In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$.

中文翻译：

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关于 Birch 和 Swinnerton-Dyer 猜想的 2 部分的弱形式

在本文中，我们研究了 Birch 和 Swinnerton-Dyer 猜想的 2-部分的弱形式，并证明了复数中心值的代数部分的 2-adic 估值的下界大号-series 用于所有最优椭圆曲线的二次扭曲族${\mathbb Q}$.