Arithmetic of Shimura Varieties and Applications

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This project concerns investigations in the theory of numbers and arithmetic geometry. This is an area of research that has applications to cyber security, through cryptography, and to some aspects of coding theory. Shimura varieties are particular kinds of higher-dimensional surfaces, and their rich geometry and arithmetic puts them among the central objects of study in modern 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. The primary goal of the Principal Investigator's research on Shimura varieties is to prove new formulas for Faltings heights, which are a way of measuring the complexity of elliptic curves and abelian varieties.The Principal Investigator will study the arithmetic of special cycles on integral models of unitary and orthogonal Shimura varieties. There are four distinct, but interrelated, parts of the project. The first part is to extend to integral models a theorem of Borcherds on the modularity of a generating series of divisors on an unitary Shimura variety. The main application of this extension will be to prove new cases of Colmez's conjectural relation between Faltings heights of CM abelian varieties and derivatives of Artin L-functions. The second part is similar to the first, but on orthogonal Shimura varieties. This will yield still more cases of Colmez's conjecture. The third part of the project is a generalization of the Gross-Kohnen-Zagier theorem to unitary and orthogonal Shimura varieties, which is expected to yield new cases of the Beilison-Bloch conjecture relating dimensions of Chow groups to orders of vanishing of L-functions. The fourth project involves the explicit description of mod p reductions of orthogonal Shimura varieties, and is necessary for the third project.