Galois coinvariants of the unramified Iwasawa modules of multiple \(\mathbb {Z}_p\)-extensions

IF 0.5 Q3 MATHEMATICS Annales Mathematiques du Quebec Pub Date : 2020-11-11 DOI:10.1007/s40316-020-00150-6
Takashi Miura, Kazuaki Murakami, Keiji Okano, Rei Otsuki
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Abstract

For a CM-field K and an odd prime number p, let \(\widetilde{K}'\) be a certain multiple \(\mathbb {Z}_p\)-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module \(X_{\widetilde{K}'}\) of \(\widetilde{K}'\) as a \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}'/K)\rrbracket \)-module. Our first main result is a description of the order of a Galois coinvariant of \(X_{\widetilde{K}'}\) in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic \(\mathbb {Z}_p\)-extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa \(\lambda \)-invariant and the ideal class group of K, we determine a necessary and sufficient condition such that \(X_{\widetilde{K}}\) is \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}/K)\rrbracket \)-cyclic. Here, \(\widetilde{K}\) is the \(\mathbb {Z}_p^2\)-extension of K.

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多个$$\mathbb的未分支Iwasawa模的Galois协变{Z}_p$$-扩展
对于CM域K和奇数素数p,设\(\widetilde{K}'\)是某个倍数\(\mathbb{Z}_p\)-在本文中,我们研究了作为(\mathbb{Z}_p\llbracket\mathrm{Gal}(\widetilde{K}'/K)\rrbracket\)-模块。我们的第一个主要结果是根据分圆的未分支Iwasawa模的特征幂级数描述\(X_{Z}_p\)-第二个结果是,如果K是一个虚二次域,如果p不在K中分裂,那么,在关于Iwasawa(λ)-不变量和K的理想子群的几个假设下,我们确定了一个充要条件,使得\(X_{Z}_p\llbracket\mathrm{Gal}(\widetilde{K}/K)\rrbracket\)-循环。这里,\(\widetilde{K}\)是\(\mathbb{Z}_p^2)-K的扩张。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
19
期刊介绍: The goal of the Annales mathématiques du Québec (formerly: Annales des sciences mathématiques du Québec) is to be a high level journal publishing articles in all areas of pure mathematics, and sometimes in related fields such as applied mathematics, mathematical physics and computer science. Papers written in French or English may be submitted to one of the editors, and each published paper will appear with a short abstract in both languages. History: The journal was founded in 1977 as „Annales des sciences mathématiques du Québec”, in 2013 it became a Springer journal under the name of “Annales mathématiques du Québec”. From 1977 to 2018, the editors-in-chief have respectively been S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea. Les Annales mathématiques du Québec (anciennement, les Annales des sciences mathématiques du Québec) se veulent un journal de haut calibre publiant des travaux dans toutes les sphères des mathématiques pures, et parfois dans des domaines connexes tels les mathématiques appliquées, la physique mathématique et l''informatique. On peut soumettre ses articles en français ou en anglais à l''éditeur de son choix, et les articles acceptés seront publiés avec un résumé court dans les deux langues. Histoire: La revue québécoise “Annales des sciences mathématiques du Québec” était fondée en 1977 et est devenue en 2013 une revue de Springer sous le nom Annales mathématiques du Québec. De 1977 à 2018, les éditeurs en chef ont respectivement été S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.
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