CAREER: Arithmetic and Geometry of Pure and Mixed Shimura Varieties

现代数论的一个中心主题是在朗兰兹程序及其p-进和p-挠类的背景下，自同构表示类和伽罗瓦表示类之间的猜想关系，两边都有代数性的概念。为了实现这种猜想关系，到目前为止，大多数已知的方法涉及使用某些代数簇及其在整数上的模型的上同调；即所谓的(纯)Shimura簇，它们上的Kuga族(在模曲线上推广Kuga-Sato簇)，它们是所谓的混合Shimura簇的特例，它们的紧致化，以及这些在整数上的好的模型。这项建议中的研究项目旨在通过沿新旧方向的广泛研究，在理解这些基本的重要几何对象方面取得进一步的进展，并开发它们的新的算术应用。数论和几何是数学中最古老的两个分支，它们的组合应用(如纠错码)在现代日常生活(例如，涉及电信和数据存储)中已变得不可或缺。纯粹的和混合的下村变种是重要的几何对象，它们以自然而神秘的方式将代数、分析和几何联系在一起，它们的理论的进步促进了数论的许多令人兴奋的最近发展。该提案中的教育项目旨在为明尼苏达大学的数论和算术几何创造一个垂直整合的学习环境，各级学生都可以从中受益，包括支持外联活动、暑期学习项目以及课程和学习研讨会的开发。

A central theme in modern number theory is the conjectural relations between classes of automorphic representations and Galois representations, with notions of algebraicity on both sides, in the context of Langlands program and its p-adic and p-torsion analogues. To realize such conjectural relations at all, most known methods so far involve the use of the cohomology of certain algebraic varieties and their models over the integers; namely, the so-called (pure) Shimura varieties, the Kuga families over them (generalizing the Kuga--Sato varieties over modular curves) which are special cases of the so-called mixed Shimura varieties, their compactifications, and good models of these over the integers. The research projects in this proposal aim at making further progress in understanding such fundamentally important geometric objects, through extensive studies along directions both old and new, and at developing their new arithmetic applications.Number theory and geometry are the 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 pure and mixed Shimura varieties are important geometric objects relating algebra, analysis, and geometry in natural yet mysterious ways, and advances in their theory have contributed to many exciting recent developments in number theory. The education projects in this proposal aim at creating a vertically integrated learning environment for number theory and arithmetic geometry at the University of Minnesota, from which students at all levels can benefit, including supports for outreach activities, summer learning projects, and the development of courses and learning seminars.