Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 2, pp. 643-668.

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 ${ℤ}_{p}\left[\left[{G}_{\infty }\right]\right],$ where ${G}_{\infty }=\phantom{\rule{4pt}{0ex}}Gal\phantom{\rule{4pt}{0ex}}\left({K}_{\infty }/k\right),$ ${K}_{\infty }$ being the cyclotomic ${ℤ}_{p}$-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.

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 ${ℤ}_{p}\left[\left[{G}_{\infty }\right]\right],$${G}_{\infty }=\phantom{\rule{4pt}{0ex}}Gal\phantom{\rule{4pt}{0ex}}\left({K}_{\infty }/k\right)$ et ${K}_{\infty }$ est la ${ℤ}_{p}$-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

DOI: 10.5802/jtnb.512
Keywords: Fitting ideals, Equivariant Main Conjecture
Nguyen Quang Do, Thong 1

1 UMR 6623 CNRS Université de Franche-Comté 16, Route de Gray 25030 Besançon Cedex - France
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Nguyen Quang Do, Thong. Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 2, pp. 643-668. doi : 10.5802/jtnb.512. http://www.numdam.org/articles/10.5802/jtnb.512/

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