2004 Fiscal Year Final Research Report Summary
Toward discretization of Nevanlinna theory
| Project/Area Number |
13304003
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
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| Co-Investigator(Kenkyū-buntansha) |
KANAI Masahiko Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70183035)
KIMURA Yoshifumi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70169944)
NAYATANI Shin Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (70222180)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Nevanlinna theory / Diophantine approximation / Diophantine approximation Diophantos / Neyanlinna analogue / ramification counting function / abc conjecture / Voita's dictionary / lemma On derivative / Diophantine analogue of derivative |
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| Research Abstract |
Nevanlinna theory is a mathematics which is based on classicical calculus. However, this theory has aspects of those Mathematics such as statistical mechanics or arithmetic geometry and this makes the application of classical Differential geometry a difficult issue. More precisely, one can apply differential geometry only after one is successful in putting our problem in a good form by making best use of its statistical or arithmetic nature. Why does Nevablinna theory have such a nature? This question motivated my research project. I aimed at constructing background geometry explaining the origin of such nature of Nevanlinna theory. The guiding principle of my study came from statistical mechanics and arithmetic geometry (Arakelov geometry). Vojta proposed the so called Vojta's dictionary between Nevanlinna theory and Diophantine approximation. The set of rational points of projective varieties defined over a number field is considered to be the Diophantine analogue of transcendental h
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olomorphic curves into the complex variety consisting of its complex points. In my project I asked the question "What is the Diophantine analogue of differentiation of holomorphic curves?" My answer to this queestio is to interpret the lemma on logarithmic derivative in Nevanlinna theory as a defining equation of the derivative of a holomorphic curve. The corresponding statement in Diophantine setting becomes the defining equation of derivatives of rational points. The Diophantine Analogue of "differentiation" thus defined is not an absolute concept. This is defined becomes meaningful only after the target of approximation is given. In Nevanlinna theory, the absolute differentiation obays the relative law, which is the consequence of Lemma on logarithmic derivative. In Diophantine setting we should take finite places into account. I proposed a definition of ramification counting function in Diophantine approximation by extending the Minkovski/Bombierri-Vaaler geometry of numbers. I then proposed a Schmidt Subspace Theorem with truncated counting function. This version of SST enables us to establish some conjectures which is equivalent to the "abc conjecture" which seems to be quite different from the original conjecture. The definition of derivatives contain the rule of counting roots of equations. In the case of transcendental holomorphic curve or the set of rational points, the rule of counting should be based on some non-trivial statistics. In fact the truncated counting function for holomorphic curves in Abelian varieties should be of level 1 regardless of the target's dimension. Such kinds of question arises if we import the Diophantine definition of derivatives in the opposite way to Nevanlinna theory. We started this direction at the end of this project. Less
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