil Lfunctions 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.
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