Geometric Theta Lifts

几何 Theta 升降机

• 批准号: 0710228
• 负责人: Patrick Morandi
• 金额: --
• 依托单位: New Mexico State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0710228&HistoricalAwards=false
• 关键词: Geometric; Theta; Lifts

This work is concerned with the geometric and arithmetic investigation of certain cycles in orthogonal and unitary locally symmetric spaces and Shimura varieties. The approach followed in this project is via the theory of dual pairs and the theta correspondence.One major theme of the proposal is to study the boundary behavior of the theta lift introduced by Kudla and Millson in order to establish an extension of the lift to the full cohomology of the underlying locally symmetric space. Furthermore, this will also yield in the non-compact situation non-vanishing results for generalized modular symbols defined by the special cycles.The other major theme is to utilize the close relationship between the Kudla-Millson lift and Borcherds' singular theta lift to investigate and generalize Borcherds' lift and also to apply the Kudla-Millon lift in unorthodox circumstances. This theme is guided by applications to arithmetic algebraic geometry and its connection to automorphic forms.The project lies at the crossroads of several classical disciplines of mathematics, namely number theory, geometry, and representation theory (the study of symmetries).Concretely, one can view the proposed work as vast generalizations in a geometric context of the investigation of the classical problem "In how many ways can one write a given positive integer as the sum of squares?". Namely, one can associate to integral solutions of equations involving sums and differences of squares geometric objects, such as curves and surfaces, inside higher-dimensional manifolds. In this way, the study of arithmetic questions leads to geometrical objects. For example, the situation when considering the sum of three squares and subtracting a fourth square naturally leads to the geometry of Einstein's theory of special relativity. The approach of this project is via the study of certain so called theta series, which are examples of modular forms. Modular forms play an increasingly central role in modern number theory. For example, they have played a decisive role in the proof of the Fermat's Last Theorem. Interesting in their own right, these subjects have contributed to major advances in cryptography and physics.Major parts of the proposed work will be carried out collaboratively with researchers in the U.S. and in Europe. In this context, the project highlights and strengthens the research profile of the PI's home institution, New Mexico State University (NMSU), a minority serving institution, geographically distant and disadvantaged. In this way, it also engages graduate students at NMSU participating in this project to the national and international research community.

本文主要研究正交和酉局部对称空间中某些圈和Shimura簇的几何和算术性质。该项目采用的方法是通过对偶对理论和theta对应。该提案的一个主要主题是研究Kudla和Millson引入的theta提升的边界行为，以建立提升到底层局部对称空间的完全上同调的扩展。另外，利用Kudla-Millson提升与Borcherds奇异theta提升之间的密切关系，研究和推广Borcherds提升，并将Kudla-Millon提升应用于非正规情形。这个主题是由算术代数几何的应用及其与自守形式的联系所引导的。这个项目位于数学的几个经典学科的交叉点，即数论，几何和表示论（对称性的研究）。具体地说，人们可以将所提出的工作看作是对经典问题的研究的几何背景下的大量概括。一个给定的正整数有多少种写法可以写成平方和？".也就是说，人们可以将涉及平方和与平方差的方程的积分解与高维流形中的几何对象（如曲线和曲面）联系起来。这样，算术问题的研究导致几何对象。例如，考虑三个平方和减去第四个平方的情况自然会导致爱因斯坦狭义相对论的几何。这个项目的方法是通过研究某些所谓的theta系列，这是模块形式的例子。模形式在现代数论中扮演着越来越重要的角色。例如，他们在费马大定理的证明中发挥了决定性的作用。这些学科本身就很有趣，它们为密码学和物理学的重大进步做出了贡献。拟议工作的主要部分将与美国和欧洲的研究人员合作进行。在这种情况下，该项目突出和加强了PI的家乡机构，新墨西哥州州立大学（NMSU）的研究概况，这是一个少数民族服务机构，地理位置遥远，处于不利地位。通过这种方式，它也吸引了NMSU的研究生参与国家和国际研究界的这个项目。