Number Theory and Applications to the Related Discrete Mathematics

A)研究有理上有限次交换场的幂积基和类群的结构。关于Hasse问题，首席研究员S.I.A.Shah(Univ.Peshawar)和Y.Motoda(Yatsushiro National College Of Technology)给出了整数环具有幂整数基或不应用M.-N.Gras和F.Tano‘e的结果的交换域的新刻画[JNT，1986]。也就是说，如果域K是八次域Q(\Sqrt{Mn}，\Sqrt{lm}，Sgrt{ι})，其中LMn是无平方整数，则K除了一个域[Arch.数学课。出席者]。因此，我们提出了一个未决问题[同上]；问题。对任意秩为3的八次2-初等阿贝尔扩张K，如果环Z-K有幂积基，那么K是否与第24次割圆域Q(Zeta_(24))重合？片山-列维斯克-首席调查员构建了一个新的自行车…家族更多具有显式基本单位制的c双二次域[CRM论文集，待定]。S.-I.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张族的构造[Acta Arith，2003]。Miyake显式地确定了Mordell曲线，并描述了它们的q-有理点集确定了相应的三次域的子集，并证明了它们都具有Mordell-Weil秩1。他还明确地将三次Fermat曲线的三次扭曲描述为一族Mordell曲线族。b)数论在算术几何和代数几何中的应用Ichikawa在算术几何范畴中构造了Teichmueller群胚，并描述了Teichmueller群胚上的Galois作用和单列表示[J.Reine Angew]。数学，2003年]。然后，他证明了Bogomolov猜想，该猜想指出，如果一条交换簇中的一条不可约曲线不同构于一条椭圆曲线，那么它的代数点在Nelon-Tate高度上是均匀离散分布的[J.Number Them，2004]。田口证明了Fortaine和Mazur的有限猜想的潜在交换情形，并证明了某些二维modp(ialois表示有理数域的存在/不存在结果)[Ramanujan J.2003，Documenta Math]。[2003]c)数论在编码理论和离散数学中的应用。田口改进了T.T.Satoh的方法，该方法快速计算有限域上椭圆曲线上有理点的个数(与T Satoh和B.SkJerna共同工作)[有限域应用，2003]。Uehara研究了代数几何编码的最小距离的确定，代数几何编码是由代数函数域构造的纠错码[九州数学，2002]。较少

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