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.

这个项目的主要目的是梳理出不同起源的数学对象之间的某些隐藏的身份：一个是使用由多项式方程定义的某些空间的几何定义的，另一个是使用具有复值的函数的无限和定义的。这样的身份已被证明有富有成效的应用，椭圆曲线密码。这种公式的第一个例子是由数学家格罗斯和扎吉尔在20世纪80年代发现的，他们研究一维空间。志村簇已被证明是现代算术几何和数论的基本对象，其应用范围从朗兰兹互易到阿贝尔簇和K3曲面的理解，再到Birch-Swinnerton-Dyer猜想（通过Gross-Zagier公式）的实例。这个项目的主要目的是研究一类特殊的志村品种，附加到正交群，这是足够简单，可以访问，但携带丰富的算术信息。史蒂夫库德洛，沿着与他的合作者，制定了一系列广泛的aptures交叉理论的这些品种，无论是经典和算术（在阿拉克洛夫意义上），将它们与特殊价值观和傅立叶系数的某些L-函数和爱森斯坦系列（及其衍生物）。本项目的重点是通过研究正交Shimura簇的积分模型及其性质，证明了这些代数的一些新的情形。除其他事项外，PI和他的合作者的工作的最终结果应该产生一个证明的数学公式的P.科尔梅兹的高度CM阿贝尔品种。