Number Theory, Its Application to Discrete Mathematics and Development

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A07) Investigation of Hasse's problem for the power integral bases and Its ApplicationOn Hasse's problem, the head investigator, S. I. A. Shah(NUCES), Y. Motoda(Yatsushiro National College of Tech.) and Uehara gave a new family of infinitely many monogenic cyclic quartic fields using a linear combination among partial differences.We investigated the Diophantine equations related to the binary recurrence sequences. We also investigated its application to the construction of an infinite family of cyclic extensions of degree p-1 having the p-ranks of the class groups of at least two[Katayama].B07) Applications of number theory to arithmetic geometry and algebraic geometrywe extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms [Ichikawa].We studied (jointly with Hyunsuk Moon) the 1-adic properties of certain modular forms and proved the non-existence and finiteness of mod 2 Galois representations of some quadratic fields [Taguchi].C07) Applications of number theory to coding theory and discrete mathematicsWe have researched on a new method of constructing Hermitian codes which are error-correcting codes constructed from Hermitian curves. we have investigated into finding an explicit expression of a basic of the trace-norm code [Uehara]. We studied Stanley-Reisner rings which are locally complete intersections. As a result, we proved that locally complete intersection Stanley-Reisner rings are complete intersections if the corresponding simplicial complexes are of dimension more than 1 and are connected [Terai].