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2003 Numerical Investigations Related to the Derivatives of the L-Series of Certain Elliptic Curves
C. Delaunay, S. Duquesne
Experiment. Math. 12(3): 311-318 (2003).

Abstract

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.

Citation

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C. Delaunay. S. Duquesne. "Numerical Investigations Related to the Derivatives of the L-Series of Certain Elliptic Curves." Experiment. Math. 12 (3) 311 - 318, 2003.

Information

Published: 2003
First available in Project Euclid: 15 June 2004

zbMATH: 1083.11041
MathSciNet: MR2034395

Subjects:
Primary: 11G40
Secondary: 11Y35 , 15A52

Keywords: analytic rank , Elliptic curves , L-series , simplest cubic fields , Tate-Shafarevich group

Rights: Copyright © 2003 A K Peters, Ltd.

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