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

Abstract

In [Zagier and Kramarz 87], the authors computed the critical value of the L-series of the family of elliptic curves 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 III 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. Thesecurves 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 TateShafarevitch groups in the rank 1 case. Finally, we give examples of curves of rank 3 and 5 for which the group III is nontrivial.

Keywords:

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