Arithmetic Moduli at Infinite Level

无限级算术模数

• 批准号: 1303312
• 负责人: Jared Weinstein
• 金额: $ 14.3万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303312&HistoricalAwards=false
• 关键词: Arithmetic; Moduli; Infinite; Level

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.

朗兰兹纲领试图将伽罗瓦表示与自同构形式联系起来。朗兰兹计划的核心是对算术模的研究，也就是在局部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变量，在局部环境下包括Lubin-Tate塔和Rapoport-Zink的空间。这样的几何对象总是用塔来构造的，例如水平为a的p次幂的模曲线塔。作为一个整体，一个算术模塔允许一个约化群的作用，研究这个约化群在塔的上同调上的作用是我们所知道的将伽罗瓦表示附加到自同构形式上的唯一方法。这个项目涉及无限级的算术模，例如沿模曲线塔的逆极限。PI最近的一个发现是，在Lubin-Tate塔的例子中，这样的极限作为对象存在于Peter Scholze的完美空间的新类别中。在很多方面，逆限制对象实际上比塔的组成层更简单。PI打算将这些发现推广到一般的算术模。这些结果将成为朗兰兹计划的新见解。特别地，这个建议代表了对GL(n)的局部朗兰兹对应的最有希望的证明，它在本质上是纯粹局部的（即，它不涉及自同构表示）。该提案包括波士顿大学研究生和本科生参与研究水平的计划，并为每个学生提供适当的项目。PI打算继续参与在广泛的社区中传播数学的项目。这些项目包括PROMYS项目，一个在波士顿大学为高中生开设的数论暑期项目，以及亚利桑那冬季学校，一个在图森为研究生开设的密集迷你课程。PI还打算在会议上介绍研究成果，包括AMS-MAA的联席会议。