Vojta's Conjecture on blowups and in Arithmetic Dynamics

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740033
• 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)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: ボエタ予想 / 数論的力学系 / 整数点 / 最大公約数 / モーデル・ラング予想

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 have succeeded in proving some cases of Vojta’s conjecture on the blowups of the projective space. I have also analyzed arithmetic properties of orbits, that is, how a point is moved by the iterates of a fixed map. In particular, I have obtained some conditions under which integral points in orbits are sparse, and I have proved that the intersection of orbits and subvarieties do not have any group-like structure as one would expect from the theory of abelian varieties.

Free Research Field

代数学 （ディオファントス幾何）