Invariants des courbes de Frey-Hellegouarch et grands groupes de Tate-Shafarevich, by Abderrahmane Nitaj

This paper has appeared in Acta Arith. 93 (2000), 303-32, and so the dvi version has been removed. Let $s$ and $t$ be rational integers without square common factors and define the generalized Frey-Hellegouarch elliptic curve $y^2=x(x-s)( x-t)$. We give an explicit description, depending on the structure of the para meters $s$ and $t, of some invariants appearing in the Birch and Swinnerton-Dyer conjecture related to these curves and their isogenous. We employ this study to find numerical examples of elliptic curves with large Tate-Shafarevitch group , in connexion with a conjecture of Goldfeld and Szpiro.

Abderrahmane Nitaj <nitaj@math.uni-sb.de>