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On Greenberg’s generalized conjecture for imaginary quartic fields. (English) Zbl 1479.11194

Let \(k\) be a number field, \(p\) a prime in \(\mathbb{Z}\) and \(\tilde{k}/k\) the compositum of all the \(\mathbb{Z}_p\)-extensions of \(k\). Let \(X(\tilde{k}):=\mathrm{Gal}(L(\tilde{k})/\tilde{k})\), where \(L(\tilde{k})\) is the maximal abelian unramified pro-\(p\)-extension of \(\tilde{k}\), then the Iwasawa module \(X(\tilde{k})\) is a finitely generated torsion module over the Iwasawa algebra \(\Lambda(\tilde{k}):=\mathbb{Z}_p[[\mathrm{Gal}(\tilde{k}/k)]]\). The Greenberg’s Generalized Conjecture (GGC for short) predicts \(X(\tilde{k})\) to be a pseudo-null \(\Lambda(\tilde{k})\)-module, i.e., its annihilator has height at least 2.
The author proves the GGC for quadratic extensions \(K/F\) of an imaginary quadratic field \(F\) such that:
1. \(p\) splits in \(F\) but does not split completely in \(K\);
2. \(p\) does not divide the class number of \(K\).
The proof follows the same path of several other papers on the subject (some of which are mentioned in the introduction of the paper under review): via class field and Galois theory, the author proves pseudo-nullity for the Iwasawa module of a \(\mathbb{Z}_p^2\)-subextension of \(\tilde{k}/k\) (which is a \(\mathbb{Z}_p^3\)-extension in the setting of the main result) and then lifts it to the \(\tilde{k}\)-level via a classical lemma of B. Perrin-Riou [Mém. Soc. Math. Fr., Nouv. Sér. 17, 130 p. (1984; Zbl 0599.14020)].

MSC:

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions

Citations:

Zbl 0599.14020
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References:

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