### Arithm'etique des courbes elliptiques a r'eduction supersinguliere en p, by Bernadette Perrin-Riou

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
$\Z_p$-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$).

Bernadette Perrin-Riou <bpr@math.u-psud.fr>