# On quadratic extensions of number fields and Iwasawa invariants fro basic Z_3 -extensions

## 抄録

Let Z<SUB>3</SUB> be the ring of 3-adic 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 μ-invariants 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.

## 収録刊行物

• Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 51(2), 387-402, 1999-04

The Mathematical Society of Japan

## 各種コード

• NII論文ID(NAID)
10002151674
• NII書誌ID(NCID)
AA0070177X
• 本文言語コード
ENG
• 資料種別
ART
• ISSN
00255645
• NDL 記事登録ID
4712209
• NDL 雑誌分類
ZM31(科学技術--数学)
• NDL 請求記号
Z53-A209
• データ提供元
CJP書誌  NDL  J-STAGE

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