Finite \(\Lambda\)-submodules of Iwasawa modules for a CM-field for \(p = 2\). (English. French summary) Zbl 1435.11137

Atsuta, Mahiro

J. Théor. Nombres Bordx. 30, No. 3, 1017-1035 (2018).

Let \(p\) be a prime, \(F\) a CM field, and \(F_\infty\) the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). We denote by \(X_{F_\infty}\) the Galois group of the maximal unramified abelian pro-\(p\)-extension of \(F_\infty\). Iwasawa proved that \(X_{F_\infty}\) is a finitely generated \(\mathbb{Z}_p[[\mathrm{Gal}(F^\infty/F)]]\)-module. If \(p\) is an odd prime, Iwasawa also proved that the minus part of \(X_{F_\infty}\) has no non-trivial finite \(\mathbb{Z}_p[[\mathrm{Gal}(F_\infty/F)]] \)-submodule. But for \(p=2\), B. Ferrero [Am. J. Math. 102, 447–459 (1980; Zbl 0463.12002)] proved that if \(F\) is an imaginary quadratic field that is not \(F=\mathbb{Q}(\sqrt{-1})\), \(\mathbb{Q}(\sqrt{-2})\) and the prime above \(2\) ramifies in \(F_\infty/\mathbb{Q}_\infty\), the maximal finite \(\mathbb{Z}_2[[\mathrm{Gal}(F_\infty/F)]]\)-submodule of \(X_{F_\infty}\) is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\) and this submodule is generated by the prime above \(2\). The purpose of this paper under review is to generalize Ferrero’s result to an arbitrary CM field.
Let \(X^{-}_{F_\infty}\) 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, the author studies the maximal finite \(\mathbb{Z}_p[[\mathrm{Gal}(F_\infty/F)]]\)-submodule of \(X^{-}_{F_\infty}\) for \(p=2\). The author also determines the size of the maximal finite \(\mathbb{Z}_2[[\mathrm{Gal}(F_\infty/F)]]\)-submodule of \(X^{-}_{F_\infty}\) under some mild assumptions.

Reviewer: Wei Feng (Beijing)

MSC:

11R23	Iwasawa theory
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers

Keywords:

Iwasawa theory; Iwasawa module; Galois module structure

Citations:

Zbl 0463.12002

References:

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