Arithmetic of Z_p-field and geometry of algebraic curves

Z_p场的算术和代数曲线的几何

• 批准号: 09640008
• 负责人: NAKAJIMA Shoichi
• 金额: $ 1.92万
• 依托单位: Gakushuin University (1998)The University of Tokyo (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640008/
• 关键词: Iwasawa theory; Algebraic number field

The aim of this project was investigating Iwasawa theory of algebraic number fields through the notion of Zp-field, and through comparing it with the arithmetic theory of elliptic curves.The results obtained in the project are concerned with(1) basic construction of Iwasawa theory(2) arithmetic of elliptic curves over algebraic number fields, and they are collected in the publication of the results of this project.In this project the following activities were also proceeded :(3) activity of "Kummer Research Group"(4) computation of examples by using computer algebra system.(3) is a project for reading papers of Kummer on number theory, expecting to find new ideas in classical great works.(4) is an attempt to find good examples for Iwasawa theory by making use of computers whose ability is extensively advanced recently.For (3) and (4) no concrete results have been published yet.However, in future we expect to publish, in some form, the results of activities (3) and (4) that were begun as part of this project.

本项目的目的是通过Zp-域的概念来研究代数数域上的岩泽理论，并将其与椭圆曲线的算术理论进行比较，所获得的结果涉及（1）岩泽理论的基本构造（2）代数数域上的椭圆曲线的算术，在本项目中还进行了以下活动：（3）“库默研究小组”活动（4）利用计算机代数系统计算实例。(3)是一个阅读库默数论论文的项目，期望在经典巨著中发现新的思想。(4)利用近年来能力得到广泛发展的计算机，试图找到岩泽理论的好例子。对于（3）和（4），还没有发表具体的结果。但是，我们希望将来以某种形式发表作为该项目一部分开始的活动（3）和（4）的结果。

项目成果

H.Ichimura, H.Sumida: "On the Iwasawa invariants of certain real abeliau fields" Tohoku Math. J.49. 203-215 (1997)

H.Ich​​imura、H.Sumida：“论某些真实阿贝廖域的岩泽不变量”东北数学。

中島匠一: "岩沢理論入門" 数理解析研究所講究録. 1026. 28-42 (1998)

Takuichi Nakajima：“岩泽理论导论”数学分析研究所的 Kokyuroku。1026. 28-42 (1998)。

H.Ichimura: "Class numbers of real quadratic function fields of genus one" Finite Fields and Their Applications. 3. 181-185 (1997)

H.Ich​​imura：“属一实二次函数域的类数”有限域及其应用。

H.Ichimura: "Local units modulo Gauss sums" J.Number Theory. 68. 36-56 (1998)

H.Ich​​imura：“局部单位模高斯和”J.数论。

S.Nakajima: "Introduction to Iwasawa theory (in Japanese)" RIMS Kokyu-roku. 1026. 28-42 (1998)

S.Nakajima：“岩泽理论简介（日语）”RIMS Kokyu-roku。