Higher Codimension Cycles on Shimura Varieties

The main focus of this project is on the theory of Shimura varieties. These are particular kinds of higher-dimensional surfaces whose rich geometry and arithmetic puts them among the central objects of study in modern mathematics. In addition to their importance in pure mathematics, they play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are important tools for understanding elliptic curves and abelian varieties, which now play a major role in cryptography.

More broadly, the project concerns the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. Graduate students supported by the award will receive training to contribute towards these projects.

The primary goal of the Principal Investigator's project is to prove new relations between special cycles and automorphic forms. For example, the Principal Investigator will use recent advances in the theory of Borcherds products and integral models to prove the modularity of generating series formed from special cycles of arbitrary codimension on integral models of orthogonal Shimura varieties.

The Principal Investigator will also prove new examples of generalized Gross-Zagier formulas, by expressing the intersection multiplicities of cycles on unitary Shimura varieties in terms of coefficients of integral kernels for automorphic L-functions.

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.