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In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves overℚ{{\mathbb{Q}}}   . We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves