Geodesic
The Analytic Rank of a Family of Elliptic Curves
Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 847-862

Voir la notice de l'article provenant de la source Cambridge University Press

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 CX 2/3-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.
DOI : 10.4153/CJM-1993-048-9
Mots-clés : 11G05, 14H52 
Mai, Liem. The Analytic Rank of a Family of Elliptic Curves. Canadian journal of mathematics, Tome 45 (1993) no. 4, pp. 847-862. doi: 10.4153/CJM-1993-048-9
@article{10_4153_CJM_1993_048_9,
     author = {Mai, Liem},
     title = {The {Analytic} {Rank} of a {Family} of {Elliptic} {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {847--862},
     year = {1993},
     volume = {45},
     number = {4},
     doi = {10.4153/CJM-1993-048-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1993-048-9/}
}
TY  - JOUR
AU  - Mai, Liem
TI  - The Analytic Rank of a Family of Elliptic Curves
JO  - Canadian journal of mathematics
PY  - 1993
SP  - 847
EP  - 862
VL  - 45
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1993-048-9/
DO  - 10.4153/CJM-1993-048-9
ID  - 10_4153_CJM_1993_048_9
ER  - 
%0 Journal Article
%A Mai, Liem
%T The Analytic Rank of a Family of Elliptic Curves
%J Canadian journal of mathematics
%D 1993
%P 847-862
%V 45
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1993-048-9/
%R 10.4153/CJM-1993-048-9
%F 10_4153_CJM_1993_048_9

[1] 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] 2. Cassels, J. S., The rational solutions of the Diophantine equation Y2 = X3 — D, Acta Math. 82(1950), 243–273. Google Scholar

[3] 3. Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39( 1977), 223–251. Google Scholar

[4] 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] 5. Ireland, K. and Rosen, I. M., A classical introduction to modern number theory, Springer- Verlag, New York, 1982. Google Scholar

[6] 6. Mai, L., The average analytic rank of a family of elliptic curves, J. Number Theory, to appear. Google Scholar

[7] 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] 8. Satgé, R, Groupes de Selmer and corps cubiques, J. Number Theory 23(1986), 294–317. Google Scholar

[9] 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] 10. Satgé, R, Un analogue du calcul de Heegner, Invent. Math. 87(1987), 425–439. Google Scholar

[11] 11. Selmer, E. S., The Diophantine equation AX3 + BY3 + CZ3 =0, Acta Math. 85(1951), 203–362. Google Scholar

[12] 12. Silverman, J., The arithmetics of elliptic curves, Springer-Verlag, New York, 1986. Google Scholar

[13] 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] 14. Weil, A., Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Annalen 168 (1967), 149–156. Google Scholar

[15] 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

Cité par Sources :