Arithmetic Moduli at Infinite Level

朗兰兹纲领致力于将伽罗瓦表示与自守形式联系起来。朗兰兹纲领的核心是算术模的研究，也就是在局部域或全局域上定义的几何对象的参数空间。算术模在全局环境下包括模曲线和Shimura簇，在局部环境下包括Lubin-Tate塔和Rapoport-Zink空间。这样的几何对象总是结构在塔，如塔的模曲线的水平一个权力的p.作为一个整体，塔的算术模承认一个行动的约化群，并研究这一行动的上同调塔是唯一的方式，我们知道如何附加伽罗瓦表示的自守形式。本课题研究无限水平上的算术模，如沿着模曲线塔的逆极限。PI最近的一个发现是，在Lubin-Tate塔的情况下，这样的极限作为Peter Scholze的新类别的完美空间中的对象存在。逆极限对象实际上比塔的组成层简单的方式有很多。PI打算将这些发现推广到一般的算术模。这些结果将被用于朗兰兹计划的新见解。特别是这个建议代表了最有希望证明GL（n）的局部朗兰兹对应，它在性质上是纯粹局部的（即，该提案包括波士顿大学研究生和本科生参与研究的计划，并为每个人提供适当的项目。PI打算继续参与在广泛的社区传播数学的计划。这些计划包括PROMYS计划，一个位于波士顿大学的高中生数论暑期课程，以及亚利桑那州冬季学校，一个在图森举行的研究生密集迷你课程。PI还打算在会议上介绍研究成果，包括AMS-MAA的联席会议。

The Langlands program endeavors to link Galois representations to automorphic forms. Central to the Langlands program is the study of arithmetic moduli, which is to say parameter spaces for geometric objects defined over a local or global field. Arithmetic moduli include modular curves and Shimura varieties in the global setting, and the Lubin-Tate tower and spaces of Rapoport-Zink in the local setting. Such geometric objects are always structured in towers, such as the tower of modular curves of level a power of p. Taken as a whole, a tower of arithmetic moduli admits an action of a reductive group, and studying this action on the cohomology of the tower is the only way we know how to attach a Galois representation to an automorphic form. This project concerns arithmetic moduli at infinite level, e.g. the inverse limit along a tower of modular curves. A recent discovery of the PI is that, in the case of the Lubin-Tate tower, such limits exist as objects in Peter Scholze's new category of perfectoid spaces. There are many ways in which the inverse limit object is actually simpler than the constituent layers of the tower. The PI intends to generalize these discoveries to general arithmetic moduli. These results will be leveraged into new insights in the Langlands program. In particular this proposal represents the most promising hope yet for a proof of the local Langlands correspondence for GL(n) which is purely local in nature (i.e., which does not involve automorphic representations).This proposal includes a plan for the research-level participation of graduate and undergraduate students at Boston University, with appropriate projects for each. The PI intends to continue his involvement in programs which disseminate mathematics throughout a broad community. These programs include the PROMYS program, a number theory summer program for high school students located in Boston University, and the Arizona Winter School, an intensive mini-course for graduate students which takes place in Tucson. The PI also intends to present research at conferences, including the joint meetings of the AMS-MAA.