Unlikely Intersections in Diophantine Geometry and Dynamics

丢番图几何与动力学中不太可能的交叉点

• 批准号: 2200981
• 负责人: Laura DeMarco
• 金额: $ 16.2万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200981&HistoricalAwards=false
• 关键词: Unlikely; Intersections; Diophantine; Geometry; Dynamics

Dynamical techniques, in which the behavior of repeated iterations of a function is studied, recently have been employed in understanding number-theoretic questions. The goal of this project is to gain a deeper understanding of this fruitful interplay, now known as "arithmetic dynamics." In particular, the project will study the connections between the field of Diophantine geometry, which uses algebraic geometry to study rational solutions of polynomial equations, and the field of arithmetic dynamics. The PI will also organize seminars and workshops to inform junior researchers and attract students to the field.The research is inspired by the theme of "unlikely intersections" in arithmetic geometry, which predicts roughly that, in the absence of an underlying structural reason, arithmetic objects cannot intersect more than dimensional considerations suggest. The projects in this research originate from the Relative Bogomolov Conjecture (RBC) concerning the distribution of points of small height in subvarieties of abelian families. RBC is a generalization of the classical Manin-Mumford and Bogomolov conjectures and remains largely open. The PI and collaborators aim to develop techniques to: (1) study the distribution of linearly related points in subvarieties of families of abelian varieties; (2) prove dynamical analogues of RBC and establish uniformity in the dynamical Bogomolov conjecture; (3) investigate the growth of heights in families of rational maps along curves and study the relationship between these estimates and integrality properties of preperiodic points in families; and (4) prove, from a statistical viewpoint, effective versions of a conjecture on "uniform boundedness of rational preperiodic points" and others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动态技术是研究函数重复迭代行为的技术，近年来已被用于理解数论问题。这个项目的目标是更深入地理解这种富有成效的相互作用，现在被称为“算术动力学”。特别是，该项目将研究丢芬图几何领域与算术动力学领域之间的联系。丢芬图几何使用代数几何来研究多项式方程的有理解。PI还将组织研讨会和讲习班，向初级研究人员提供信息，并吸引学生进入该领域。这项研究的灵感来自于算术几何中的“不太可能的相交”主题，它粗略地预测，在没有潜在的结构原因的情况下，算术对象的相交不会超过维度考虑所建议的。本研究项目源于有关阿贝尔科亚种中小高度点分布的相对Bogomolov猜想（RBC）。RBC是经典的Manin-Mumford猜想和Bogomolov猜想的推广，在很大程度上仍然是开放的。PI及其合作者的目标是开发技术：(1)研究阿贝尔变种族亚变种中线性相关点的分布；(2)证明了RBC的动力学类似物，建立了动态Bogomolov猜想的均匀性；(3)研究沿曲线的有理映射族中高度的增长，并研究这些估计与族中周期前点的完整性性质之间的关系；(4)从统计的角度证明了“有理前周期点的一致有界性”等猜想的有效形式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。