On quadratic extensions of number fields and Iwasawa invariants fro basic Z_3 extensions
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Abstract
Let Z<SUB>3</SUB> be the ring of 3adic integers. For each number field F, let F<SUB>∞, 3</SUB> denote the basic Z<SUB>3</SUB>extension over F, • let λ<SUB>3</SUB>(F) and μ<SUB>3</SUB>(F) denote respectively the Iwasawa λ and μinvariants of F<SUB>∞, 3</SUB>/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}<SUB></SUB> denote the infinite set of totally imaginary quadratic extensions in C over k (so that \mathscr{L}<SUB></SUB>coincides with the set \mathscr{L}<SUP></SUP> in the text when k is totally real); let \mathscr{L}<SUB>+</SUB> denote the infinite set of quadratic extensions in C over k in which every infinite place of k splits (so that \mathscr{L}<SUB>+</SUB> coincides with the set \mathscr{L}<SUP>+</SUP> 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}<SUB></SUB>λ<SUB>3</SUB>(K)=λ<SUB>3</SUB>(k), μ<SUB>3</SUB>(K)=μ<SUB>3</SUB>(k)} has an explicit positive density in \mathscr{L}<SUB></SUB>. The paper also proves that a subset of {L∈ \mathscr{L}<SUB>+</SUB>λ<SUB>3</SUB>(L)= μ<SUB>3</SUB>(L)=0} has an explicit positive density in \mathscr{L}<SUB>+</SUB> 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.
Journal

 Tokyo Sugaku Kaisya Zasshi

Tokyo Sugaku Kaisya Zasshi 51(2), 387402, 199904
The Mathematical Society of Japan