Study of Cyclotomic Iwasawa Theory.

在2001-2003年期间，我主要研究了环切Iwasawa理论，正规积分基问题以及它们之间的一些关系。在下面，p表示一个素数。(A)设K为实阿贝尔域，K_∞/K为分环z_p扩展，K_n为第n层。设A_n为K_n的理想类群的Sylow p子群，A_∞为A_n的自然内射极限。设A_0是A_0在A_∞上的像。证明了投降核A_∞/A_0与K_∞相关的某伽罗瓦群同构，并给出了A_∞/A_0 ={0}的一个条件。(B)设K为一个虚阿贝尔域，其中ζ_p∈K，且K_∞K_n如(A)所示。设a是最大实子域K^+_n的自由平方整数。我描述了当m大于或等于n时，循环扩展K_m(a^<1/p>)/K_m具有与K相关的p进l函数的相对正积分基（NIB）。(C)设F是一个数字域，其中ζ_p不属于F，且K = F（ζ_p）。证明了p次的非分枝环扩展N/F有NIB当且仅当NK/K中有NIB。当p = 3且F是虚二次域时，这是Brinkhuis已经知道的。我给出了这个结果的一些应用。(D)Gomez Ayala给出了初等度Kummer延生具有NIB的充分必要条件。我将此准则推广到一般循环Kummer扩展。作为一个应用，我展示了整数环在数域上的阿贝尔扩展的“投降”定理。

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.