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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over genus fields of real quadratic fields


Author: Chung Pang Mok
Journal: Trans. Amer. Math. Soc. 374 (2021), 1391-1419
MSC (2020): Primary 11G05, 11G40
DOI: https://doi.org/10.1090/tran/8254
Published electronically: November 25, 2020
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Abstract: In this paper, we remove certain hypotheses in the theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over narrow genus class fields of real quadratic fields. Along the way, we establish that certain normalized special values of $ L$-functions are squares of rational numbers, a result that is of independent interest, and can be regarded as instances of the rank zero case of the Birch and Swinnerton-Dyer conjecture modulo squares.


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Additional Information

Chung Pang Mok
Affiliation: School of Mathematical Sciences, Soochow University, 1 Shi-Zi Street, Suzhou 215006, Jiangsu Province, China
Email: zpmo@suda.edu.cn

DOI: https://doi.org/10.1090/tran/8254
Keywords: Elliptic curves, special values of $L$-functions.
Received by editor(s): September 12, 2019
Received by editor(s) in revised form: July 3, 2020
Published electronically: November 25, 2020
Article copyright: © Copyright 2020 American Mathematical Society