Unlikely Intersections in Diophantine Geometry and Dynamics

动力学技术，其中的行为反复迭代的功能进行了研究，最近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，现在被称为“算术动力学”。“特别是，该项目将研究丢番图几何领域之间的联系，它使用代数几何来研究多项式方程的有理解，以及算术动力学领域。PI还将组织研讨会和讲习班，为初级研究人员提供信息，并吸引学生进入该领域。该研究的灵感来自算术几何中的“不太可能的交叉”主题，该主题大致预测，在缺乏潜在结构原因的情况下，算术对象不可能超过维度考虑所建议的交叉。本研究的课题来源于相对博戈莫洛夫猜想（RBC），该猜想涉及阿贝尔族的子簇中小高度点的分布。RBC是经典Manin-Mumford和Bogomolov定理的推广，并且在很大程度上仍然是开放的。PI和合作者的目标是开发技术：（1）研究阿贝尔簇族的子簇中线性相关点的分布;（2）证明RBC的动力学类似物并建立动力学Bogomolov猜想的一致性;（三）研究有理映射族沿着曲线的高度的增长，并研究这些估计与预周期点的完整性之间的关系，家庭;（4）从统计学的观点证明了“有理预周期点的一致有界性”猜想的有效版本等。该奖项反映了NSF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。

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.