权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Congruence relations between class numbers of cyclotomic fields

分圆域类数之间的同余关系

基本信息

批准号：
02640063
负责人：
UEHARA Tsuyoshi
金额：
$ 1.41万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1990
资助国家：
日本
起止时间：
1990 至 1991
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-02640063/
关键词：
Cyclotomic Field Class Number Congruence Relation Quadratic Field Ideal Class Group Unit Group Factorizing Algorithm Elliptic Curve 円単数ヤコビ和 RSA暗号

项目摘要

In this project, we have studied mainly about congruence relations for class numbers and units of cyclotomic fields, in particular of quadratic fields.We firstly established (cf. [1]) a general linear congruence relation modulo an odd prime number between the class numbers and units of the quadratic fields whose discriminants are divisible by 8m, where m>O is an odd square-free rational integer. From this relation we deduced various congruences for class numbers and units of two quadratic fields and indicated how they included known results by many authors. Using Stickelberger's relation we secondly gave (cf. [2)) a congruence modulo 8 between Jacobi sums and the class number of a real quadratic field, and as an application of this congruence we proved a new type of congruence for the class numbers of a real quadratic field and the corresponding imaginary one. In the proof we developed a new method of calculation of 2-adic logarithm of Jacobi sums. We believe that this method is fundamental to obtain a general congruence modulo some power of a prime number between class numbers of cyclotomic fields. Thirdly we were concerned with algorithm of factorization of large natural numbers into prime numbers, and studied a sieve method using polynomials of degree 3. Finally, during our studies of a factorizing algorithm by elliptic curves, we found two simple proof of the points of an elliptic curve over any field forming an abelian group ; one use only elementary calculus, and the other is done by computer.Each investigator obtained good result for a respective research plan.

本课题主要研究分圆域特别是二次域的类数和单位的同余关系。[1])证明了判别式可被8 m整除的二次域的类数与单位之间模奇素数的一般线性同余关系，其中m> 0是无奇平方有理数.从这个关系，我们推导出各种同余类数和单位的两个二次域，并指出他们如何包括已知的结果，许多作者。使用Stickelberger的关系，我们第二次给（cf。[2))给出了真实的二次域的Jacobi和与类数的模8同余关系，并作为该同余关系的应用，证明了真实的二次域与虚二次域的类数的一类新的同余关系.在证明中，我们提出了一种新的计算雅可比和的2-adic对数的方法。我们相信，这种方法是基本的，以获得一般的同余模的一些权力之间的一个素数的类数的分圆域。第三，研究了大自然数分解为素数的算法，研究了一种利用三次多项式的筛法。最后，在研究椭圆曲线的分解算法时，我们发现了域上椭圆曲线上的点构成交换群的两个简单证明，一个只用初等微积分，另一个用计算机完成，每个研究者在各自的研究计划中都得到了很好的结果。