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

Bertolini, Massimo; Darmon, Henri; Rotger, Victor

doi:10.1090/S1056-3911-2015-00675-0

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 \[ L(E,\varrho ,1) \ne 0\quad \Rightarrow \quad (E(H)\otimes \varrho )^{\mathrm {Gal}(H/\mathbb {Q})} = 0,\] 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
MR Author ID: 249679
Email: massimo.bertolini@uni-due.de

Henri Darmon
Affiliation: Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1111, Montréal, Canada
MR Author ID: 271251
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
MR Author ID: 698263
Email: victor.rotger@upc.edu

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.