In Zagier and Kramarz, the authors computed the critical value of the L-series of the family of elliptic curves {\small $x^3+y^3=m$} and they pointed out some numerical phenomena concerning the frequency of curves with a positive rank and the frequency of occurrences of the Tate-Shafarevich groups {\small$\TSg$} in the rank 0 case (assuming the Birch and Swinnerton-Dyer conjecture). In this paper, we give a similar study for the family of elliptic curves associated to simplest cubic fields. These curves have a nonzero rank and we discuss the density of curves of rank 3 that occurs. We also remark on a possible positive density of nontrivial Tate-Shafarevitch groups in the rank 1 case. Finally, we give examples of curves of rank 3 and 5 for which the group {\small $\TSg$} is nontrivial.
@article{1087329234,
author = {Delaunay, C. and Duquesne, S.},
title = {Numerical Investigations Related to the Derivatives of the L-Series of Certain Elliptic Curves},
journal = {Experiment. Math.},
volume = {12},
number = {1},
year = {2003},
pages = { 311-318},
language = {en},
url = {http://dml.mathdoc.fr/item/1087329234}
}
Delaunay, C.; Duquesne, S. Numerical Investigations Related to the Derivatives of the L-Series of Certain Elliptic Curves. Experiment. Math., Tome 12 (2003) no. 1, pp. 311-318. http://gdmltest.u-ga.fr/item/1087329234/