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

• Published in: Journal of the American Mathematical Society 2017-07, Vol.30 (3), p.601-672
• Main Authors: Darmon, Henri, Rotger, Victor
• Format: Article
• Language: English
• Subjects: 35 Partial differential equations; 35H Close-to-elliptic equations; Anàlisi matemàtica; Artin representations; Classificació AMS; Differential equations, Elliptic; Elliptic curves; Equacions diferencials el·líptiques; Equacions funcionals; equivariant Birch and Swinnerton-Dyer conjecture; Euler Systems; Gross-Kudla-Schoen diagonal cycles; Matemàtiques i estadística; p-adic families of modular forms; Àrees temàtiques de la UPC
• ISSN: 0894-0347; 1088-6834
• DOI: 10.1090/jams/861

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.