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

TL;DR

This paper proves the Birch and Swinnerton-Dyer conjecture for certain elliptic curves with complex multiplication over number fields, under specific conditions, and extends results to CM abelian varieties.

Contribution

It provides a complete proof of the BSD conjecture for CM elliptic curves under particular assumptions and generalizes to CM abelian varieties.

Findings

Proof of BSD conjecture for CM elliptic curves with non-zero L-value

Verification of the equivariant refinement by Gross

Extension of results to CM abelian varieties

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, 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 $A/K$.