The Arithmetic and Geometry of Integral Models of Orthogonal Shimura Varieties

这个项目的主要目的是梳理出起源非常不同的数学对象之间的某些隐藏身份：一个是使用多项式方程定义的特定空间的几何定义的，另一个是使用具有复值的无穷和函数定义的。这种恒等式已被证明在椭圆曲线密码学中有丰富的应用。这种公式的第一个实例是由数学家Gross和Zagier在20世纪80年代发现的，他们研究的是一维空间。PI将把他们的工作扩展到更高的维度。志村变分已被证明是现代算术几何和数论中的基本对象，其应用范围从朗兰兹互易到对阿贝尔变分和K3曲面的理解，再到Birch-Swinnerton-Dyer猜想的实例（通过Gross-Zagier公式）。本课题的主要目的是研究一类特殊的志村变项，这些志村变项附着在正交群上，简单到可以接近，但却携带着丰富的算术信息。史蒂夫·库德拉（Steve Kudla）和他的合作者一起，对这些变种的交点理论提出了一系列广泛的猜想，既有经典的，也有算术的（Arakelov意义上的），将它们与某些l函数和爱森斯坦级数（及其导数）的特殊值和傅里叶系数联系起来。本课题的重点是通过研究正交志村变的积分模型及其性质，证明这些猜想的一些新情况。除此之外，PI和他的合作者的工作的最终结果应该会产生P. Colmez关于CM阿贝尔变体高度的猜想公式的证明。

The main purpose of this project is to tease out certain hidden identities between mathematical objects of very disparate origins: One defined using the geometry of certain spaces defined by polynomial equations, and one defined using infinite sums of functions with complex values. Such identities have proven to have fruitful applications to elliptic curve cryptography. The first instance of such a formula was discovered by the mathematicians Gross and Zagier in the 1980s, who worked with one dimensional spaces. The PI will extend their work to higher dimensions.Shimura varieties have proven to be fundamental objects in modern day arithmetic geometry and number theory, with applications ranging from Langlands reciprocity to the understanding of abelian varieties and K3 surfaces to instances of the Birch-Swinnerton-Dyer conjecture (through the Gross-Zagier formula). The main purpose of this project is to study a special class of such Shimura varieties, attached to orthogonal groups, which are simple enough to be accessible, yet carry rich arithmetic information. Steve Kudla, along with his collaborators, has formulated a series of wide ranging conjectures on the intersection theory of these varieties, both classical and arithmetic (in the Arakelov sense), relating them to special values and Fourier coefficients of certain L-functions and Eisenstein series (and their derivatives). The key part of this project is concerned with proving some new cases of these conjectures, through the study of integral models of orthogonal Shimura varieties and their properties. Among other things, the eventual results of the work of the PI and his collaborators should yield a proof of a conjectural formula of P. Colmez on the heights of CM abelian varieties.