The Arithmetic and Geometry of Integral Models of Orthogonal Shimura Varieties

正交志村品种积分模型的算术和几何

• 批准号: 1803623
• 负责人: Keerthi Madapusi
• 金额: $ 4.44万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1803623&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Integral; Models; Orthogonal

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.

该项目的主要目的是梳理出起源截然不同的数学对象之间的某些隐藏的恒等式：一种是使用由多项式方程定义的某些空间的几何形状来定义的，另一种是使用具有复数值的函数的无限和来定义的。事实证明，此类身份在椭圆曲线密码学中具有卓有成效的应用。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。Shimura 簇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理解阿贝尔簇、K3 曲面到 Birch-Swinnerton 的实例。戴尔猜想（通过 Gross-Zagier 公式）。该项目的主要目的是研究此类 Shimura 变种的特殊类别，它们隶属于正交群，它们足够简单易于访问，但携带丰富的算术信息。史蒂夫·库德拉 (Steve Kudla) 与他的合作者一起，对这些品种的交集理论（包括经典的和算术的（阿拉克洛夫意义上的））提出了一系列广泛的猜想，并将它们与某些 L 函数和爱森斯坦的特殊值和傅立叶系数联系起来。系列（及其衍生品）。该项目的关键部分是通过研究正交志村簇的积分模型及其性质来证明这些猜想的一些新案例。除其他外，PI 及其合作者工作的最终结果应能证明 P. Colmez 关于 CM 阿贝尔簇高度的猜想公式。