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arXiv · 1712.01271

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

Li Cai·Chao Li·Shuai Zhai
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Abstract

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.

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Li Cai, Chao Li, Shuai Zhai. 2017-12-03. On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves. https://arxiv.org/abs/1712.01271

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