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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2009, Issue 633


Rank-one quadratic twists of an infinite family of elliptic curves

Dongho Byeon / Daeyeol Jeon / Chang Heon Kim
Published Online: 2009-06-16 | DOI: https://doi.org/10.1515/CRELLE.2009.060


A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E/ℚ has (analytic) rank 1. This assertion has been confirmed by Vatsal [Math. Ann. 311: 791–794, 1998] and the first author [Acta Arith. 114: 391–396, 2004] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E/ℚ using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

About the article

Received: 2007-07-17

Revised: 2008-03-21

Published Online: 2009-06-16

Published in Print: 2009-08-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2009, Issue 633, Pages 67–76, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2009.060.

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