Study of Cyclotomic Iwasawa Theory.

圆切岩泽理论研究。

基本信息

批准号：
13640036
负责人：
ICHIMURA Humio
金额：
$ 1.54万
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：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

During 2001-2003, I studied, as follows, cyclotomic Iwasawa theory, normal integral basis problem and some relation between them. In what follows, p denotes a prime number.(A)Let K be a real abelian field, K_∞/K the cyclotomic Z_p-extension, and K_n its n-th layer. Let A_n be the Sylow p-subgroup of the ideal class group of K_n, and A_∞ the natural injective limit of A_n. Let A_0 be the image of A_0 in A_∞. I proved that the capitulation cokernel A_∞/A_0 is isomorphic to a certain Galois group associated to K_∞, and gave a condition for A_∞/A_0 ={0}.(B)Let K be an imaginary abelian field with ζ_p ∈ K, and K_∞ K_n be as in (A). Let a be a square free integer of the maximal real subfield K^+_n. I described for what m greater than or equal n, the cyclic extension K_m(a^<1/p>)/K_m has a relative normal integral basis (NIB) in terms of the p-adic L-function associated to K.(C)Let F be a number field with ζ_p 【not a member of】 F, and K = F(ζ_p). I proved that an unramified cyclic extension N/F of degree p has a NIB if and only in NK/K has a NIB. When p = 3 and F is an imaginary quadratic field, this is already known by Brinkhuis. I gave some applications of this result.(D)Gomez Ayala gave a necessary and sufficient condition for a Kummer extension of prime degree to have a NIB. I generalised this criterion for a general cyclic Kummer extension. As an application, I Showed a "capitulation" theorem for rings of integers of abelian extensions over a number field.

在2001年至2003年期间，我研究了以下内容，循环岩川理论，正常的积分基础问题以及它们之间的某些关系。在接下来的内容中，p表示一个素数。（a）让k为一个真正的阿贝尔场，k_∞/k cyclotomic z_p-extension和k_n的n层。令A_N为k_n理想类组的Sylow p-subgroup，而A_∞的自然注入限度为a_n。令A_0为A_∞中A_0的图像。我证明了CokernelA_∞/A_0是与K_∞相关的某个Galois组的同构，并给出了A_∞/A_0 = {0}的条件。（b）让K为具有ζ_P∈K的想象中的Abelian Abelian字段，并且K_ k_n k_n as in（a）。令A为最大实际子字段K^+_ n的无方块整数。我描述了大于或等于n的元素，在与K.（c）相关的p-AdiC l功能方面，环状扩展K_M（A^<1/p>）/k_m具有相对正常的积分基础（NIB）。我规定，如果且仅在nk/k中，一个未受到的循环延伸n/f具有笔尖。当p = 3和f是一个虚构的二次场时，这已经是Brinkhuis所知道的。我给出了一些结果的应用。（d）戈麦斯·阿亚拉（Gomez Ayala）给出了必要且充分的条件，即可延伸质学位，以便具有笔尖。我将此标准概括为一般的循环kummer扩展。作为一个应用程序，我显示了一个“投降”定理，用于在一个数字字段上的Abelian扩展名的整数环。