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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.