Modularity and p-adic Langlands

翻译后摘要：模块化和p-进朗兰兹算术的最基本的对象之一是绝对伽罗瓦群的数域。一个丰富的来源表示这样的群体是p-adic上同调的代数簇。Fontaine-Mazur猜想精确地预言了哪些p-adic伽罗瓦表示应该以这种方式出现。什么是值得注意的猜想是，最微妙的条件，在其制定只涉及限制伽罗瓦表示分解群以上的素数p.在某些情况下，丰泰-马祖尔联合收割机结合他们的哲学与朗兰兹，并预测伽罗瓦表示来自模块化的特征形。最近的工作对这一猜想已给予新的推动力的连接与Breuil的p进朗兰兹计划，特别是代表所产生的更高的重量模块化的形式。该项目的目的是通过探索这种对应关系的新情况以及它们与几何和模块化应用的联系来追求这种联系。十多年前，怀尔斯证明了费马大定理。他通过将椭圆曲线与模形式联系起来做到了这一点。后者是复杂的功能，承认一个令人难以置信的大量对称性。怀尔斯的突破涉及使用p进伽罗瓦表示附加到椭圆曲线。他的结果可以被看作是一个特殊的情况下，一个更普遍的哲学由于方丹和马祖尔，其中预测，一定类的p进伽罗瓦表示总是产生于模形式。最近Breuil提出了一个p-adic模拟的朗兰兹对应。这是一个全新的成分，它似乎对模块化问题有很强的应用。该项目旨在发展Breuil的对应关系，并探索其与伽罗瓦表示的几何和模块性的关系。

Abstract: Modularity and p-adic Langlands One of the most fundamental objects of arithmetic is the absolute Galois group of a number field. A rich source of representations for such groups is the p-adic cohomology of algebraic varieties. The Fontaine-Mazur conjecture predicts precisely which p-adic Galois representations ought to arise in this way. What is remarkable about the conjecture is that the most subtle condition in its formulation involves only the restriction of the Galois representation to the decomposition groups of primes above p. In certain situations Fontaine-Mazur combine their philosophy with that of Langlands, and predict which Galois representations come from modular eigenforms. Recently work on this conjecture has been given new impetus by a connection with Breuil's p-adic Langlands program, especially for representations which arise from higher weight modular forms. The project's aim is to pursue this connection by exploring new cases of this correspondence as well as their connections to geometry and applications to modularity. A little over ten years ago Wiles proved Fermat's Last Theorem. He did this by relating elliptic curves to modular forms. The latter are complex functions which admit an incredibly large number of symmetries. Wiles' breakthrough involved the use of the p-adic Galois representation attached to an elliptic curve. His result can be viewed as a special case of a more general philosophy due to Fontaine and Mazur, which predicts that a certain class of p-adic Galois representations always arise from modular forms. Recently Breuil has proposed a p-adic analogue of the Langlands correspondence. This is a completely new ingredient which seems to have very strong applications to questions about modularity. The project aims to develop Breuil's correspondence and explore its relation to geometry and modularity of Galois representations.