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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

References

1. Birch, B. J. and Stephens, N. M., The parity of the rank of the Mordell-Weil group, Topology 5(1966), 295–299.Google Scholar
2. Cassels, J. S., The rational solutions of the Diophantine equation Y2 = X3 — D, Acta Math. 82(1950), 243–273.Google Scholar
3. Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39( 1977), 223–251.Google Scholar
4. Gouvea, F. and Mazur, B., The squarefree sieve and the rank of elliptic curves, J. Amer. Math. Soc. (1) 4(1991), 1–23.Google Scholar
5. Ireland, K. and Rosen, I. M., A classical introduction to modern number theory, Springer- Verlag, New York, 1982.Google Scholar
6. Mai, L., The average analytic rank of a family of elliptic curves, J. Number Theory, to appear.Google Scholar
7. Rubin, K., The work ofKolyvagin on the arithmetics of elliptic curves, Lecture Notes in Mathematics 1399, New York, Springer-Verlag, 1989.Google Scholar
8. Satgé, R, Groupes de Selmer and corps cubiques, J. Number Theory 23(1986), 294–317.Google Scholar
9. Satgé, R, Quelques résultats sur les entiers qui sont sommes des cubes de deux rationels, Soc. Math. France, Astérisque 147-148(1987), 335–341.Google Scholar
10. Satgé, R, Un analogue du calcul de Heegner, Invent. Math. 87(1987), 425–439.Google Scholar
11. Selmer, E. S., The Diophantine equation AX3 + BY3 + CZ3 =0, Acta Math. 85(1951), 203–362.Google Scholar
12. Silverman, J., The arithmetics of elliptic curves, Springer-Verlag, New York, 1986.Google Scholar
13. Taniyama, Y, L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Japan 9(1957), 330–336.Google Scholar
14. Weil, A., Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Annalen 168 (1967), 149–156.Google Scholar
15. Zagier, D. and Kramarz, G., Numerical investigations related to the L-series of certain elliptic curves, J. Ind. Math. Soc. 52(1987), 51–69.Google Scholar