Finite $\Lambda$-submodules of Iwasawa modules for a CM-field for $p=2$

Atsuta, Mahiro

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1017-1035.

Let $p$ be a prime, $X_{F_{\infty }}^{-}$ the minus quotient of the Iwasawa module, which we define to be the Galois group of the maximal unramified abelian pro-$p$-extension over the cyclotomic ${\mathbb{Z}}_p$-extension over a CM field $F$. If $p$ is an odd prime, it is well known that $X_{F_{\infty }}^{-}$ has no non-trivial finite ${\mathbb{Z}}_p〚\mathrm{Gal}(F_{\infty }/F)〛$-submodule. But $X_{F_{\infty }}^{-}$ has non-trivial finite ${\mathbb{Z}}_p〚\mathrm{Gal}(F_{\infty }/F)〛$-submodule in some cases for $p=2$. In this paper, we study the maximal finite ${\mathbb{Z}}_p〚\mathrm{Gal}(F_{\infty }/F)〛$-submodule of $X_{F_{\infty }}^{-}$ for $p=2$. We determine the size of the maximal finite ${\mathbb{Z}}_2〚\mathrm{Gal}(F_{\infty }/F)〛$-submodule of $X_{F_{\infty }}^{-}$ under some mild assumptions.

Soit $F$ un corps CM et $p$ un nombre premier. Soit $X_{F_{\infty }}^{-}$ le quotient “moins” du groupe de Galois de la pro-$p$-extension abélienne non ramifiée maximale de la ${\mathbb{Z}}_p$-extension cyclotomique de $F$. Si $p$ ne vaut pas 2, il est bien connu que $X_{F_{\infty }}^{-}$ n’a pas de sous-module fini non-trivial. Mais pour $p=2$, il peut arriver que $X_{F_{\infty }}^{-}$ contient un sous-module fini non-trivial. Dans cet article, nous étudions le sous-module fini maximal de $X_{F_{\infty }}^{-}$ pour $p=2$, et nous déterminons ce module sous certaines légères hypothèses.

Classification: 11N56, 14G42
Keywords: Iwasawa theory, Iwasawa module, Galois module structure

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