Diagonal Cycles and Euler Systems Ii: the Birch and Swinnerton-dyer Conjecture for Hasse-weil-artin L-functions

This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over 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 ≥ 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.