  In the present paper, we prove, for a large class of elliptic curves defined
over $\mathbb{Q}$, 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 $\mathbb{Q}$ without complex
multiplication.

Number Theory