Torsion Boundary Cohomology of PEL-type Shimura Varieties

在这个项目中，PI建议研究(一般)PEL型Shimura簇的边界上同调，即从紧支撑上同调到普通上同调的典型上同调的锥体，具有扭转(或积分)自同构系数。其结果将是更好地理解PEL型Shimura变种的整个扭转上同调，这可能回答许多关于自由性、可升性和同余的问题，并可能解释扭转内部上同调中有趣的(潜在)病理(PI在他与Junecue Suh的联合工作中注意到)。PI希望表明，这种病理确实发生在一般情况下，但有算术上有意义的(或许令人惊讶的)解释。PI还希望这个项目中开发的技术将有助于研究其他有趣的问题，如theta对应的算术性。几何和数论是数学中两个最古老的分支，它们的结合应用(如纠错码)在现代日常生活中已变得不可或缺(例如，涉及电信和数据存储)。所谓的下村族是重要的几何对象，因为它们以一种自然而神秘的方式将分析、几何和数论联系在一起，而下村族理论的进步促成了数论中许多最令人兴奋的最新发展。这个项目旨在探索这一重要理论中的一些相对较新的领域，其中许多基本问题尚未得到回答。该协会相信，这个项目的进展将在几何和数论的几个非常不同的分支之间建立新的联系。该项目还将支持传播这一领域的知识和新想法的活动。

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.