On quadratic extensions of number fields and Iwasawa invariants for basic \(\mathbb{Z}_3\)-extensions. (English) Zbl 0927.11052

Let \(\mathbb{Z}_3\) be the ring of 3-adic integers. Assume that a number field, namely a finite extension over the rational field, is always contained in the complex field. For each number field \(F\), let \(F_{\infty,3}\) denote the basic \(\mathbb{Z}_3\)-extension over \(F\), \(\lambda_3(F)\) the Iwasawa \(\lambda\)-invariant of \(F_{\infty,3}\), \(\mu_3(F)\) the Iwasawa \(\mu\)-invariant of \(F_{\infty,3}/F\). Given any number field \(k\), let \(Q_-\) denote the infinite set of totally imaginary quadratic extensions over \(k\), and \(Q_+\) the infinite set of quadratic extensions over \(k\) in which every infinite place of \(k\) splits. After studying the distribution of certain quadratic extensions over \(k\), that of certain cubic extensions over \(k\), and the relation between the two distributions, in this paper it is mainly proved that, if \(k\) is totally real, then a subset of \(\{K\in Q_-\mid \lambda_3(K)= \lambda_3(k)\), \(\mu_3(K)= \mu_3(k)\}\) has an explicit positive density in \(Q_-\).
The author also proves that the subset of \(\{L\in Q_+\mid \lambda_3(L)= \mu_3(L)= 0\}\) has an explicit positive density in \(Q_+\) if 3 does not divide the class number of \(k\) but is divided by only one prime ideal of \(k\). Some consequences of the above results are added in the last part of the paper.


11R23 Iwasawa theory
11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
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