Refinement of Iwasawa theory and its applications

岩泽理论的完善及其应用

基本信息

批准号：
14340016
负责人：
KURIHARA Masato
金额：
$ 3.71万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340016/
关键词：
Iwasawa theory Iwasawa main conjecture Iwasawa module ideal class group Fitting ideal Stickelbergerイデアル

项目摘要

The central theme of Iwasawa theory is the main conjecture which states that the characteristic ideal of an algebraic object like Iwasawa modules is essentially generated by the p-adic L-function. Namely, the characteristic polynomial of a certain Galois action and the orders of certain arithmetic objects like ideal class groups are determined by the zeta values. In this research, we could prove that the p-adic L-functions (in a wide sense) have more information than the characteristic polynomials and the orders. More precisely, we could show that the Fitting ideals which have more information than the characteristic ideals are determined by the p-adic L-functions.First of all, we defined the Stickelberger ideal for an imaginary abelian field K in a different way from that of Iwasawa and Sinnott, and proposed a conjecture which claims that this ideal coincides with the 0-th Fitting ideal of the class group of K Our Stickelberger ideal is defined by using several Stickelberger elements of subfields and extension fields of K, so this ideal is regarded as an analytic object, and our conjecture can be regarded as a refinement of usual main conjecture. We proved this conjecture in several cases. We also proposed an analogous conjecture for the Iwasawa module of the cyclotomic Z_{p} extension of K, and proved it completely.We also generalized a structure theorem by Kolyvagin and Rubin for abelian fields to general CM fields. We proved that the higher Fitting ideals of the class group are generated by elements constructed from Stickelberger elements. This gives a typical example of the refinement of Iwasawa theory in our sense.

Iwasawa 理论的中心主题是主要猜想，该猜想指出像 Iwasawa 模这样的代数对象的特征理想本质上是由 p 进 L 函数生成的。也就是说，某个伽罗瓦作用的特征多项式和某些算术对象（如理想类群）的阶数是由zeta值决定的。在这项研究中，我们可以证明p进L函数（广义上）比特征多项式和阶数具有更多的信息。更准确地说，我们可以证明比特征理想具有更多信息的拟合理想是由 p 进 L 函数决定的。首先，我们以与此不同的方式定义了虚交换域 K 的 Stickelberger 理想。 Iwasawa 和 Sinnott 提出了一个猜想，声称该理想与 K 类群的第 0 次拟合理想一致。我们的 Stickelberger 理想是通过使用以下几个 Stickelberger 元素来定义的： K的子域和扩展域，所以这个理想被视为一个解析对象，我们的猜想可以被视为对通常主要猜想的细化。我们通过几个例子证明了这个猜想。我们还对K的分圆Z_{p}扩展的Iwasawa模提出了类似的猜想，并得到了充分的证明。我们还将Kolyvagin和Rubin关于阿贝尔域的结构定理推广到了一般的CM域。我们证明了类群的更高拟合理想是由 Stickelberger 元素构造的元素生成的。这是我们意义上岩泽理论完善的一个典型例子。