1996 Fiscal Year Final Research Report Summary
Value Distribution Theeory and Algebraic Geometry
Project/Area Number 
08304007

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  総合 
Research Field 
Geometry

Research Institution  Nagoya University 
Principal Investigator 
KOBAYASHI Ryoichi Nagoya University, Graduate School of Polymathematics, Professor, 大学院多元数理科学研究科, 教授 (20162034)

CoInvestigator(Kenkyūbuntansha) 
TSUJI Hajime Tokyo Institute of Technology School of Science, Associate Professor, 理学部, 助教授 (30172000)
ENOKI Ichiro Ohsaka University, Department of Mathematics Graduate School of Science, Associa, 大学院・理学研究科, 助教授 (20146806)
MABUCHI Toshiki Ohsaka University, Department of Mathematics Graduate School of Science, Profess, 大学院・理学研究科, 教授 (80116102)
BANDO Shigetoshi Touhoku University, Department of Mathematics Graduate School of Science, Profes, 大学院・理学研究科, 教授 (40165064)
OHSAWA Takeo Graduta School of Polymathematics, Nagoya University, Professor, 大学院多元数理科学研究科, 教授 (30115802)

Project Period (FY) 
1996

Keywords  holomorphic curve / value distribution theory / Diophantine approximation / Radon transform / Abeliam variety / second main conjecture / Tening on the bagasilhmic derivative / Vojta's dictionary 
Research Abstract 
The ultimate purpose of this project is to establish the geometry which fully explain the similarity between value distribution theory of holomorphic curves in projective algebraic varieties and the diophantine approximations on arithmetically defined projective varicties. The major difficulty in this research is the lack of definition of the notion of differentiation (in the direction of Spec Z of rational points of arithmetic varieties. Through the investigation under this project, we arrived at the fundamental idea which may be described as follows. Instead of trying to define the notion of differentiation of rational points in the direction of Spec Z itself, we try to find a system of functional equations among valuedistrivutiontheoretic functions (e.g., Weil functions). Then, using Vojta's dictionary, we translate the functional equations to equations in diophantine approximations. These equations will "define" the diffrentials of rational points. In the course of justifying this idea, we obtained the following results : (1) We invented the Radon transform transforming a holomorphic curve in higher dimensional varieties into a system of meromnorphic functions. (2) We can include the group structure of Abelian varieties into the framework of Radon transform, if we replace "rational functions" by holomorphic maps into hyperbolic Riemann surfaces. As a result, we can show that the second main conjecture holds for holomorphic curves in Abelian varieties. (3) We showed that the asymptotic behavior of the Weil functions of jets of a given holomorphic curve is the same for all jets, if a curve under consideration is not contained in a special proper algebraic subset, which is the obstruction to the second main conjecture. This is considered to be system of functional equations we are looking for.

Research Products
(12 results)