Geometric Theta Lifts

New Mexico State University, Las Cruces NM

Investigators

Abstract

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.