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

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”Coleman和Paulin首先打算研究特征曲线上局部与全局兼容性（远离p点）失败的后果。它的失效应该与特征曲线的局部几何密切相关。特别地，它的失效应该对应于特殊和主级数分量之间的交叉。这是Ribet的升能级定理和降能级定理的p进几何类比。作为第二个项目，他们打算利用Fontaine的(phi, gamma)-模块概念和Colmez的p进局部朗兰兹对应，通过发展p进模形式的理论，在p点建立局部到全局的原理。这可能有助于为p进自同构表示提出一个更概念化的定义，至少对于GL2是这样。古典朗兰兹哲学认为，在实际分析、算术和几何之间存在着深刻的对偶性。连接这些不同数学领域的粘合剂是表征理论。朗兰兹最初的见解是算术数据（伽罗瓦表示）和真实分析数据（自同构表示）之间应该有一种自然的对应关系，在许多情况下由几何来调解。伽罗瓦表示来自几何，因此猜想自同构有很强的限制-有许多不能是几何的。近年来，这种表示应该对应于自同构表示的一些p进模拟，这一点已经变得很明显。Coleman和Paulin打算在模形式的情况下研究这种新的p进对应。这将有助于为迅速兴起的p进朗兰兹哲学提出一个更概念化的方法。

"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.