Unlikely Intersections in Diophantine Geometry and Dynamics

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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.