## 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 - DOI: https://doi.org/10.1090/jams/861
- Published electronically: June 10, 2016
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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.## References

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## Bibliographic Information

**Henri Darmon**- Affiliation: Department of Mathematics, McGill University, Montréal H3A-0B9, Canada
- MR Author ID: 271251
- Email: darmon@math.mcgill.ca
**Victor Rotger**- Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
- MR Author ID: 698263
- Email: victor.rotger@upc.edu
- 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: https://doi.org/10.1090/jams/861
- MathSciNet review: 3630084