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Pseudo-null submodules of the unramified Iwasawa module for \(\mathbb Z_p^2\)-extensions. (English) Zbl 1234.11145

Let \(k\) be a number field, \(p\) a prime in \(\mathbb{Z}\) and \(\tilde{k}\) the compositum of all the \(\mathbb{Z}_p\)-extensions of \(k\). Let \(\Lambda=\mathbb{Z}_p[[\text{Gal}(\tilde{k}/k)]]\) be the Iwasawa algebra associated with \(\tilde{k}/k\) and \(X(\tilde{k})\) the Galois group of the maximal abelian unramified pro-\(p\)-extension of \(\tilde{k}\). Then \(X(\tilde{k})\) is a finitely generated torsion \(\Lambda\)-module and the Generalized Greenberg’s Conjecture (GGC) states that it is pseudo-null (i.e., its annihilator has height at least 2).
The paper presents some criteria for \(X(\tilde{k})\) to have (or not have) non trivial pseudo-null submodules when \(\tilde{k}/k\) is a \(\mathbb{Z}_p^2\)-extension. These criteria apply to quadratic imaginary fields \(k\) and depend on the behaviour of the prime \(p\), the irreducibility of generators of certain characteristic ideals and the subgroup generated by primes dividing \(p\) in the ideal class group of a degree \(p\) extension \(k_1/k\) contained in \(\tilde{k}\). The last two sections provide an explicit method (based on the intersection of the decomposition fields of some primes of \(k\)) for determining the field \(k_1\) and computations for \(p=3\) which lead to the verification of the GGC for some imaginary quadratic fields.

MSC:

11R23 Iwasawa theory
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