On the p-adic Langlands program

数论研究的是整数和整数方程的可解性。它是数学最古老的分支之一。一个特别著名的问题是1637年的费马大定理，它说非零整数的两个n次幂的和不可能是非零整数的n次幂，只要n至少等于3。直到1994年左右，怀尔斯和泰勒才解决了这个问题，他们在椭圆曲线（几何对象）和模形式（分析对象和对称理论）之间建立了深刻的联系。这种深刻的联系是朗兰兹纲领的一个特殊例子，朗兰兹纲领由许多非常普遍和相互关联的猜想组成，这些猜想特别有助于解释数论中的许多现象。******我的工作是关于朗兰兹计划的p进式推广，近年来吸引了很多人的兴趣。到目前为止，它只在最简单的情况下（维度2）得到了正确的理解。仅凭这一点，就解决了以前似乎遥不可及的数论问题。在提议的工作中，我的目标是阐明n b>2的p进朗兰兹程序的n维情况。特别地，我建议研究Shimura变量上同调中出现的全局候选表示，以及它们的局部解析向量。

Number theory is the study of whole numbers and the solvability of equations in whole numbers. It is one of the oldest branches of mathematics. One particularly famous problem is Fermat's Last Theorem from 1637, which says that the sum of two n-th powers of non-zero whole numbers cannot be the n-th power of a non-zero whole number, as soon as n is at least 3. It was only solved around 1994 by Wiles and Taylor who established a deep connection between elliptic curves (objects of geometry) and modular forms (objects of analysis and the theory of symmetries). This deep connection is a special instance of the Langlands Program, which consists of a number of very general and interlinked conjectures that, in particular, help to explain many phenomena in number theory.******My work concerns a p-adic generalisation of the Langlands program which has been attracting a lot of interest in recent years. So far, it has only been properly understood in the simplest possible case (dimension 2). This alone has led to the solution of problems in number theory that previously seemed out of reach. In the proposed work I aim to shed light on the n-dimensional case of the p-adic Langlands program for n > 2. In particular, I propose to study the global candidate representations that arise in the cohomology of Shimura varieties, as well as their locally analytic vectors.