2018 Fiscal Year Final Research Report

Nonarchimedean geometry, algebraic, arithmetic dynamics and related fields

Research Project

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Project/Area Number	15K04817
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Algebra
Research Institution	Doshisha University
Principal Investigator	Kawaguchi Shu 同志社大学, 理工学部, 教授 (20324600)
Project Period (FY)	2015-04-01 – 2019-03-31
Keywords	非アルキメデス幾何 / 代数・数論力学系
Outline of Final Research Achievements	For a connected smooth projective curve X of genus g, global sections of any line bundle L with degree at least 2g + 1 give an embedding of the curve into projective space. With Kazuhiko Yamaki, we consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry from the viewpoint of faithful tropicalizations. We also consider a higher dimensional case using adjoint line bundles. With Liang-Chung Hsia, we study arithmetic properties of a one-parameter family of Henon maps. We study inverse degrees of quadratic triangular maps on affine space defined over Q-algebras. With Shigeru Mukai and Ken-Ichi Yoshikawa, we study relationship between the difference of elliptic j-functions and the Borcherds Phi funciton.
Free Research Field	代数幾何学
Academic Significance and Societal Importance of the Research Achievements	代数幾何学では，多項式の共通零点で定義される代数多様体の性質を調べる．多項式の係数を複素数とする複素代数多様体は深く研究されている．一方で，複素代数多様体の族を小さい範囲で考えるときには，係数をピュイズー級数体にとると良いことがある．ピュイズー級数体は非アルキメデス体とよばれるものであり，一般に非アルキメデス体上の代数多様体は，多面体を組み合わせたようなトロピカル代数多様体と結びつく．整構造を保つトロピカル化の研究は，このような非アルキメデス幾何の基礎的な研究である．代数・数論力学系におけるアフィン平面上の写像として重要なへノン写像の族に関する性質も基礎的な研究である．