On quadratic extensions of number fields and Iwasawa invariants for basic Z<SUB>3</SUB>-extensions

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  • On quadratic extensions of number fields and Iwasawa invariants for basic Z3-extensions

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Abstract

Let Z3 be the ring of 3-adic integers. For each number field F, let F∞, 3 denote the basic Z3-extension over F, • let λ3(F) and μ3(F) denote respectively the Iwasawa λ- and μ-invariants of F∞, 3/F. Here a number field means a finite extension over the rational field Q contained in the complex field C;F=C, \ [F:Q]<∞. Now let k be a number field. Let \mathscr{L}- denote the infinite set of totally imaginary quadratic extensions in C over k (so that \mathscr{L}-coincides with the set \mathscr{L}- in the text when k is totally real); let \mathscr{L}+ denote the infinite set of quadratic extensions in C over k in which every infinite place of k splits (so that \mathscr{L}+ coincides with the set \mathscr{L}+ in the text when k is totally real). After studying the distribution of certain quadratic extensions over k, that of certain cubic extensions over k, and the relation between the two distributions, this paper proves that, if k is totally real, then a subset of {K∈ \mathscr{L}-|λ3(K)=λ3(k), μ3(K)=μ3(k)} has an explicit positive density in \mathscr{L}-. The paper also proves that a subset of {L∈ \mathscr{L}+|λ3(L)= μ3(L)=0} has an explicit positive density in \mathscr{L}+ 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.

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