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

  • Jie Shu ORCID logo EMAIL logo and Shuai Zhai ORCID logo


In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over . 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.

Award Identifier / Grant number: 11701092

Funding statement: Jie Shu is supported by the National Natural Science Foundation of China (Grant No. 11701092).


We would like to thank John Coates for encouragement, useful discussions and polishings on the manuscript, thank Ye Tian and Xin Wan for helpful advice and comments, and thank Yongxiong Li for helpful comments and carefully reading the manuscript. We also thank the referee for helpful advice.


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Received: 2020-08-08
Revised: 2021-01-13
Published Online: 2021-02-27
Published in Print: 2021-06-01

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