Grant-in-Aid for Scientific Research(C)(2)

科学研究资助金(C)(2)

基本信息

批准号：
11640040
负责人：
KURIHARA Masato
金额：
$ 2.11万
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 (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

B. Mazur began to construct in 1970's Iwasawa theory for elliptic curves which generalizes the work of Iwasawa on ideal class groups to Selmer groups and Tate Shafarevich groups. If an elliptic curves has ordinary reduction at every prime above p, we have a sufficient Iwasawa theory which describes the relation between the Selmer group and the p-adic L-function. But if it does not have ordinary reduction, nothing had been known for a long time. In our research, at first we considered an elliptic curve over the rational number field which has supersingular reduction at p. We determined the Galois wodule structure of the Selmer groups over the intermediate fields of the cyclotomic Z-extension of the rational number field, in particular we determined the orders of the p-components of the Tate-Shafarevich groups over them. Our new discovery is that they can be described by using fractional invariants though in usual Iwasawa theory one uses integer invariants. We constructed a conjectuve which describes how the orders grow in general case, and proved it in some special cases. We found that the distribution relation and the Galois module structure of the local Mordell Weil group are important to understand this phenomenon. We also constructed a homomovphism which sends the zeta element by K. Kato to the modular element by Mazuv and Tate, which gives the relation between the p-adic analytic side and the p-adic algebraic side.

B. Mazur在1970年代开始建造椭圆形曲线理论，该理论概括了Iwasawa对理想阶级群体的工作和Selmer群体和Tate Shafarevich群体的工作。如果椭圆形曲线在P上方的每个素数上都有普通的降低，那么我们就有足够的Iwasawa理论，描述了Selmer组与P-Adic L功能之间的关系。但是，如果没有普通的减少，那么很长一段时间都没有知道。在我们的研究中，首先，我们考虑了在p处具有超差减少的理性数字场上的椭圆曲线。我们确定了Selmer组的GALOIS WODULE结构，这是理性数字场的环形Z扩展的中间场，特别是我们确定了Tate-Shafarevich基团的P-Components的顺序。我们的新发现是可以通过使用分数不变式来描述它们，尽管在通常的Iwasawa理论中，人们使用整数不变性。我们构建了一个猜想，该猜想描述了订单在通常的情况下如何增长，并在某些特殊情况下证明了这一点。我们发现，局部Mordell Weil组的分布关系和GALOIS模块结构对于理解这种现象很重要。我们还构建了一种同源性，该同性恋将K. Kato的Zeta元素发送到Mazuv和Tate的模块化元素，这给出了P-ADIC分析侧与P-Adic代数方面之间的关系。