Height pairings on unitary and orthogonal Shimura varieties

主要研究Shimura簇的积分模型的算法，特别强调显式Arakelov理论和特殊圈的交重数与Eisenstein级数系数之间的关系。这种关系的原型是Gross-Zagier定理，它的证明是通过计算模曲线上Heegner点的算术交重数，然后将这些交重数与Rankin-Selberg卷积L函数的核函数的傅立叶系数进行比较来进行的。PI将证明么正Shimura簇上圈的相似关系，并利用这些关系证明高权模形式的Gross-Zagier型定理。Shimura簇是一种特殊的高维曲面，其丰富的几何和算法使其成为现代数学的中心研究对象之一。它们在朗兰兹计划中扮演着重要的角色，该计划在理论物理中越来越受到关注，并且是理解椭圆曲线和阿贝尔簇的基本工具，这些曲线和阿贝尔簇在密码学中有应用。首席研究员对Shimura品种的研究部分是因为它们与Birch和Swinnerton-Dyer的猜想有关，这是克莱数学研究所的百万美元千禧年奖问题之一。在这一猜想的方向上，已知的最有力的结果来自格罗斯和扎吉尔的工作，他们的方法依赖于对被称为模曲线的一维下村变种的详细研究。PI将研究这些结果的高维版本，以便更好地理解Birch和Swinnerton-Dyer猜想及其推广和变体。

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.