Generalizations of Serre's Conjecture

这个项目研究了p进朗兰兹程序的一些模p方面，以及它们在数论中更经典的问题中的应用。正如朗兰兹纲领涉及到几个研究领域（伽罗瓦表示、自同构形式、局部和全局域上约化群的表示理论），p进格朗兰兹纲领研究了伽罗瓦表示的变形、p进格霍奇理论、约化群的p进格表示理论和p进系数的上同调之间的关系。然而，p进规划只是处于萌芽阶段，很少有一般的猜想，更不用说定理了。考虑p进朗兰兹规划中涉及的p进对象的约简模p是很自然的，当我们这样做时，首先会导致涉及Serre猜想的权重部分的推广的问题。PI提出将这一猜想推广到任意约化群，并证明了一些特殊情况。PI还提出证明权大于2的模形式的Sato-Tate猜想的第一种情况。朗兰兹程序是数学的一个分支，它包括数论（研究整数方程）、表示理论和分析的思想。该程序的数论部分由大量相互关联的猜想组成，这些猜想将方程式的整数解与其他表面上不相关的数学对象联系起来。最近，很明显，朗兰兹纲领的这一部分应该概括为所谓的“p进朗兰兹纲领”。到目前为止，这个新程序还没有一个精确的推测公式，PI建议研究组成该程序的猜想的正确公式，证明它们的情况，并将它们应用于数论中更经典的问题。

This project investigates some of the mod p aspects of the p-adic Langlands program, and their applications to more classical problems in number theory. Just as the Langlands program relates several areas of study (Galois representations, automorphic forms, and the representation theory of reductive groups over local and global fields), the p-adic Langlands program studies the relations between deformations of Galois representations, p-adic Hodge theory, the p-adic representation theory of reductive groups, and cohomology with p-adic coefficients. However, the p-adic program is only in an embryonic stage, and there is very little by way of general conjectures, let alone theorems. It is natural to consider the reduction mod p of the p-adic objects involved in the p-adic Langlands program, and when one does so one is led firstly to questions involving generalizations of the weight part of Serre's conjecture. The PI proposes to generalize this conjecture to arbitrary reductive groups, and to prove some special cases. The PI also proposes to prove the first cases of the Sato-Tate conjecture for modular forms of weight greater than 2.The Langlands program is a branch of mathematics that includes ideas from number theory (the study of equations in whole numbers), representation theory and analysis. The number theoretic part of the program consists of a vast set of interlinked conjectures relating the solutions of equations in whole numbers to other apparently unrelated mathematical objects. Recently, it has become clear that this part of the Langlands program should generalise to what is known as a "p-adic Langlands program". This new program does not, as yet, have a precise conjectural formulation, and the PI proposes to investigate the correct formulations of the conjectures that should make up the program, to prove cases of them, and apply them to more classical problems in number theory.