In this project, the PI proposes to study the boundary cohomology of (general) PEL-type Shimura varieties, namely the cone of the canonical morphism from the compactly supported cohomology to the ordinary cohomology, with torsion (or integral) automorphic coefficients. A consequence will be a better understanding of the whole torsion cohomology of PEL-type Shimura varieties, which might answer many questions about freeness, liftability, and congruences, and might explain intriguing (potential) pathologies in the torsion interior cohomology (which the PI noticed in his joint work with Junecue Suh). The PI hopes to show that such pathologies do occur in general, but with arithmetically meaningful (and maybe surprising) explanations. The PI also hopes that techniques developed in this project will be useful for studying other interesting questions, such as the arithmeticity of theta correspondences.
Geometry and number theory are two oldest branches of mathematics, and combined applications of them (such as error correcting codes) have become indispensable in modern daily life (involving, for example, telecommunication and data storage). The so-called Shimura varieties are important geometric objects because they relate analysis, geometry, and number theory in a natural yet mysterious way, and advances in the theory of Shimura varieties have contributed to many of the most exciting recent developments in number theory. This project aims at exploring some relatively new territories in this important theory, where many basic questions have yet to be answered. The PI believes that progresses in this project will establish new links among several very different branches of geometry and number theory. The project will also support activities disseminating the knowledge and new ideas in this field.
