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

这个奖项是根据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球兼容失败(远离p)对特征曲线的影响。它的失效应该与特征曲线的局部几何密切相关。特别是，它的失效应该对应于特殊串联元件和主串联元件之间的交叉。这是Ribet水平提升和水平下降定理的p-进几何类比。作为第二个项目，他们打算通过利用Fontaine的(Phi，Gamma)-模的概念和Colmez的p-进的局部朗兰兹对应来发展P-进制模形式的理论，从而在p处建立局部到全局的原理。这可能有助于提出一个更概念性的定义，至少对于GL2而言，p-进自同构表示。经典的朗兰兹哲学认为，实分析、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捆绑在一起的粘合剂是表示理论。朗兰兹最初的见解是，在算术数据(伽罗瓦表示法)和真实分析数据(自同构表示法)之间应该有一种自然的对应关系，在许多情况下是通过几何来调节的。伽罗瓦表示来自几何，因此猜想自同构有很强的限制--有许多不是几何的。近年来，很明显，这样的表示应该对应于自同构表示的某种p-进模拟。Coleman和Paulin打算在模形式的情况下研究这种新的p-进对应。这将有助于提出一种更具概念性的方法来看待迅速兴起的p-adic朗兰兹哲学。

"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.