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The Analytic Rank of a Family of Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

Liem Mai
Liem Mai*
Affiliation:
Centre de Recherches Mathématiques, Université de Montréal, CP 6128-A, Montreal, Quebec H3C 3J7
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Abstract

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We study the family of elliptic curves Em X3 + Y3 = m where m is a cubefree integer.

The elliptic curves Em with even analytic rank and those with odd analytic rank are proved to be equally distributed. It is proved that the number of cubefree integers m ≤ X such that the analytic rank of Em is even and ≥ 2 is at least CX2/3-ε, where ε is arbitrarily small and C is a positive constant, for X large enough. Therefore, if we assume the Birch and Swinnerton-Dyer conjecture, the number of all cubefree integers m ≤ X such that the equation X3 + Y3 = m have at least two independent rational solutions is at least CX2/3-ε.

Keywords

11G0514H52
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 847 - 862
Copyright
Copyright © Canadian Mathematical Society 1993

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