On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves

In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton-Dyer for many of these infinite families of quadratic twists. Recently, Xin Wan has used our results to prove for the first time the full Birch-Swinnerton-Dyer conjecture for some explicit infinite families of elliptic curves defined over Q without complex multiplication.