Integral p-adic Hodge Theory and Its Application to Automorphic Forms

积分p-adic Hodge理论及其在自守形式中的应用

• 批准号: 0901360
• 负责人: Tong Liu
• 金额: $ 16.33万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901360&HistoricalAwards=false
• 关键词: Integral; adic; Hodge; Theory; Its

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The methods of p-adic Hodge theory have emerged as a powerful tool for studying Galois representations arising from algebraic varieties over p-adic fields. This proposal investigates integral p-adic Hodge theory, which is the study of integral and torsion structures associated to semi-stable Galois representations and how they relate to automorphic forms and various integral and torsion p-adic cohomology groups of algebraic varieties. The first goal in this proposal is to provide new classifications of lattices in semi-stable representations and torsion semi-stable representations, and to study invariants (such as Weil-Delignerepresentations) attached to them. The second goal is to use these results to study Galois representations associated to automorphic forms. Examples of such applications are to prove the compatibility between the local Langlands correspondence and Fontaine's construction for Hilbert modular forms, and the bounded denominator conjecture for certain modular forms on noncongruence subgroups (joint with W. Li and L. Long).This research is in the area of number theory, one of the oldest branches of mathematics, which is concerned with questions about the integers. From the time of Euclid, research in number theory has been active and productive but difficult. The research described in this proposal aims to understand the behavior and properties of integer solutions to systems of polynomials when focusing on properties relative to a fixed prime, via some analytic, algebraic, and geometric tools. Recent advances in cryptography and coding theory have depended on solutions to problems of this type.

“该奖项是根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金的。”p-adic Hodge 理论的方法已成为研究 p-adic 域上代数簇的伽罗瓦表示的有力工具。该提案研究了积分 p-adic 霍奇理论，该理论研究与半稳定伽罗瓦表示相关的积分和扭转结构，以及它们如何与自守形式以及代数簇的各种积分和扭转 p-adic 上同调群相关。该提案的第一个目标是提供半稳定表示和扭转半稳定表示中的晶格的新分类，并研究附加到它们的不变量（例如 Weil-Deligner 表示）。第二个目标是利用这些结果来研究与自同构形式相关的伽罗瓦表示。此类应用的示例是证明局部朗兰兹对应和 Fontaine 的希尔伯特模形式构造之间的兼容性，以及非同余子群上某些模形式的有界分母猜想（与 W. Li 和 L. Long 联合）。这项研究属于数论领域，这是数学最古老的分支之一，涉及整数问题。自欧几里得时代以来，数论研究一直活跃且富有成果，但困难重重。本提案中描述的研究旨在通过一些分析、代数和几何工具，在关注相对于固定素数的属性时，了解多项式系统整数解的行为和属性。密码学和编码理论的最新进展依赖于此类问题的解决方案。