Number Theory and Its Apprication to Discrete Mathematics

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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