Arithmétique des courbes elliptiques a réduction supersinguliere en p

Abstract

We review the main conjecture for an elliptic curve on $\Q$ having good supersingular reduction at $p$ and give some consequences of it. Then we define the notion of $\lambda$-invariant and of $\mu$- invariant in this situation, generalizing a work of Kurihara and deduce from it the behaviour of the order of the group of Shafarevich-Tate along the cyclotomique ${\mathbb{Z}}_p$-extension. By examples, we give some arguments which, by allying theorems and numeral calculations, allow to calculate the order of the $p$-primary part of the group of Shafarevich-Tate in not yet known cases (non trivial Shafarevich-Tate group, curves of rank greater than $1$).