Arakelov Geometry and Algebraic Dynamics

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The project in this award lies at the interface of algebraic dynamics and arithmetic geometry, motivated by an analogy between rational points on projective varieties and orbits of dynamical systems. Understanding the rational solutions to polynomial equations is a classical and fundamental problem, with advances such as Fermat's Last Theorem and the proof of the Mordell Conjecture being among the most celebrated examples of progress in modern mathematics. These questions find analogues as well as broader frameworks in the area of dynamical systems. The project funded through this proposal focuses on dynamical systems on projective varieties defined over number fields and function fields. A key aim is to extend knowledge on the arithmetic of points of small canonical height with respect to rational functions to more general polarized dynamical systems. The funding for this project will support the infrastructure of a growing group at UIC specializing in questions at the intersection of number theory, dynamics, and logic. It will also support collaborations between the PI and other researchers whose work applies non-archimedean analysis and Arakelov geometry to number-theoretic problems. The PI plans to organize a workshop at BIRS and to be a project leader for a future Women in Numbers research team.Three types of problems will be investigated. The first centers on the arithmetic of points of small canonical height for polarized dynamical systems in dimension larger than 1. Specific directions include the Torsion Conjecture for abelian varieties, along with its function field analogue, as well as a sparseness conjecture for torsion points on abelian varieties that are S-integral relative to a non-torsion ample divisor. Under this heading also falls a project studying certain distinguished local canonical heights on abelian varieties and their relation to the global Neron-Tate height. The second thread connects Arakelov invariants on higher genus curves to key Diophantine conjectures about their rational points, capitalizing on recent work linking these invariants to dynamics and analysis on Jacobian varieties. A central component of this thread focuses on the self-intersection of the admissible relative dualizing sheaf introduced by Zhang, and links this quantity to metric invariants of the underlying curve. The third project concerns pluripotential theory on Berkovich analytic spaces associated to projective varieties. Here a particular ultimate goal is the development of suitable quantitative equidistribution statements for small points in arbitrary dimension, yielding natural applications to the first aforementioned project.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.