2009 Fiscal Year Final Research Report
A new bridge toward the abc conjecture via p-adic elliptic Diophantine approximation
| Project/Area Number |
19540053
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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 Noriko (河野 典子) Nihon University, 理工学部, 教授 (90215195)
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| Project Period (FY) |
2007 – 2009
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| Keywords | 数論 / ディオファントス近似 / abc予想 / 対数一次形式 / 楕円曲線 / p進楕円対数 |
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| Research Abstract |
We prove a new Diophantine approximation for linear forms in elliptic logarithms (in Crelle, vol.628, with S.David). As far as the height of the linear forms is concerned, our result is the first optimal one. We thus solve a conjecture of S.Lang dating back to 1964. We also show a p-adic version in a case, namely, a lower bound for linear forms in two p-adic elliptic logarithms. We refine here previous estimates in the dependency on the height of algebraic coefficients of the linear forms (the p-adic result is in press in Kyushu Journal of Mathematics, vol.64, No.2, 2010). A lower bound for linear form, in n terms of p-adic elliptic logarithms, is in preparation. This generalization would be useful to determine the set of S-integer points on elliptic curves defined over a number field, whenever we know a basis of the Mordell-Weil group.
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