CAREER: Arithmetic Dynamical Systems on Projective Varieties

This project centers on problems in a recent new area of mathematics called arithmetic dynamics. This subject synthesizes problems and techniques from the previously disparate areas of number theory and dynamical systems. Motivations for further study of this subject include the power of dynamical techniques in approaching problems in arithmetic geometry and the richness of dynamics as a source of compelling problems in arithmetic. The funding for this project will support the training of graduate students and early career researchers in arithmetic dynamics through activities such as courses and workshops, as well as collaboration between the PI and researchers in adjacent fields. The project’s first area of focus is the setting of abelian varieties, where the PI plans to tackle various conjectures surrounding the fields of definition and S-integrality of points of small canonical height. One important component of this study is the development of quantitative lower bounds on average values of generalized Arakelov-Green’s functions, which extend prior results in the dimension one case. The PI intends to develop such results for arbitrary polarized dynamical systems, opening an avenue for a wide variety of arithmetic applications. A second area of focus concerns the relationship between Arakelov invariants on curves over number fields and one-dimensional function fields, and arithmetic on their Jacobian varieties. Here the project aims to relate the self-intersection of Zhang’s admissible relative dualizing sheaf to the arithmetic of small points on Jacobians, as well as to other salient Arakelov invariants such as the delta invariant. The third goal is to study canonical heights of subvarieties, especially in the case of divisors. A main focus here is the relationship between various measurements of the complexity of the dynamical system and the heights of certain subvarieties. The final component of the project aims to relate the aforementioned generalized Arakelov-Green’s functions to pluripotential theory, both complex and non-archimedean. 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.