Vojta's conjecture, integral points on algebraic varieties, and arithmetic dynamics

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17522
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nihon University
• Principal Investigator: YASUFUKU Yu 日本大学, 理工学部, 准教授 (00585044)
• Research Collaborator: LEVIN Aaron ; TUCKER Thomas ; WANG Julie Tzu-Yueh
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: Vojta予想 / 整数点 / 数論的力学系 / ブローアップ / 最大公約数

Outline of Final Research Achievements

Diophantine geometry is a study on integral and rational solutions to multivariable polynomials, and Vojta’s conjecture is one of the most important conjectures in this field. During the span of this grant, I proved a special case of Vojta’s conjecture on surfaces which can be obtained as multiple blowups of the projective plane. Moreover, we characterized when the set of integral points for complements of the projective plane can be large, using affine algebraic geometry and the notion of weights of fibration introduced by Campana. In the field of arithmetic dynamics where we study iterations of self-maps, we showed that the denominator of each point in an orbit under a certain cubic polynomial contains a new prime in its prime factorization.

Free Research Field

代数学 (ディオファントス幾何)

Academic Significance and Societal Importance of the Research Achievements

射影平面における整数点集合の特徴づけでは，アフィン代数幾何という1980年代から主に日本で研究されてきた分野を，初めて整数論に応用した．これをきっかけにアフィン代数幾何が整数論の世界でもより注目されることになり，新たな応用が生まれる可能性もある点で学術的意義の高い結果である．また，軌道の点の素因数分解を用いて暗号を構築できるので，楕円曲線暗号の次となり得る暗号の安全性研究の土台となる点で，社会的意義もある．