Generalizations of Serre's Conjecture

这个项目研究了p进朗兰兹程序的一些mod p方面，以及它们在数论中更经典问题上的应用。正如朗兰兹纲领将几个研究领域（伽罗瓦表示、自守形式和局部域和整体域上的约化群的表示理论）联系起来一样，p-adic朗兰兹纲领研究伽罗瓦表示的变形、p-adic霍奇理论、约化群的p-adic表示理论和p-adic系数的上同调之间的关系。然而，p-adic程序只是在一个萌芽阶段，很少有一般的解释，更不用说定理了。这是很自然的考虑减少模p的p进对象中涉及的p进朗兰兹计划，当一个人这样做的第一个是导致问题涉及推广的重量部分塞尔猜想。 PI提出将这个猜想推广到任意约化群，并证明了一些特殊情况。PI还提议证明Sato-Tate猜想的第一个例子，其模的重量大于2。朗兰兹计划是数学的一个分支，包括数论（整数方程的研究），表示论和分析的思想。该计划的数论部分由一组庞大的相互关联的图表组成，这些图表将整数方程的解与其他显然无关的数学对象联系起来。最近，它已经变得很清楚，这一部分的朗兰兹计划应推广到什么是所谓的“p-adic朗兰兹计划”。这个新的程序还没有一个精确的数学公式，PI建议研究组成程序的数学公式的正确公式，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.