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Let E be an elliptic curve over Q and let % be an odd, irreducible twodimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-WeilArtin L-series L(E, %, s), namely, the implication L(E, %, 1) 6= 0 ⇒ (E(H)⊗ %)Gal(H/Q) = 0, where H is the finite extension of Q cut out by %. The proof relies on padic families of global Galois cohomology classes arising from BeilinsonFlach elements in a tower of products of modular curves.

Beilinson-Flach elements and Euler systems II: The Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin 𝐿-series