On the p-adic Langlands program

数论是研究整数和方程在整数中的可解性的学科。它是数学中最古老的分支之一。一个特别著名的问题是1637年的费马最后定理，该定理指出，只要n至少为3，非零整数的两个n次方的和就不可能是非零整数的n次方。直到1994年左右，Wiles和Taylor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。这种深层次的联系是朗兰兹计划的一个特例，朗兰兹计划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-进推广，这在最近几年引起了许多人的兴趣。到目前为止，只有在最简单的情况下才能正确地理解它(维度2)。仅此一项就导致了以前似乎遥不可及的数论问题的解决。在所提出的工作中，我的目的是阐明n&gt；2的p-进的朗兰兹程序的n维情形。特别是，我建议研究在Shimura簇的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.