A conjecture of Gross and Zagier: Case e (Q)tor ≃ Z/2Z ⊕ Z/2Z, Z/2Z ⊕z/4Z or Z/2Z ⊕ Z/6Z

Abstract

© 2020 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)tor ≃ Z/2Z ⊕ Z/2Z, Z/2Z ⊕Z/4Z or Z/2Z ⊕ Z/6Z