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分圆域（塔）的方法，很快就被推广到任意数域。最近，椭圆曲线的岩泽理论被构建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标是将岩泽理论推广到形式群，其中包括乘法和椭圆曲线群作为特殊情况。为此，我们首先研究了（1）形式群的一般理论，并且作为该理论的重要例子，（2）数域岩泽理论（乘法群），（3）椭圆曲线岩泽理论（椭圆曲线群）。对于(1)，我们主要考虑本田的理论，该理论在整数环上对形式群进行分类。我们还尝试使用 Maple 对形式群进行数值计算。对于（2），我们主要关心的是格林伯格猜想及其与正规积分基的关系。对于（3），我们获得了椭圆曲线的 mu 不变量的结果，扩展了 R.Greenberg 之前的结果。

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。

H.Ichimura: "On power integral bases of unramified cyclic extension of prime degree"Journal of Algebra. 235. 104-112 (2001)

H.Ich​​imura：“论素数次无分支循环扩展的幂积分基”代数杂志。

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