Number Theory and Applications to the Related Discrete Mathematics

本课题的研究工作主要完成了以下三个方面的工作：（1）研究有理数域上有限次交换域的幂积分基和子群的结构。I. A. Shah（Univ. Peshawar）和Y. Motoda（八代国立工业大学）给出了整数环具有幂整基或不应用M. N. Gras and F.. Tano 'e [JNT，1986年]。也就是说，如果一个域K是一个八度域Q（\sqrt{mn}，\sqrt{lm}，sgrt{i}），其中lmn是一个无平方的整数，那么K除了一个域之外没有幂整基。因此，我们提出了一个开放的问题[同上] ;问题。对任意秩为3的八次2-初等阿贝尔扩张K，若环Z K有幂整基，则K与第24阶分圆域Q（\zeta_{24}）重合吗？Katayama-Levesque-首席研究员构建了一个新的bicycli家族 关于我们 c双二次领域与明确的基本系统的单位[CRM会议录，出现]。S.- I. Katayama和C. Levesque应用这些结果的同时丢番图方程和建设的某些家庭的循环扩展[学报Arith。2003]. Miyake明确地确定了Mordell曲线，描述了它们的Q-有理点集，确定了相应三次域的子集，并证明了它们的Mordell-Weil秩一般都为1。他还明确描述了三次扭曲的费马曲线的程度三作为一个家庭的莫德尔曲线。B）应用数论算术几何和代数几何市川构建了Teichmueller groupoid的范畴内的算术几何，他描述了伽罗瓦行动和monodromy表示的Teichmueller groupoid [J. Reine Angew.数学、2003年]。接下来，他证明了Bogomolov猜想，其中指出，如果一个不可约曲线在交换品种是不同构的椭圆曲线，那么它的代数点分布均匀离散的Neron-Tate高度[J.数论，2004].Taguchi证明了Fortaine和Mazur的有限性猜想的潜在阿贝尔情况，并证明了某些二维modp的存在/不存在结果（有理数域的Ialois表示（与H. Moon）[Ramanujan J. 2003，Documenta Math. 2003].C）数论在编码理论和离散数学中的应用.Taguchi改进了. T的方法Satoh的快速p-adic计算有限域上椭圆曲线上有理点的数目（与T Satoh和B的联合工作。Skjernaa）[有限域应用，2003年]。Uehara研究了代数几何码的最小距离的确定，代数几何码是由代数函数域构造的纠错码[Kyushu J. Math.，2002年]。少

The three aims of our project have been accomplished by the investigators as follows,A)Investigation of the power integral bases and the structures of the class groups of abelian fields of finite degree over the rationalsOn Hasse's problem, the head investigator, S. I. A. Shah (Univ. Peshawar) and Y. Motoda (Yatsushiro National College of Technology) gave a new characterization of abelian fields whose rings of integers have power integral bases or do not applying a result of M.-N. Gras and F..Tano'e [JNT, 1986 ]. Namely if afield K is an octic field Q(\sqrt{mn}, \sqrt{lm}, sgrt{ι}), where lmn is a square-free integer, then K has no power integral basis except for one field [Arch. Math. To appear]. Thus we proposed an open problem[ibid] ;Problem. For any octic 2-elementary abelian extensionK of rank 3, if the ring Z K has a power integral basis, then does K coincide with the 24-th cyclotomic field Q(\zeta_{24})? Katayama-Levesque-the head investigator constructed a new family of bicycli … More c biquadratic fields with explicit fundamental system of units[CRM Proceedings, to appear]. S.-I. Katayama and C. Levesque applied these results to the simultaneous diophantine equations and to the construction of certain family of cyclic extensions[Acta Arith., 2003].Miyake explicitly determined the Mordell curves and described the sets of the Q-rational points of them ascertain subsets of corresponding cubic fields and showed that both of them have Mordell-Weil rank 1 generically. He also explicitly described cubic twists of the Fermat curve of degree three as a family of Mordell curves.B)Applications of number theory to arithmetic geometry and algebraic geometryIchikawa constructed the Teichmueller groupoids in the category of arithmetic geometry, and he described the Galois action and the monodromy representation on the Teichmueller groupoids [J. Reine Angew. Math., 2003]. Next he proved the Bogomolov conjecture which states that if an irreducible curve in an abelian variety is not isomorphic to an elliptic curve, then its algebraic points are distributed uniformly discretely for the Neron-Tate height [J. number Theory, 2004].Taguchi proved the potentially abelian case of the finiteness conjecture of Fortaine and Mazur and proved some existence/non-existence results on certain 2-dimensional modp (ialois representations of the rational number field(joint work with H. Moon) [Ramanujan J. 2003, Documenta Math. 2003].C)Applications of number theory to coding theory and discrete mathematics.Taguchi improved the method of.T. Satoh of the fastp-adic calculation of the number of rational points on elliptic curves over finite fields (joint work with T Satoh and B. Skjernaa) [Finite Field Appl., 2003]. Uehara researched into the determination of the minimum distance of algebraic geometry codes, which are error-correcting codes constructed by algebraic function fields [Kyushu J. Math., 2002]. Less