Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

Beilinson-Flach elements and Euler systems II: The Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin $ L$-series


Authors: Massimo Bertolini, Henri Darmon and Victor Rotger
Journal: J. Algebraic Geom. 24 (2015), 569-604
DOI: https://doi.org/10.1090/S1056-3911-2015-00675-0
Published electronically: March 23, 2015
MathSciNet review: 3344765
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Abstract | References | Additional Information

Abstract: Let $ E$ be an elliptic curve over $ \mathbb{Q}$ and let $ \varrho $ be an odd, irreducible two-dimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-Weil-Artin $ L$-series $ L(E,\varrho ,s)$, namely, the implication

$\displaystyle L(E,\varrho ,1) \ne 0\quad \Rightarrow \quad (E(H)\otimes \varrho )^{\mathrm {Gal}(H/\mathbb{Q})} = 0,$

where $ H$ is the finite extension of $ \mathbb{Q}$ cut out by $ \varrho $. The proof relies on $ p$-adic families of global Galois cohomology classes arising from Beilinson-Flach elements in a tower of products of modular curves.


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Additional Information

Massimo Bertolini
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Ellernstr. 29, 45326 Essen, Germany
Email: massimo.bertolini@uni-due.de

Henri Darmon
Affiliation: Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1111, Montréal, Canada
Email: darmon@math.mcgill.ca

Victor Rotger
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Cata- lunya, Despatx 413, C. Jordi Girona 1-3, 08034 Barcelona, Spain
Email: victor.rotger@upc.edu

DOI: https://doi.org/10.1090/S1056-3911-2015-00675-0
Received by editor(s): September 7, 2014
Received by editor(s) in revised form: December 10, 2014
Published electronically: March 23, 2015
Article copyright: © Copyright 2015 University Press, Inc.

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
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