Geometric Theta Lifts

本文主要研究了局部对称空间和Shimura簇中某些圈的几何和算术性质。在这个项目中遵循的方法是通过对偶对理论和theta对应。该提议的一个主要主题是研究Kudla和Millson引入的theta提升的边界行为，以建立该提升到基础局部对称空间的全上同调的扩张。另外，利用Kudla-Millson升力与Borcherds奇异theta升力之间的密切关系，研究和推广了BorCherds升力，并将Kudla-Millon升力应用于非正统情形。这一主题以算术代数几何的应用及其与自同构形的联系为指导。该项目位于数论、几何学和表示论(对称性研究)等几个经典数学学科的十字路口。具体地说，人们可以将拟议的工作视为研究经典问题的几何背景下的广泛概括：一个给定的正整数可以用多少种方法写成平方和？也就是说，在高维流形中，人们可以联系到涉及平方几何对象(如曲线和曲面)的和和差的方程的整体解。通过这种方式，算术问题的研究引出了几何对象。例如，当考虑三个平方之和并减去第四个平方时的情况自然会导致爱因斯坦狭义相对论的几何学。这个项目的方法是通过对某些所谓的theta级数的研究，这些级数是模块化形式的例子。模形式在现代数论中扮演着越来越重要的角色。例如，它们在费马大定理的证明中起了决定性的作用。这些学科本身很有趣，为密码学和物理学的重大进步做出了贡献。拟议中的主要工作将与美国和欧洲的研究人员合作进行。在这方面，该项目突出并加强了国际和平协会的总部机构--新墨西哥州立大学(NMSU)--这是一个地理位置遥远、处境不利的少数群体服务机构的研究概况。通过这种方式，它还让NMSU的研究生参与到国内和国际研究社区中来参与这一项目。

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.