Special Cycles on Shimura Varieties and Derivative of L-Series

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The investigator is mainly working on three projects. The first one is a fundamental and basic intersection problem between the Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over Z. The goal is to prove a beautiful intersection formula conjectured by J. Bruinier and the investigator. It has at least two applications. One is a highly non-trivial generalization of the celebrated Chowla-Selberg formula, aconjecture of Colmez on Faltings' height of CM abelian varieties. The second is to obtain a nice upper bound for the denominator of the CM values of Igusa invariants|refining a conjecture of Lauter. The second application is also practically important in Cohn-Lauter cryptosystem using genus two curves. The second project is a joint one with Bruinier They try to study how the CM value of twisted Bocherds products behave as the CMcycle change. They also want to find a factorization formula for CM values. Twisted Borcherds products are a family of canonical Hilbert modular functions with coeficients in the real quadratic field. The third project is to solve a general intersection problem in a degenerated Hilbert modular surface and use it to give another proof of the beautiful formula of Gross and Keating on intersection of three modular correspondences.Broad Impact: Mathematics has always been essential to the development of science and technology. Now Mathematics has become increasingly important to business, finance, biology, and social studies too. Basic research in mathematics, which is usually `pure' mathematics (without application), has increasingly found unexpected and deep application in unexpected areas. For example, the number theory I studies used to call the queen ofMathematics, meaning that it is very beautiful and very attractive but useless in practice. But now it is extremely important in coding theory and cryptosystem, both are essentialto our electric communication and electronic business transaction as well as our national security. Arithmetic and algebraic geometry which is also in part of my proposed research has now applications in engineering such as face recognition and economics. My proposed research contributes to basic and deep understanding of some arithmetic and geometric subjects in number theory and arithmetic geometry, and will in turn contribute to our society's well-being. One of my proposed project has direct applications to cryptosystem,which is essential to our national security and national economy.