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

Darmon, Henri; Rotger, Victor

doi:10.1090/jams/861

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.

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