Height pairings on unitary and orthogonal Shimura varieties

Boston College, Chestnut Hill MA

Investigators

Abstract

The Principal Investigator will study the arithmetic of integral models of Shimura varieties, with particular emphasis on explicit Arakelov theory and the relations between intersection multiplicities of special cycles and the coefficients of Eisenstein series. The prototype of such a relation is the Gross-Zagier theorem, whose proof proceeds by computing the arithmetic intersection multiplicities of Heegner points on modular curves, and then comparing these multiplicities with the Fourier coefficients of the kernel function for the Rankin-Selberg convolution L-function. The PI will prove similar relations for cycles on unitary and orthogonal Shimura varieties, and use these relations to prove Gross-Zagier type theorems for modular forms of higher weight. 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 essential tools for understanding elliptic curves and abelian varieties, which have applications to cryptography. The Principal Investigator's research into Shimura varieties is motivated partly by their connections to the conjecture of Birch and Swinnerton-Dyer, one the the Clay Mathematics Institute's million-dollar Millenium Prize Problems. The strongest known results in the direction of this conjecture come from work of Gross and Zagier, whose methods relied on the detailed study of one-dimensional Shimura varieties called modular curves. The PI will study higher dimensional versions of these results in order to better understand the Birch and Swinnerton-Dyer conjecture, and its generalizations and variants.