Study of Iwasawa Theory for Cyclotomic Fields.

岩泽圆场理论的研究。

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640041/
• 关键词: power integral basis; normal integral basis; Greenberg conjecture; 整正規基底; 整巾基底; 素数次不分岐Kummer拡大; 整正規底

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.

在1999-2000期间，我研究了（i）（i）有限字段上某些FERMAT函数字段的类数字，（ii）固定质数分配的真实二次次数的家族，以及（iii）Unramifield Kummer prime度的积分基础。在这里，我总结了主磁场F的有限扩展e/f的主结果（iii）的结果，一个人说它具有幂积分基础（简称为pib）时，当O_e = o_f [α]对于某些α∈O_e时。在这里，o_e（resp.o_f）是e（resp.f）的环积分。如果e/f是galois，则当O_e在OFT OUND OD o_f [GAL（E/F）]上自由时，它具有正常的积分基础（简称NIB）。我获得了以下两个结果1。众所周知，如果具有NIB，则不受影响的Kummer扩展具有PIB。我首先获得了该结果的“定量”版本，然后使用PIB构建了许多此类示例，但使用iWasawa Theory的几个结果，没有NIB.2。令p为奇数素数，k一个想象的阿贝尔场，其中包含一个统一的原始p-词根，k_∞/k cyclotomic z_p- extension。对于k_∞/k.i的每一层k_n，都描述了p p的k_n上未受到的kummer扩展的客观化，以用iwasawa不变式具有PIB。特别是，如果“格林伯格猜想”拥有K。

项目成果

Ichimura, H.: "On power integral bases of unramified cyclic extensions of prime degree."J.Algebra. 235. 104-112 (2001)

Ichimura, H.：“关于素数次无分支循环扩展的幂积分基础。”J.代数。

小屋 良祐: "On a duality theorem of abelian varieties over higher dimensional local fields"Kodai Mathematical Journal. (発売予定).

Ryosuke Kodai：“关于高维局部域上的阿贝尔簇的对偶定理”Kodai Mathematical Journal（待发行）。

市村文男,隅田浩樹: "A note on integral bases of unramified cyclic extensions of prime degree II"Manuscripta Mathematica. (印刷中). (2001)

Fumio Ichimura、Hiroki Sumida：“关于素数次无分支循环扩展的积分基的说明”Manuscripta Mathematica（2001 年出版）。

Ichimura, H.: "A note on integral bases of unramified cyclic extensions of prime degree."Abh.Math.Univ.Humburg. 70. 275-279 (2000)

Ichimura, H.：“关于素数次无分支循环扩展的积分基的注释。”Abh.Math.Univ.Humburg。

Ichimura, H.: "A note on quadratic fields in which a fixed prime number splits completely II."Proc Japan Acad.. 75. 150-151 (1999)

Ichimura, H.：“关于固定素数完全分裂的二次域的注释 II。”Proc Japan Acad.. 75. 150-151 (1999)