Numerical investigations related to the L-series of certain elliptic curves.

Zagier, Don; Kramarz, G.

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Numerical investigations related to the L-series of certain elliptic curves.

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Zagier, Don
Max Planck Institute for Mathematics, Max Planck Society;

Kramarz, G.
Max Planck Society;

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Zagier, D., & Kramarz, G. (1987). Numerical investigations related to the L-series of certain elliptic curves. The Journal of the Indian Mathematical Society. New Series, 52, 51-69.

Cite as: https://hdl.handle.net/21.11116/0000-0004-395B-7

Abstract

As the title indicates, the authors make computations around the Birch and Swinnerton-Dyer conjecture for the family x\sp 3+y\sp 3=m, m\in \bbfN, in the range m≤ 70000. They compute approximations of L(1) and L'(1). They show that the density of curves with a zero of order ≥ 2 (resp. ≥ 3) does not seem to be 0 as expected by the experts. Using these approximations as well as the Birch and Swinnerton-Dyer conjecture, the authors compute the order of the Tate-Shafarevich group. The results are presented graphically. Finally, a tentative analogue of the Cohen-Lenstra heuristic is presented and discussed but, as the authors observe the recipe does not seem to work in this case.