Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa

Nguyen Quang Do, Thong

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 643-668.

Pour un nombre premier impair $p$ et une extension abélienne $K/k$ de corps de nombres totalement réels, nous utilisons la Conjecture Principale Équivariante démontrée par Ritter et Weiss (modulo la nullité de l’invariant ${\mu }_p$) pour calculer l’idéal de Fitting d’un certain module d’Iwasawa sur l’algèbre complète ${\mathbb{Z}}_p\left[\left[G_{\infty }\right]\right],$ où $G_{\infty }=Gal(K_{\infty }/k)$ et $K_{\infty }$ est la ${\mathbb{Z}}_p$-extension cyclotomique de $K$. Par descente, nous en déduisons la $p$-partie de la version cohomologique de la conjecture de Coates-Sinnott, ainsi qu’une forme faible de la $p$-partie de la conjecture de Brumer

For an odd prime number $p$ and an abelian extension of totally real number fields $K/k,$ we use the Equivariant Main Conjecture proved by Ritter and Weiss (modulo the vanishing of the ${\mu }_p$ invariant) to compute the Fitting ideal of a certain Iwasawa module over the complete group algebra ${\mathbb{Z}}_p\left[\left[G_{\infty }\right]\right],$ where $G_{\infty }=Gal(K_{\infty }/k),$ $K_{\infty }$ being the cyclotomic ${\mathbb{Z}}_p$-extension of $K$. By descent, this gives the $p$-part of (a cohomological version of) the Coates-Sinnott conjecture, as well as a weak form of the $p$-part of the Brumer conjecture.

Mots clés : Fitting ideals, Equivariant Main Conjecture

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