## 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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