Serre-type conjectures and mod p Langlands correspondences

数论是研究整数和整数方程的可解性的学科。它是数学最古老的分支之一。一个特别著名的问题是费马大定理，它涉及某些简单方程的整数不可解性。直到15年前，通过在椭圆曲线（几何对象）和模形式（分析对象和对称性理论）之间建立深刻的联系，这个问题才得到解决。这种深层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命题组成，特别有助于解释数论中的许多现象。我的工作涉及朗兰兹纲领的某些方面，这些方面将模形式（及其推广）与伽罗瓦表示联系起来。在过去的十年里，人们对所谓的p-adic朗兰兹纲领产生了很大的兴趣。到目前为止，它只在最简单的情况下得到了正确的理解。仅这一点就导致了以前似乎遥不可及的数论问题的解决。在我的工作中，我的目标是阐明的“下一个”情况下的p-adic朗兰兹计划。

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, which concerns the insolvability in the integers of certain simple equations. It was only solved some 15 years ago, by establishing 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 some aspects of the Langlands program that link modular forms (and their generalisations) to Galois representations. In the last decade there has been a lot of interest in the so-called p-adic Langlands program. So far, it has only been properly understood in the simplest possible case. This alone has led to the solution of problems in number theory that previously seemed out of reach. In my work I aim to shed light on the "next" case of the p-adic Langlands program.