2007 Fiscal Year Final Research Report Summary
Number Theory, Its Application to Discrete Mathematics and Development
| 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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| Keywords | Hasse's Problem / Integral basis / Cyclic extension of degree p-1 / Modular forms / Galois representation / Hermitian code / Trace-norm code / Stanley-Reisner ring |
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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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[Presentation] ハッセの問題と共役差積2008
Author(s)
中原 徹
Organizer
九州大学代数学セミナー[招待講演]
Place of Presentation
九州大学理学部
Year and Date
2008-02-22
Description
「研究成果報告書概要(和文)」より
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[Presentation] ハッセの問題について2007
Author(s)
元田 康夫, 中原 徹, 上原 健, SHAH, Syed, Inayat, Ali
Organizer
研究集会「代数的整数論とその周辺」[招待講演]
Place of Presentation
京都大学数理解析研究所
Year and Date
2007-12-13
Description
「研究成果報告書概要(和文)」より
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[Presentation] On a Problem of Hasse2007
Author(s)
S. I. A., Shah, Y., Motoda, T., Nakahara, T., Uehara
Organizer
Algebraic Number Theory and its related Topics(RIMS Kyoto Univ.)
Place of Presentation
Kyoto
Year and Date
2007-12-13
Description
「研究成果報告書概要(欧文)」より
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