Research of Iwasawa Theory fof Formal Groups

岩泽形式群理论研究

• 批准号: 12640038
• 负责人: NAKAJIMA Shoichi
• 金额: $ 2.18万
• 依托单位: Gakushuin University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640038/
• 关键词: Iwasawa Theory; Formal Group; Elliptic Curve; 岩澤理論(Iwasawa Theory); 形式群(Formal Group); 形式群(Formal Grorp); 岩澤理論

Iwasawa Theory started as a method for investigating (a tower of) cyclotomic fields and soon was generalized to arbitrary number fields. Recently, Iwasawa Theory for elliptic curves has been constructed and has developed extensively. Iwasawa Theory for number fields is related to multiplicative group and Iwasawa Theory for elliptic curves is naturally related to the group structure of elliptic curves. Our aim in this project was to generalize Iwasawa Theory for formal groups, which include multiplicative and elliptic curve groups as special cases. For that purpose, we first investigated(1) general theory of formal groupsand, as important examples of the theory,(2) Iwasawa Theory of number fields (multiplicative group),(3) Iwasawa Theory of elliptic curves (elliptic curve group).For (1) we mainly considered Honda's theory which classified formal groups over the integer ring. We also tried numerical computation for formal groups using Maple. For (2) our main concern was Greenberg's conjecture and its relation to normal integral bases. For (3) we obtained results for mu-invariants of elliptic curves, extending the former results of R.Greenberg.

岩泽理论最初是一种研究割圆域(塔)的方法，并很快被推广到任意数域。近年来，关于椭圆曲线的岩泽理论已经建立并得到了广泛的发展。数域的岩泽理论与乘法群有关，椭圆曲线的岩泽理论自然与椭圆曲线的群结构有关。我们在这个项目中的目的是将岩泽理论推广到形式群，其中包括作为特例的乘法和椭圆曲线群。为此，我们首先研究了(1)形式群的一般理论，作为该理论的重要例子，(2)数域(乘群)的岩泽理论，(3)椭圆曲线的岩泽理论(椭圆曲线群)。我们还尝试了用Maple对形式群进行数值计算。因为(2)我们主要关注Greenberg猜想及其与正规整基的关系。对于(3)，我们得到了椭圆曲线的Mu不变量的结果，推广了R.Greenberg的结果。

项目成果

H.Ichimura, F.Kawamoto: "An infinite family of totally real number fields"Acta Arithmetica. 106-2. 171-181 (2003)

H.Ich​​imura、F.Kawamoto：“全实数域的无限族”算术学报。

H.Naito: "Local fields generated by 3-division points on elliptic curves"Proc.Japan Academy. 78-9. 173-178 (2002)

H.Naito：“由椭圆曲线上的 3 分点生成的局部场”Proc.Japan Academy。

on a normal integral bases problem over cyclotomic Zp-extension, II

关于分圆 Zp 扩展的正规积分基问题，II

• 发表时间: 2002
• 期刊: J.Number Theory 96
• 作者: Yoshitaka Hachimori;Yoshichika Iizuka;Yoshitaka Hachimori;Yoshichika Iizuka;Shoichi Nakajima;Humio Ichimura
• 通讯作者: Humio Ichimura

Local fields generated by 3-division points of elliptic curves

椭圆曲线三分点生成的局部场

• 发表时间: 2002
• 期刊: Proc.Japan Acad. 78
• 作者: Yoshitaka Hachimori;Yoshichika Iizuka;Yoshitaka Hachimori;Yoshichika Iizuka;Shoichi Nakajima;Humio Ichimura;Hirotada Naito
• 通讯作者: Hirotada Naito

Humio Ichimura: "On a guotient of the unramified Iwasawa module over an abelian number field, II"Pacific Journal of Math.. 206 No1. 129-137 (2002)

Humio Ichimura：“关于阿贝尔数域上的未分支 Iwasawa 模块的 guotient，II”《太平洋数学杂志》.. 206 No1。