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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: 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]