Experimental Mathematics

Numerical Investigations Related to the Derivatives of the L-Series of Certain Elliptic Curves

C. Delaunay and S. Duquesne

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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 information

Experiment. Math., Volume 12, Number 3 (2003), 311-318.

First available in Project Euclid: 15 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11Y35: Analytic computations 15A52

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


Delaunay, C.; Duquesne, S. Numerical Investigations Related to the Derivatives of the L -Series of Certain Elliptic Curves. Experiment. Math. 12 (2003), no. 3, 311--318. https://projecteuclid.org/euclid.em/1087329234

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