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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2001-2003年期间，我主要研究了分圆岩泽理论、正规积分基问题以及它们之间的一些关系。在下文中，p表示素数。（A）设K是真实的Abel域，K_∞/K是Z_p分圆扩张，K_n是第n层扩张。设A_n是K_n的理想类群的Sylow p-子群，A_∞是A_n的自然内射极限.设A_0是A_0在A_∞中的像。证明了投降上核A_∞/A_0同构于某个与K_∞相关的Galois群，并给出了A_∞/A_0 ={0}的条件。（B）设K是一个虚阿贝尔域，且K_p ∈ K，K_∞ K_n与（A）相同.设a是极大真实的子域K^+ n的无平方整数.我描述了当m大于或等于n时，循环扩张K_m（a^<1/p>）/K_m具有与K相关的p进L函数的相对正规积分基（NIB）。（C）设F是数域，且K = F（n_p）.证明了p次非分歧循环扩张N/F有NIB当且仅在NK/K中有NIB。当p = 3且F是一个虚二次域时，Brinkhuis已经知道了这一点。我给出了这个结果的一些应用。（D）Gomez Ayala给出了素数度的库默扩张有NIB的一个充要条件.我推广了这个标准的一般循环库默扩展。作为应用，给出了数域上阿贝尔扩张的整数环的“投降”定理。