| Co-Investigator(Kenkyū-buntansha) |
MIYAKE Katsuya Tokyo Metropolitan University, Department Mathematics, Professor, 大学院・理学研究科, 教授 (20023632)
ICHIKAWA Takashi Saga University, Faculty Science Engineering, Professor, 理工学部, 教授 (20201923)
UEHARA Tsuyoshi Saga University, Faculty Science Engineering, Professor, 理工学部, 教授 (80093970)
TAGUCHI Youichiro Kyushu University, Department Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (90231399)
KATAYAMA Shin-ichi Tokushima University, Department Math Science, Professor, 総合科学部, 教授 (70194777)
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| Research Abstract |
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 rationals
On 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
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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 geometry
Ichikawa 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
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