Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

Jie Shu; Shuai Zhai

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arXiv · 2102.11808

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

Jie Shu·Shuai Zhai

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Abstract

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove the existence of explicit infinite families of quadratic twists with analytic ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank $1$. In addition, we show that these families of quadratic twists satisfy the $2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

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Jie Shu, Shuai Zhai. 2021-02-23. Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves. https://arxiv.org/abs/2102.11808

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