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p-adic analysis of algebraic numer fields and algorithm on algebraic number fields or finite fields

代数数域的 p-adic 分析以及代数数域或有限域上的算法

基本信息

批准号：
12640029
负责人：
KUDO Aichi
金额：
$ 1.6万
依托单位：
NAGASAKI 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 2001
项目状态：
已结题

项目摘要

A. Kudo studied algorithms for computing p-adic properties of Bernoulli numbers, for prime factor ization of integers and for calculation of Carmichael numbers. He proved that the p-adic Euler constant γ_p of p-adic zeta function is equivalent to -(B_<p-1>/(p-1) - 1/p) for modulo p (p【greater than or equal】5), where B_n denotes the n-th Bernoulli number, and calculated the value (modp) of them for p【less than or equal】20,000,000. It is derived that in this range γ_p*0 (modp) for p≠5, 13, 563. Also he calculated all Carmichael numbers less than 10^<17> with their prime factorizations.T. Washio studied class number and Hasse invariants of elliptic curves over finite fields. By means of determination of the number of rational points, he derived a sufficient condition for an elliptic curve over a finite prime field GF(p) to be supersingular, and gave a new equality of binomial coefficients.Y. Sueyoshi studied 4-class ranks of quadratic number fields and matrices over finite field GF(2). Using Redei matrices, he proved new inequality relations between the narrow 4-class ranks of quadratic number fields. He further gave a characterization of Redei matrices with minimal rank. As an application of these results, he also proved the fact that if the ideal class group of imaginary quadratic number field K contains a subgroup of type (4,4, 2,2) and 4 is not contained in the prime discriminant of K, then the 2-class field tower of K is infinite.Y. Maruyama studied discrete optimization problems using the theory of finite automata. He intro duced the notion of a new sequential decision process called bitone sequential decision process and gave a strong representation theorem for a discrete decision process.

A.工藤研究算法计算p-adic性能的伯努利数，素因子化的整数和计算的卡迈克尔号码。他证明了p-adic zeta函数的p-adic欧拉常数γ_p<p-1>对于模p（p[大于或等于]5）等价于-（B_ /（p-1）- 1/p），其中B_n表示第n个伯努利数，并计算了当p[小于或等于] 20，000，000时它们的值（modp）。在此范围内，当p ≤ 5，13，563时，γ_p*0（modp）为零。他还计算了所有小于10^的卡迈克尔数<17>及其素因子分解。鹫尾研究了有限域上椭圆曲线的类数和Hasse不变量。通过确定有理点的个数，他得到了有限素域GF（p）上的椭圆曲线是超奇异的一个充分条件，并给出了一个新的二项式系数等式。Sueyoshi研究了有限域GF（2）上二次数域和矩阵的4类秩。使用Redei矩阵，他证明了新的不等式关系之间的狭窄4类秩的二次数域。他进一步给出了一个表征Redei矩阵的最小秩。作为这些结果的应用，他还证明了：如果虚二次数域K的理想类群包含一个（4，4，2，2）型子群，且4不包含在K的素判别式中，则K的2-类域塔是无穷的. Maruyama研究离散优化问题使用有限自动机理论。他引入了一种新的序贯决策过程的概念，称为双序贯决策过程，并给出了离散决策过程的强表示定理。