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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 , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | 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 / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

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 , P K the Heegner point in E (K) , and III (E / K) the Shafarevich–Tate group of E over K. Let 2 u K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P K has infinite order in E (K) , then the integer c ⋅ m ⋅ u K ⋅ | 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 / 3 ℤ. [ABSTRACT FROM AUTHOR]

Abstract: When an elliptic curve of square-free conductor N has a rational point of odd prime order , Dummigan (2005) in explicitly constructed a rational point of order l on the optimal curve E, isogenous over to , under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves such that a positive proportion of quadratic twists of has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) . [ABSTRACT FROM AUTHOR]