| Project/Area Number |
18540040
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Saga University |
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Principal Investigator |
NAKAHARA Toru Saga University, Fac. Sci. Engrg., Professor (50039278)
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| Co-Investigator(Kenkyū-buntansha) |
UEHARA Tsuyoshi Saga Univ., Fac. Sci. Engrg., Professor (80093970)
ICHIKAWA Takashi Saga Univ., Fac. Sci. Engrg., Professor (20201923)
TERAI Naoki Saga Univ., Fac. Edu., Associate Prof. (90259862)
KATAYAMA Shin-ichi Tokushima Univ., Fac. IAS, Professor (70194777)
TAGUCHI Yuichiro Kyushu Univ., Dep. Math., Associate Prof (90231399)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,880,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥480,000)
Fiscal Year 2007: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Hasse's Problem / Integral basis / Cyclic extension of degree p-1 / Modular forms / Galois representation / Hermitian code / Trace-norm code / Stanley-Reisner ring / Siegel 保型形式 / ハッセの問題 / 整数環の巾底 / 不定方程式 / 超幾何微分方程式 / 保型形式のフーリエ係数 / LDPC符号 / 超グラフ |
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| Research Abstract |
A07) Investigation of Hasse's problem for the power integral bases and Its Application
On 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 geometry
we 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 mathematics
We 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].
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