Title: Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton–Dyer conjecture
Abstract: Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L -function of E over K , when combined with the Birch and Swinnerton–Dyer conjecture, gives a conjectural formula for the order of the Shafarevich–Tate group of E over K . In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K , the 3-part of the conjectural formula is true.