Intersection Theory and Height Pairings in Arithmetic Geometry

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Many questions in arithmetic geometry are motivated by the philosophy that algebraic information can be obtained by geometric methods. Height functions classify the arithmetic complexity of geometric objects, and a height pairing is a function on the heights of two objects that relates to how those geometric objects intersect. Height pairings have been instrumental in many foundational results in the field including Faltings’ celebrated proof of the Mordell Conjecture concerning rational points on curves. The PI will continue in this tradition by applying modern variations on height pairings to major open problems including an effective (computable) version of the Mordell Conjecture. The PI will regularly organize workshops and seminars to provide students, postdocs, and other mathematicians in the area substantial opportunities for instruction, discussion and collaboration. Graduate students supported by the award will receive training to contribute towards these projects. The investigator and his students will study arithmetical intersection theory of special cycles on Shimura varieties and diagonal cycles on split three folds. The PI expects to achieve the following objectives with this proposal: (1) making essential progress on conjectures by Gan--Gross--Prasad conjecture and Kudla; (2) further developing Arakelov geometry with the goal of proving the effective Mordell conjecture. These expected achievements will make a significant contribution to the study of the arithmetic geometry. It is also very likely that new questions and speculations will arise in the course of studying questions raised by this proposal, stimulating future research programs in the field.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.