Number Theory and Applications to the Related Discrete Mathematics

本课题研究人员主要完成了以下三个目标：A)在理性Hasse问题上有限次阿贝尔域的幂积分基和类群结构的研究。在Hasse问题上，主要研究人员s.i.a. Shah（白沙瓦大学）和Y. Motoda（日本东京理工大学）给出了整数环具有幂积分基或不应用m - n结果的阿贝尔域的新特征。Gras和F……Tano'e [JNT, 1986]。即，如果场K是一个光场Q(\sqrt{mn}, \sqrt{lm}, sgrti)，其中lmn是一个无平方整数，则K除了一个场[Arch]外没有幂积分基。数学。[出现]因此，我们提出了一个开放性问题[同上]；问题。对于任何秩为3的octic 2-初等阿贝尔扩展K，如果环zk有幂积分基，那么K是否与第24环切场Q重合（{}\zeta ＿24{）？katayama - levesque -首席研究员构建了一个新的双二次域族…更多的c双二次域具有明确的基本单位制[CRM Proceedings，即将出现]。-我。Katayama和C. Levesque将这些结果应用于联立丢番图方程和某些循环扩展族的构造[数学学报]。[j]。Miyake明确地确定了Mordell曲线，并描述了其q有理点的集合，确定了相应的三次域的子集，并证明了它们都具有一般的1阶Mordell- weil秩。他还明确地将三次费马曲线的三次扭曲描述为莫德尔曲线族。ichikawa构造了算术几何范畴内的Teichmueller群似，并描述了Teichmueller群似上的伽罗瓦作用和单态表示[J]。莱恩·安格。数学。[j]。其次，他证明了Bogomolov猜想，该猜想指出，如果一个阿贝尔变量中的不可约曲线与椭圆曲线不同态，那么它的代数点对于Neron-Tate高度是均匀离散分布的[J]。数论，2004]。Taguchi证明了Fortaine和Mazur有限猜想的潜在阿贝尔情形，并证明了有理数域的某些二维模表示的存在性/不存在性结果（与H. Moon合作）[Ramanujan J. 2003, Documenta Math. 2003]. c)数论在编码理论和离散数学中的应用。田口改良了t。有限域上椭圆曲线上有理点数目的快速进进计算（与T . Satoh和B. Skjernaa合著）[有限域应用]。[j]。Uehara研究了代数几何码最小距离的确定，代数几何码是由代数函数域构造的纠错码[Kyushu J. Math]。[j]。更少}

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