Reciprocity, functoriality and the p-adic Langlands programme

朗兰兹计划包含一个猜想和定理网络，在数学和物理的不同领域之间建立联系。朗兰兹计划的核心部分是朗兰兹互易性，它涉及非常不同类型的数学对象：一方面，“伽罗瓦表示”编码了代数方程的解的对称性，另一方面，“自同构形式”是非凡的函数，它满足常见三角函数正弦和余弦的周期性特性。在这些不同类型的数学对象之间建立桥梁，可以将表面上棘手的问题从一端传输到另一端，在那里它们变得可解。例如，寻找代数方程(“丢番图方程”)的整数解，例如费马最后定理，以及描述算术数据的统计行为(Sato-Tate猜想)。这一提议将创造新的工具来理解伽罗瓦表示、自同构形以及它们之间的桥梁。该提案的中心主题是调查和开发几何和数论之间的新互动。

The Langlands programme encompasses a web of conjectures and theorems establishing links between different areas of mathematics and physics.A central part of the Langlands programme is Langlands reciprocity, which relates very different kinds of mathematical objects: on the one hand, "Galois representations" which encode symmetries of solutions to algebraic equations, and on the other hand "automorphic forms" which are remarkable functions which satisfy periodicity properties generalising those of the familiar trigonometric functions sine and cosine.Building bridges between these different kinds of mathematical objects allows apparently intractable questions to be transported from one side to another, where they become solvable. Examples include finding integer solutions to algebraic equations ("Diophantine equations"), such as Fermat's last theorem, and describing the statistical behaviour of arithmetic data (the Sato-Tate conjecture). This proposal will create new tools to understand Galois representations, automorphic forms and the bridge between them. The central theme of the proposal is to investigate and exploit new interactions between geometry and number theory.