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Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-functions

Authors: Henri Darmon and Victor Rotger
Journal: J. Amer. Math. Soc. 30 (2017), 601-672
MSC (2010): Primary 11G05; Secondary 11G40
Published electronically: June 10, 2016
MathSciNet review: 3630084
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Abstract: This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank $0$, for elliptic curves over $\mathbb {Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $4$. When the associated $L$-function vanishes (to even order $\ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $p$-adic $L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.

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Additional Information

Henri Darmon
Affiliation: Department of Mathematics, McGill University, Montréal H3A-0B9, Canada
MR Author ID: 271251

Victor Rotger
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
MR Author ID: 698263

Keywords: Elliptic curves, Artin representations, equivariant Birch and Swinnerton-Dyer conjecture, Gross-Kudla-Schoen diagonal cycles, $p$-adic families of modular forms, Euler Systems
Received by editor(s): September 21, 2014
Received by editor(s) in revised form: October 16, 2015, and April 27, 2016
Published electronically: June 10, 2016
Additional Notes: The first author was supported by an NSERC Discovery grant.
The second author was supported by Grants MTM2012-34611 and MTM2015-63829-P
Article copyright: © Copyright 2016 American Mathematical Society