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.

Langlands计划努力将Galois代表与汽车形式联系起来。 Langlands计划的核心是对算术模量的研究，即在本地或全球领域定义的几何对象的参数空间。算术模量包括全球环境中的模块化曲线和shimura品种，以及在本地环境中的lubin-tate塔和Rapoport-Zink的空间。这样的几何物体始终以塔结构，例如p级的模块化曲线塔。从整体上讲，算术模量塔承认了一个还原群体的作用，并且研究了这一行动对塔的共同体，这是我们知道如何将Galois代表的唯一途径连接到汽车形式。该项目涉及无限级别的算术模量，例如沿模块化曲线塔的逆极限。 PI最近发现的是，在Lubin-Tate塔的情况下，这种限制是彼得·索尔兹（Peter Scholze）新的Perfectoid空间类别中的物体中存在的。逆极限对象实际上比塔的组成层更简单。 PI打算将这些发现概括为一般算术模量。这些结果将利用兰兰兹计划中的新见解。特别是，该提案代表了迄今为止纯粹本地人本地兰格兰的通信证明的最有希望的希望（即，这不涉及自动形态代表）。该提案包括研究研究生和本科学生的研究级别的计划，并提供适合每种研究的项目。 PI打算继续参与整个广泛社区的数学传播计划。这些计划包括Promys计划，这是一个针对位于波士顿大学的高中学生的数字理论夏季课程，以及亚利桑那冬季学校，这是在图森举行的针对研究生的密集的小型课程。 PI还打算在会议上进行研究，包括AMS-MAA的联合会议。