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Let d > 0 be a fundamental discriminant of a real quadratic field. Let h (d) be the class number and ε d the fundamental unit of the real quadratic field Q (d). In this paper, we prove that if there is an elliptic curve E over Q whose Hasse-Weil L -function L E / Q (s) has a zero of order g at s = 1 , then there is an effectively computable constant κ > 0 satisfying h (d) log ⁡ ε d > 1 κ (log ⁡ d) g − 3 ∏ p | d , p ≠ d (1 − ⌊ 2 p ⌋ p + 1). [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 / 2 ℤ ⊕ ℤ / 2 ℤ , ℤ / 2 ℤ ⊕ ℤ / 4 ℤ or ℤ / 2 ℤ ⊕ ℤ / 6 ℤ. [ABSTRACT FROM AUTHOR]

Let d be a fundamental discriminant, χ d be the Dirichlet character associated to the quadratic field Q (d) and L (s , χ d) be the Dirichlet L-function. In [Go] , Goldfeld obtained an effective lower bound for L (1 , χ d) with uncalculated constants. For d < 0 , the constants are computed in [Oe] and for d > 0 , the constants are computed in [BK] by using elliptic curves with complex multiplication. In this paper, we show that the result of [BK] is worked out for elliptic curves without complex multiplication too and compute the corresponding constants. [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]

In this paper, we show that for any given integer k ≥ 2 , there are infinitely many cube-free integers n having exactly k prime divisors such that n is a sum of two rational cubes. This is a cubic analogue of the work of Tian [Ti] , which proves that there are infinitely many congruent numbers having exactly k prime divisors for any given integer k ≥ 1 . [ABSTRACT FROM AUTHOR]

Abstract: Let be an elliptic curve with j-invariant 0 defined over a number field K. In this paper, we give a simple condition on K which determines whether all cubic twists of have the same root number or not. This is a cubic twist analogue to the work [D-D] of Dokchitser and Dokchitser on quadratic twists of elliptic curves. [Copyright &y& Elsevier]

Abstract: Using Heegner points on elliptic curves, we construct points of infinite order on certain elliptic curves with a -rational torsion point of odd order. As an application of this construction, we show that for any elliptic curve E defined over which is isogenous to an elliptic curve defined over of square-free conductor N with a -rational 3-torsion point, a positive proportion of quadratic twists of E have (analytic) rank r, where . This assertion is predicted to be true unconditionally for any elliptic curve E defined over due to Goldfeld (1979) , but previously has been confirmed unconditionally for only one elliptic curve due to Vatsal (1998) . [Copyright &y& Elsevier]

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]

Abstract: Let E be an elliptic curve over 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. [Copyright &y& Elsevier]