Local to Global Compatibility, p-adic Local Langlands and p-adic Level Lowering/Raising

局部到全局兼容性、p-adic 局部 Langlands 和 p-adic 电平降低/升高

• 批准号: 0901603
• 负责人: Robert Coleman
• 金额: $ 5.27万
• 依托单位: University of California-Berkeley
• 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 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901603&HistoricalAwards=false
• 关键词: Local; Global; Compatibility; adic; Langlands

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."Coleman and Paulin first intend to study the consequences of the failure of local to global compatibility (away from p) on the eigencurve. Its failure should be intimately related to the local geometry of the eigencurve. In particular it's failure should correspond to crossings between special and principal series components. This is a a p-adic geometric analogue of Ribet's level raising and level lowering theorems. As a second project they intend to establish a local to global principle at p by developing the theory of p-adic modular forms using Fontaine's notion of (phi, gamma)- modules and Colmez's p-adic local langlands correspondence. This may help to suggest a more conceptual definition for a p-adic automorphic representation, at least for GL2.The classical Langlands philosophy suggests that there is a profound duality between real analysis, arithmetic and geometry. The glue binding these different mathematical realms is representation theory. Langlands original insight was that there should be a natural corespondence between arithmetic data (Galois representations) and real analytic data (automorphic representations), mediated in many cases by geometry. The galois representations coming from geometry and therefore conjecturally automorphic have strong restrictions - there are many which cannot be geometric. In recent years it has become apparent that such representations should correspond to some p- adic analogue of an automorphic representation. Coleman and Paulin intend to study this new p-adic correspondence in the case of modular forms. This will help to suggest a more conceptual approach to the rapidly emerging p-adic langlands philosophy.

“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。“科尔曼和保林首先打算研究局部到全局相容性（远离p）在特征曲线上失败的后果。 它的失效应该与特征曲线的局部几何形状密切相关。 特别是它的故障应对应于特殊和主要系列组件之间的交叉。 这是一个p-adic几何模拟Ribet的水平提高和水平降低定理。 作为第二个项目，他们打算建立一个本地到全球的原则，在p发展理论的p-adic模块形式使用方丹的概念（phi，伽玛）-模块和科尔梅兹的p-adic本地langlands对应。 这可能有助于为p进自守表示提出一个更概念化的定义，至少对GL 2来说是这样。经典的朗兰兹哲学认为，在真实的分析、算术和几何之间存在着深刻的二元性。 将这些不同的数学领域粘合在一起的是表示论。 朗兰兹最初的见解是，算术数据（伽罗瓦表示）和真实的分析数据（自守表示）之间应该有一种自然的对应关系，在许多情况下以几何为媒介。 伽罗瓦表示来自几何，因此是几何自守的，有很强的限制-有许多不可能是几何的。 近年来，很明显，这样的表示应该对应于自守表示的某种p进模拟。科尔曼和Paulin打算在模形式的情况下研究这种新的p-adic对应。 这将有助于提出一个更概念化的方法，迅速新兴的p-adic langlands哲学。