A conjecture of Gross and Zagier: Case e (Q) tor ≅ Z&/3Z

Abstract

© 2019 World Scientific Publishing Company.Let E be an elliptic curve defined over Q of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, PK the Heegner point in E(K), and III(E/K) the Shafarevich-Tate group of E over K. Let 2uK be the number of roots of unity contained in K. Gross and Zagier conjectured that if PK has infinite order in E(K), then the integer c m uK |III(E/K)|1 2 is divisible by |E(Q)tor|. In this paper, we show that this conjecture is true if E(Q)torZ/3Z;