The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication

Abstract

Let $E/F$ be an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in an imaginary quadratic field $K$. We give a complete proof of the conjecture of Birch and Swinnerton-Dyer for $E/F$, as well as its equivariant refinement formulated by Gross $\href{https://doi.org/10.1007/978-1-4899-6699-5_14}{[39]}$, under the assumption that $L(E/F, 1) \neq 0$ and that $F(E_{tors})/K$ is abelian. We also prove analogous results for CM abelian varieties $A/K$.