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Published January 2022 | Submitted
Journal Article Open

The even parity Goldfeld conjecture: Congruent number elliptic curves


In 1979 Goldfeld conjectured: 50% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first instance - the congruent number elliptic curves.

Additional Information

© 2021 Elsevier Inc. Received 13 April 2021, Accepted 31 May 2021, Available online 29 June 2021. It is a pleasure to thank John Coates, Wei He, Shinichi Kobayashi, Jinzhao Pan, Dinakar Ramakrishnan, Alex Smith, Richard Taylor and Wei Zhang for helpful discussions and instructive comments. The authors cordially thank Chris Skinner and Shou-Wu Zhang for inspiring conversations. The article owes its existence to a generous suggestion of Dorian Goldfeld. The authors would like to express their sincere gratitude to Dorian Goldfeld also for his enticing conjecture. A.B. is partially supported by the NSF grant DMS #2001409, and Y.T. by the NSFC grants #11688101 and #11531008.

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August 22, 2023
October 23, 2023