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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-functions
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by Henri Darmon and Victor Rotger
J. Amer. Math. Soc. 30 (2017), 601-672
Published electronically: June 10, 2016


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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Bibliographic Information
  • Henri Darmon
  • Affiliation: Department of Mathematics, McGill University, Montréal H3A-0B9, Canada
  • MR Author ID: 271251
  • Email:
  • Victor Rotger
  • Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
  • MR Author ID: 698263
  • Email:
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 601-672
  • MSC (2010): Primary 11G05; Secondary 11G40
  • DOI:
  • MathSciNet review: 3630084