Arithmetic Intersection, Modular Forms, and Complex Multiplication

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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 integers. 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, a conjecture 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, in which they try to study how the CM value of twisted Bocherds products behave as the CM cycle change. They also want to find a factorization formula for CM values. Twisted Borcherds products are a family of canonical Hilbert modular functions with coefficients 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. The 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 the society's well-being. One of the proposed project has direct applications to cryptosystem, which is essential to national security and national economy. Number Theory is becoming extremely important in coding theory and cryptosystem. Arithmetic and algebraic geometry which is also in part of the proposed research has now applications in engineering such as face recognition and economics.