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 0 , for elliptic curves over Q Q viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4 4 . When the associated L L -function vanishes (to even order ≥ 2 ≥ 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 p -adic L L -function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain p p -adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.