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 notions of {$\lambda$}-invariant and {$\mu$}-invariant in this situation, generalizing a work of Kurihara and deduce the behaviour of the order of the Shafarevich-Tate group up the cyclotomic {$\Z_p$}-extension. On examples, we give some arguments which, by combining theorems and numeral calculations, allow to calculate the order of the p-primary part of the Shafarevich-Tate group in cases that are not yet known (nontrivial Shafarevich-Tate group, curves of rank greater than 1).
@article{1067634729,
author = {Perrin-Riou, Bernadette},
title = {Arithm\'etique des courbes elliptiques \`a r\'eduction supersinguli\`ere en p},
journal = {Experiment. Math.},
volume = {12},
number = {1},
year = {2003},
pages = { 155-186},
language = {fr},
url = {http://dml.mathdoc.fr/item/1067634729}
}
Perrin-Riou, Bernadette. Arithmétique des courbes elliptiques à réduction supersingulière en p. Experiment. Math., Tome 12 (2003) no. 1, pp. 155-186. http://gdmltest.u-ga.fr/item/1067634729/