Geometric Theta Lifts

本文研究了正交酉局部对称空间和志村变域中某些环的几何和算术研究。在这个项目中遵循的方法是通过对偶对理论和θ对应。该提案的一个主要主题是研究Kudla和Millson引入的theta升力的边界行为，以建立升力到底层局部对称空间的完全上同的扩展。此外，对于由特殊环定义的广义模符号，在非紧化情况下，也会得到不消失的结果。另一个主要主题是利用Kudla-Millson举升和Borcherds奇异举升之间的密切关系来研究和推广Borcherds举升，并在非正统情况下应用Kudla-Millon举升。这个主题是由应用到算术代数几何及其连接到自同构形式的指导。该项目位于几个经典数学学科的交叉点，即数论、几何和表示理论（对称的研究）。具体地说，我们可以把提出的工作看作是对经典问题“给定的正整数有多少种写法可以写成平方和？”的几何背景下的广泛推广。也就是说，人们可以将涉及高维流形中几何对象（如曲线和曲面）的和和和差的方程的积分解联系起来。这样，对算术问题的研究导致了对几何对象的研究。例如，考虑三个平方和减去第四个平方和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。在此背景下，该项目突出并加强了PI所在机构新墨西哥州立大学（NMSU）的研究概况，这是一所少数民族服务机构，地理位置遥远且处于不利地位。通过这种方式，它还吸引了参与该项目的新密歇根州立大学研究生参与国内和国际研究界。

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.