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In the present paper, we prove, for a large class of elliptic curves definedover $\mathbb{Q}$, the existence of an explicit infinite family of quadratictwists with analytic rank $0$. In addition, we establish the $2$-part of theconjecture of Birch and Swinnerton-Dyer for many of these infinite families ofquadratic twists. Recently, Xin Wan has used our results to prove for the firsttime the full Birch--Swinnerton-Dyer conjecture for some explicit infinitefamilies of elliptic curves defined over $\mathbb{Q}$ without complexmultiplication.

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