Integral p-adic Hodge theory and its application to automorphic Galois representations

积分p进Hodge理论及其在自守Galois表示中的应用

• 批准号: 1406926
• 负责人: Tong Liu
• 金额: $ 16.02万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406926&HistoricalAwards=false
• 关键词: Integral; adic; Hodge; theory; its

This research is in the area of number theory, one of the oldest branches of mathematics, which is concerned with questions about the integers. The research described in this proposal aims to understand the properties of the integer solutions to systems of polynomials, via some analytic, algebraic, and geometric tools. Recent advances in cryptography and coding theory have depended on solutions to these problems. For instance, most cell phone calls are protected by a code based on elliptic curves over a prime, one of the primary objects of study in the PI's field.The PI will continue his work on integral p-adic Hodge theory. The first aim is to improve the understanding of integral structures (lattices) in semi-stable p-adic Galois representations, and then use this improvement to study the reduction of these representations. As the expected consequences, certain parts of automorphic lifting theorems and Serre's conjecture will be extended to more general settings. The second aim is to study Galois representations associated to automorphic representations and certain algebraic varieties. In particular, the PI and his collaborators will focus on the Galois representations arising from modular forms of noncongruence subgroups and certain torsion representations constructed (by Scholze) from Shimura varieties.

这项研究是在数论领域，数学的最古老的分支之一，这是有关整数的问题。在这个建议中描述的研究旨在通过一些分析，代数和几何工具来理解多项式系统的整数解的性质。密码学和编码理论的最新进展依赖于这些问题的解决方案。 例如，大多数手机通话都受到基于素数上椭圆曲线的代码的保护，这是PI领域的主要研究对象之一。PI将继续他在积分p进霍奇理论方面的工作。第一个目标是提高半稳定p-adic Galois表示中的积分结构（格）的理解，然后使用这种改进来研究这些表示的约简。作为预期的结果，自守提升定理和塞尔猜想的某些部分将被推广到更一般的设置。第二个目的是研究与自守表示和某些代数簇相关的伽罗瓦表示。特别是，PI和他的合作者将专注于伽罗瓦表示所产生的模块形式的非同余子群和某些扭转表示（由Scholze）从志村品种。