Notes on the dual of the ideal class groups of CM-fields

Kurihara, Masato

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 971-996.

Dans cet article, pour une extension abélienne $K/k$ de corps de nombres de type CM, nous proposons une conjecture qui décrit complètement l’idéal de Fitting de la partie moins du dual de Pontryagin du groupe de classes de rayon $T$ de $K$, pour un ensemble $T$ d’idéaux premiers, comme $\mathrm{Gal}(K/k)$-module. Nous soulignons que nous considérons ici le groupe de classes au sens propre, sans laisser de côté les idéaux ramifiés (l’objet que nous étudions n’est pas le quotient du groupe de classes par le sous-groupe engendré par les classes des idéaux premiers ramifiés). Nous prouvons que notre conjecture est une conséquence de la conjecture de nombres de Tamagawa équivariante, et prouvons la version de notre conjecture en théorie d’Iwasawa.

In this paper, for a CM abelian extension $K/k$ of number fields, we propose a conjecture which describes completely the Fitting ideal of the minus part of the Pontryagin dual of the $T$-ray class group of $K$ for a set $T$ of primes as a $\mathrm{Gal}(K/k)$-module. Here, we emphasize that we consider the full class group, and do not throw away the ramifying primes (the object we study is not the quotient of the class group by the subgroup generated by the classes of ramifying primes). We prove that our conjecture is a consequence of the equivariant Tamagawa number conjecture, and also prove the Iwasawa theoretic version of our conjecture.

Classification : 11R29, 11R23, 11R37
Mots-clés : Class groups, Fitting ideals

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