2003 Fiscal Year Final Research Report Summary
Diophantine Inequality in Diophantine Geometry and Gap Principle
| 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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| Keywords | Diophantine approximation β / Diophantine problem / integer solution / rational point / elliptic curve / linear forms in logarithms / p-adic linear forms in logarithms / hyperelliptic curve |
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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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Research Products
(15 results)
