| Project/Area Number |
09640074
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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 | RIKKYO UNIVERSITY |
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Principal Investigator |
AOKI Noboru RIKKYO UNIV.COLLEGE OF SCIENCE.ASSISTANT PROF., 理学部, 助教授 (30183130)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1998: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1997: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Gauss sum / Jacobi sum / Fermat curve / elliptic curve / congruent number / Selmer group / Cassels pairing / Tate-Shafarevich group / テイト・シャファレヴィッチ群 / 円分体 / フェルマ-曲線 |
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| Research Abstract |
In this research, I have studied rational points on algebraic curves. The main targets are Fermat curves and elliptic curves.
As for Fermat curves, I treated the purity problem on Gauss sums and Jacobi sums which appear in the zeta functions of those curves. I was able to generalize certain known results, and proposed a new conjecture on the purity of Gauss sums. I showed that the conjecture is valid in some cases.
As for elliptic curves, I treated special elliptic curves'which are closely related to the congruent number problem. The main result is an explicit formula for the size of the 2-Selmer groups. As far as I know, such an explicit formula has never been obtained in the literature. The key point in the proof is an explicit calculation of the Cassels pairing restricted to a subgroup of the Tate-Shafarevich group.
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