Congruence relations between class numbers of cyclotomic fields

在本课题中，我们主要研究了环切场，特别是二次场的类数和单位的同余关系。我们首先建立了（cf.[1]）在类数和可被8m整除的二次域的单位之间模一个奇素数的一般线性同余关系，其中m >o是一个无奇平方的有理整数。从这个关系中，我们推导出了两个二次域的类数和单位的各种同余，并指出了它们如何包含许多作者已知的结果。利用Stickelberger关系，给出了（参见[2]）实二次域的Jacobi和与类数之间的一个同余模8，并作为这个同余的应用，证明了实二次域的类数与相应的虚二次域的类数的一种新型同余。在证明中，我们提出了一种计算雅可比和二进对数的新方法。我们认为，这种方法对于在切环场的类数之间得到一个素数的一般同余模是基本的。第三，研究了大自然数分解为素数的算法，研究了一种利用3次多项式的筛选法。最后，在研究椭圆曲线分解算法的过程中，我们找到了椭圆曲线上的两个点在任何形成阿贝尔群的域上的简单证明；一个只用初等微积分，另一个用计算机完成。每位研究者在各自的研究计划中都获得了良好的结果。

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.