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

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“p-adic霍奇理论的方法已经成为研究由p-adic域上的代数簇产生的伽罗瓦表示的有力工具。该计划研究积分p-adic霍奇理论，这是与半稳定伽罗瓦表示相关的积分和扭转结构的研究，以及它们如何与自守形式和代数簇的各种积分和扭转p-adic上同调群相关。在这个建议的第一个目标是提供新的分类格在半稳定表示和扭转半稳定表示，并研究不变量（如Weil-Deligne表示）附加到他们。第二个目标是利用这些结果来研究与自守形式相关的伽罗瓦表示。这类应用的例子是证明Hilbert模形式的局部Langlands对应和方丹构造之间的相容性，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.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。从欧几里得时代起，数论的研究就一直很活跃，成果颇丰，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，了解多项式系统的整数解的行为和性质，重点是相对于固定素数的性质。密码学和编码理论的最新进展依赖于这类问题的解决方案。

"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.