| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
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| Research Abstract |
The abc conjecture implies several famous results in number theory. For example, Mordell conjecture (solved by G. Faltings), Siegel-zero problem (the connection was made by A. Granville) and so on. A lower bound for linear forms in p-adic logarithms is one tool to obtain a non-trivial upper bound which is in a similar shape of the abc conjecture. The observation was dealt by R. Tijdeman, C. Stewart, Kunrui Yu, etc. Although it is not enough to achieve the conjectural estimate (the possible estimate is still an exponential one), it is well-known that the linear bin in p-adic logarithms is nothing but an important way to abridge Diophantine approximations and the abc conjecture. Being motivated by such a reason, we are interested in linear forms in logarithms. We consider here an elliptic version and obtain an estimate for linear forms in elliptic logarithms, by means of the method of G. Chudnovsky carried out on the formal group of the elliptic curves (Sinnou David and N. Hirata-Kohno,"
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Linear forms in elliptic logarithms", to appear in Journal fur die reine and angewandte Mathematik, and a preprint : N. Hirata-Kohno,"Linear forms in p-adic elliptic logarithms"). In the formal article a conjecture of S. Lang is solved. Useful estimates of elliptic logarithmic functions are also calculated with Sinnou David : "Logarithmic Functions and Formal Groups of Elliptic Curves", in : Diophantine Equations, Tata Institute of Fundamental Research, Studies in Mathematics, Narosa Publishing House, 2008, 243-256. We adapt this method to get Diophantine approximations of values of Gauus'hypergeometric series as a joint work with M. Huttner, where we regard the integral representation of the hypergeometric series as an ablelian logarithmic function at one point at the infinity.
In N. Hirata-Kohno, S.)Laishram, T. N. Shorey and R. Tijdeman "An extension of a theorem of Euler", Acta Arithmetica vol. 129 (1), 2007, 71.102, sufficient conditions of non-existence of the integer solutions to certain exponential Diophantine equations are discussed. Less
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