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Transactions of the American Mathematical Society

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Heegner points on modular curves


Authors: Li Cai, Yihua Chen and Yu Liu
Journal: Trans. Amer. Math. Soc. 370 (2018), 3721-3743
MSC (2010): Primary 11G05, 11G07
DOI: https://doi.org/10.1090/tran/7053
Published electronically: December 14, 2017
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Abstract: In this paper, we study the Heegner points on more general modular curves other than $ X_0(N)$, which generalizes Gross' work ``Heegner points on $ X_0(N)$''. The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such a kind of Heegner points, we construct certain families of quadratic twists of $ X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $ 2$-part of their BSD conjectures hold.


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Additional Information

Li Cai
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email: lcai@math.tsinghua.edu.cn

Yihua Chen
Affiliation: Academy of Mathematics and Systems Science, Morningside center of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: yihuachenamss@163.com

Yu Liu
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email: yliu@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/tran/7053
Received by editor(s): May 31, 2016
Received by editor(s) in revised form: August 18, 2016, and August 21, 2016
Published electronically: December 14, 2017
Additional Notes: The first author was supported by the Special Financial Grant from the China Postdoctoral Science Foundation 2014T70067.
Article copyright: © Copyright 2017 American Mathematical Society

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