Quadratic twists of elliptic curves

The paper generalizes, for a wide class of elliptic curves defined over , the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curve , including showing that there is a large class of explicit quadratic twists whose complex -series does not vanish at, and for which the full Birch-Swinnerton-Dyer conjecture is valid.