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Study of Iwasawa Theory for Cyclotomic Fields.

岩泽圆场理论的研究。

基本信息

批准号：
11640041
负责人：
ICHIMURA Humio
金额：
$ 1.09万
依托单位：
Yokohama City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

During 1999〜2000, I studied (i) the class numbers of certain Fermat function fields over finite fields, (ii) the family of real quadratic fields in which a fixed prime number splits, and (iii) integral bases of unramifield Kummer extensions of prime degree. Here, I summarize the results of the main one (iii).For a finite extension E/F of a number field F, one says that it has a power integral basis (PIB for short) when O_E=O_F[α] for some α∈O_E. Here, O_E(resp.O_F) is the ring integers of E(resp.F). If E/F is Galois, it has a normal integral basis (NIB for short) when O_E is free over the group ring O_F[Gal (E/F)]. I obtained the following two results.1. It is known that an unramified Kummer extension of prime degree has a PIB if it has a NIB.I first obtained a "quantitative" version of this result, and then constructed many such examples with PIB but no NIB using several results of Iwasawa theory.2. Let p be an odd prime number, K an imaginary abelian field containing a primitive p-th root of unity, and K_∞/K the cyclotomic Z_p-extension. For each layer K_n of K_∞/K.I described the obstruction for unramified Kummer extensions over K_n of degree p to have a PIB in terms of Iwasawa invariants. In particular, I showed that they have a PIB for sufficiently large n if the "Greenberg conjecture" holds for the maximal real subfields of K.

1999〜2000年间，我研究了（i）有限域上某些费马函数域的类数，（ii）固定素数分裂的实二次域族，以及（iii）素数次的unramifield Kummer扩展的积分基。在这里，我总结一下主要的（iii）的结果。对于数域F的有限扩展E/F，当O_E=O_F[α]对于某些α∈O_E时，可以说它具有幂积分基（简称PIB）。这里，O_E(resp.O_F)是E(resp.F)的环整数。如果E/F是伽罗瓦，当O_E在群环O_F[Gal(E/F)]上自由时，它具有正规积分基(简称NIB)。我得到了以下两个结果： 1．众所周知，素数次的未分支库默扩展如果有NIB，则也有PIB。我首先获得了这个结果的“定量”版本，然后使用Iwasawa理论的几个结果构建了许多带有PIB但没有NIB的此类例子。 2．令 p 为奇素数，K 为包含原 p 单位根的虚交换域，K_∞/K 为分圆 Z_p 扩展。对于 K_∞/K.I 的每一层 K_n，我描述了在 p 度 K_n 上无分支 Kummer 扩展的阻碍，以根据 Iwasawa 不变量获得 PIB。特别是，我证明，如果“格林伯格猜想”对于 K 的最大实子域成立，那么它们具有足够大的 n 的 PIB。