1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of . We know by the Ferrero-Washington theorem (cf. [2], [15]) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. [1], [3], [5]-[8], [13]). This paper is a continuation of our previous papers [3], [5]-[7] and [12], that is to say, we investigate Greenberg’s conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg’s conjecture for most of these k’s.
@article{bwmeta1.element.bwnjournal-article-aav69i3p277bwm,
author = {Takashi Fukuda and Hisao Taya},
title = {The Iwasawa l-invariants of Zp-extensions of real quadratic fields},
journal = {Acta Arithmetica},
volume = {69},
year = {1995},
pages = {277-292},
zbl = {0828.11060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav69i3p277bwm}
}
Takashi Fukuda; Hisao Taya. The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields. Acta Arithmetica, Tome 69 (1995) pp. 277-292. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav69i3p277bwm/
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