Moduli of Galois Representations

数论是数学最古老的分支之一。数论最基本的研究是多项式方程的整数解，特别是几何学中的方程。例如，一个直角三角形的三条边的长度是由勾股定理联系起来的。虽然找到所有边长为有理数的直角三角形是很简单的，但可能令人惊讶的是，确定哪些整数可以是具有有理数边的直角三角形的面积仍然是一个未解决的问题。数论有许多重要的实际应用。例如，大多数手机通话都受到基于椭圆曲线的密码的保护，这是朗兰兹计划的主要研究对象之一。近几十年来，在理解朗兰兹自守形式和伽罗瓦表示之间的互逆性方面取得了重大进展。用于证明互易定律的技术通常需要考虑伽罗瓦表示的p-adic族。将进行三个项目，以调查伽罗瓦表示的p-adic家庭。第一个项目将产生某些伽罗瓦表示的模栈，其中允许残差表示变化。这将打开证明结果的可能性，在朗兰兹计划建立他们一般在这些空间。第二个项目，关于Breuil的局部-全局相容性猜想的类型在p-adic朗兰兹计划，将是一个示范这种技术。第三个项目是在非一般情况下研究GL（n）的Serre猜想的权重部分，其中n大于2。

Number theory is one of the oldest branches of mathematics. At its most fundamental, number theory is the study of whole number solutions to polynomial equations, and especially to equations motivated by geometry. For example, the lengths of the three sides of a right triangle are related by the Pythagorean theorem. While it is straightforward to find all right triangles whose side lengths are rational numbers, it perhaps surprisingly remains an unsolved problem to determine which whole numbers can be the area of a right triangle with rational sides. Number theory has many important practical applications. For instance, most cellular telephone calls are protected by a code based on elliptic curves, one of the primary objects of study in the Langlands program, the principal investigator's area of research.Significant progress has been made in recent decades towards understanding Langlands's conjectural reciprocity between automorphic forms and Galois representations. Techniques for proving reciprocity laws generally require the consideration of p-adic families of Galois representations. Three projects to investigate p-adic families of Galois representations will be undertaken. The first project will produce moduli stacks of certain Galois representations in which the residual representation is allowed to vary. This will open up the possibility of proving results in the Langlands program by establishing them generically on these spaces. The second project, on Breuil's local-global compatibility conjecture for types in the p-adic Langlands program, will be a demonstration of this technique. A third project is to study the weight part of Serre's conjecture for GL(n), for n larger than 2, in non-generic cases.