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力学系のゼータ関数とその数論的力学系への応用

动力系统的Zeta函数及其在算术动力系统中的应用

基本信息

批准号：
22KJ0286
负责人：
竹平航平
金额：
$ 1.54万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2023
资助国家：
日本
起止时间：
2023-03-08 至 2025-03-31
项目状态：
未结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22KJ0286/
关键词：
数論的力学系ゼータ関数高さ関数 multiplier 高さゼータ関数

项目摘要

今年度は下記の研究を行った。(1)射影直線上の力学系における、multiplierの母関数として定義されるゼータ関数の極、零点の挙動の研究：特別研究員採用前までの研究において、multiplierと呼ばれる力学系の局所的挙動を反映する不変量の母関数として定義されるゼータ関数が、ある線形写像の固有多項式を用いて表示できることを、十分一般の仮定のもとで示していた。この線形写像は、射影直線上の1次微分形式のなす層のテンソル冪のコホモロジー群に、力学系を定める写像から誘導されるものであり、特定の場合には、詳細な明示計算が可能である。そこで、今年度はそのような線形写像の明示計算を行い、ゼータ関数の極、零点に対応する固有値の挙動の研究を行ない、力学系のパラメータを増大させた場合における最大固有値の漸近挙動を明らかにした。(2)力学系の高さゼータ関数に対する研究：数論的力学系において、Call-Silvermanによるdynamical canonical heightと呼ばれる、ある種の高さ関数が知られている。Hsia(1997)は、このdynamical canonical heightに関する一般化Dirichlet級数として定義される、力学的高さゼータ関数について、力学系が函数体上で定義され、かつ、力学系の挙動が穏やかな場合にゼータ関数の複素平面上への有理型接続や、ゼータ関数の代数性を証明した。本年度の研究では、このゼータ関数の主要項と考えられる、あるアデール上の積分で定義される関数について、Hsiaの仮定が成立しないある力学系の族を用いて明示的に計算し、解析接続を示した。また、この明示計算には、超越数論で活発に研究される、Mahlerの関数方程式を満たす部分があり、超越数論との関連性も期待される。

This year, we will conduct a study on the project. (1) on the projective straight line, the Department of Mechanics and the multiplier Department of Mechanics define the number of people in the Department of Mechanics and the Department of Mechanics. in the projective straight line, the data of the Department of Mechanics of the Department of Mechanics and the Department of Mechanics of the Department of Mechanics in the projective straight line are used to define the number of computers of the Department of Mechanics and the Department of Mechanics of the Department of Mechanics. in the projective straight line, the data of the Department of Mechanics and the Department of Mechanics of the multiplier are used to reflect the data of the Department of Mechanics, and the inherent multinomial of the image of the system. It is very general to show you how to use it. In the projective straight line, the first differential form, the projective straight line, the first differential form, the projective straight line, the projective line, the The image of this year's program shows that it is necessary to make an explicit calculation of the number of activities, zero percent of the total number of students, and the Department of Mechanics, the department of mechanics and the department of mechanics. the program is the largest one in terms of performance. (2) Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Call-Silverman Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mechanics, Department of Mathematics, Department of Mechanics, Department of Mathematics, Department of Mathematics, Department of Mechanics, Department of Mechanics, Department of Hsia (1997), general Dirichlet data definition, mechanical properties, mechanical properties, mechanical functions, mechanical properties, and so on, in the complex plane, the rational connection and the algebraic properties of the functions of the department of mechanics. This year, the main items of this year's research and development program are the active definition of the number of units, the establishment of the Hsia program, the calculation and analysis of the family of the Department of Mechanics, which is clearly indicated in the software. Express calculation, transcendental mathematical research, Mahler mathematical equation, partial mathematical calculation, transcendental mathematical connectivity and expectation.