| Project/Area Number |
12640042
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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 | Nihon University |
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Principal Investigator |
HIRATA-KOHNO Noriko Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (90215195)
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| Project Period (FY) |
2000 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
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| Keywords | Diophantine approximation β / Diophantine problem / integer solution / rational point / elliptic curve / linear forms in logarithms / p-adic linear forms in logarithms / hyperelliptic curve / abc予想 / 対数一時形式 / p進距離 / デイオファントス近似 / アーベル多様体 / 不定方程式 / 数の幾何学 / 楕円対数 / Wirsing System / Resultant不等式 / 超越近似 |
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| Research Abstract |
1) Work with J. H. EVERTSE : We investigate Wirsing system which is defined as a system of DIophantine Inequalities and gave some conditions such that the system has only finitely many solutions in algebraic numbers of bounded degree. We also show some relations between the system and Resultant inequalities.
2) Work with Sinnou DAVID : We prove a new lower bound for linear forms in elliptic logarithms. As far as the height of the linear forms is concerned, our result is the best possible. We thus completely solve a conjecture of S. Lang dating back to the sixties. Our general result includes a "simultaneous version" and is totally quantitative : it takes into account the height of the point, the height of the elliptic curve and the degree of the field of definition of the given data. The previously best known estimate in this context was due to the head investigator and goes back to the early nineties.
3) Work with Marc HUTTNER : We present Diophantine Approximations concerning values of Gauss' hypergeometric function. Our estimate relies on a natural application of the method of Chudnovsky for the quantitative theory of linear forms in elliptic logarithms. We regard Gauss' hypergeometric function as an abelian logarithmic function.
4) p-adic alanogs : We give an estimate of linear forms in p-adic logarithms in elliptic case. We define for this estimate a p-adic elliptic logarithmic function viewed as a local reversed function of the Lutz-Weil p-adic elliptic function.
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