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Abstract
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.
Received: 2007-07-17
Revised: 2008-03-21
Published Online: 2009-06-16
Published in Print: 2009-August
© Walter de Gruyter Berlin · New York 2009
