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纲领及其p-adic和p-torsion类似物的背景下，自守表示类和伽罗瓦表示类之间的代数关系，两边都有代数性的概念。 为了实现这种代数关系，迄今为止，大多数已知的方法涉及使用某些代数簇及其模型在整数上的上同调;即，所谓的（纯）志村品种，他们的Kuga家庭（推广Kuga-Sato簇在模曲线上）是所谓的混合Shimura簇的特殊情况，它们的紧化，以及整数上的好模型。 本计划的研究项目旨在通过广泛的新旧研究，进一步理解这些具有根本重要性的几何对象，并开发它们在算术上的新应用。数论和几何是数学的两个最古老的分支，在现代日常生活中（涉及例如电信和数据存储），纠错码（诸如纠错码）已经变得不可或缺。 纯和混合志村品种是重要的几何对象有关代数，分析和几何在自然但神秘的方式，和进步，他们的理论作出了贡献，许多令人兴奋的最新发展数论。 本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纵向一体化的学习环境，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.