Unlikely Intersections in Diophantine Geometry and Dynamics

最近，研究函数重复迭代行为的动力学技术已被用于理解数论问题。这个项目的目标是更深入地了解这种富有成效的相互作用，即现在所知的“算术动力学”。特别是，该项目将研究丢番图几何领域和算术动力学领域之间的联系。丢番图几何利用代数几何来研究多项式方程的有理解。国际数学联合会还将组织研讨会和工作坊，为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学中“不太可能的交叉点”的主题，该主题大致预测，如果没有潜在的结构原因，算术对象的交叉点不可能超过维度考虑所表明的程度。这项研究的项目源于相对Bogomolov猜想(RBC)，该猜想涉及到小高点在阿贝尔科亚种中的分布。RBC是经典的Manin-Mumford和Bogomolov猜想的推广，在很大程度上仍然是开放的。PI和合作者的目标是发展以下技术：(1)研究Abel变种家族亚变种中线性相关点的分布；(2)证明RBC的动态相似，并在动态Bogomolov猜想中建立一致性；(3)研究有理映射族中高度的沿曲线的增长，并研究这些估计与族中前周期点的完整性性质的关系；以及(4)从统计学的角度证明关于“有理准周期点的一致有界性”等猜想的有效版本。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.