Number Theory and Its Apprication to Discrete Mathematics

本项目的三个目标由研究人员完成如下：A.有限次交换场的类群和整数环的结构的研究首席研究员Levesque-Katayama证明了一个新的由两个实二次域K组成的合成族，我们可以为K的单位群和子域写出Hasse单位指数，以及类数[KLN]的一些关系。Miyake和Kishi成功地将类数可被3[Km]整除的二次域上的所有非分支循环三次扩张参数化。元田元弘(Yatsushiro National College Of Technology)利用虚二次域刻画了整数环有幂基模和无Z模的阿贝尔域，其中Z表示有理整数环[MN]。面积B。将数论应用到算术几何和代数几何中，Ichikawa得到了Teichmuller g…更多的Roupoid是基点在模空间无穷远处的基本群群，用于分类尖点Riemann曲面。利用代数曲线上的算术Schottky-Mumford一致化理论，他构造了算术几何范畴中的Teichmuller群胚[I]。Kato将分圆域上经典的显式高互易定律推广到剩余域不一定是完美的p-完全离散赋值域[KK].C区数论在编码理论和离散数学中的应用上原得到了一类由仿射平面曲线产生的代数几何码的最小距离的结果[U].日韩和韩日数论及其在相关领域的应用联合研讨会在东北大学举行(1999.11.24-11.26)在仙台和首尔的韩国大学(2001.2.19)-2.20)，80人参加，共28场演讲。两国调查人员和研究生的酬金或旅费都由这笔赠款支付。另一方面，凭借这笔赠款，经评审的期刊：当代数学高级研究(Adv.Stud.沉思。数学)，第二卷出版，分发给世界各地的130个研究所和/或大学。此外，为了在加拿大和匈牙利共和国进行受邀演讲，航空公司的费用只由资助人支付，因此，这笔助学金对于开展这一研究项目是不可忽视的。较少

The three aims of our project were accomplished by the investigators as follows ;Area A.Investigation of the structures of the class groups and the rings of integers of abelian fields of finite degreeThe head investigator-Levesque-Katayama exhibited a new family of certain composita of two real quadratic fields K, for which we can write the Hasse unit index for the unit groups of K and of the subfields, and some relation for the class numbers [KLN]. Miyake with Kishi succeeded to parametrize all of unramified cyclic cubic extensions over those quadratic fields whose class numbers are divisible by 3 [KM]. The head investigator-Motoda (Yatsushiro National College of Technology) have characterized some abelian fields whose rings of integers have a power basis or do not as a Z-free module via an imaginary quadratic field, where Z denotes the ring of rational integers [MN].Area B.Applications of number theory to arithmetic geometry and algebraic geometry Ichikawa obtained that Teichmuller g … More roupoids are fundamental groupoids with base points at infinity of the moduli space classifying pointed Riemann surfaces. By using the arithmetic Schottky-Mumford uniformization theory on algebraic curves, he constructed Teichmuller groupoids in the category of arithmetic geometry [I]. Kato gave generalizations of the classical explicit higher reciprocity law in a cyclotomic field to p-adically complete discrete valuation fields whose residue fields are not necessarily perfect [Kk].Area C.Applications of number theory to coding theory and discrete mathematics Uehara obtained a result on the minimum distance of a class of algebraic-geometric codes arising from affine plane curves [U].Japan-Korea and Korea-Japan joint seminar on Number Theory and its Application to the related Area were held at Tohoku Univ.(1999.11.24-11.26) in Sendai and Korea Univ in Seoul (2001.2.19-2.20) with 80 participants and 28 talks in total. The honorariums or the travel expenses for investigators and graduate students in both countries were paid by this grant. On the other hand, by virtue of the grant, the refereed Journal : Advanced Studies in Contemporary Mathematics (Adv. Stud. Contemp. Math.), Vol.2 was published, which is distributed to 130 institutes and/or universities all over the world. Moreover to give invited talks in Canada and Hungarian Republics, the airline expences only were paid by the grant.Then this grant-in-aid is indespensable to develop this research project. Less