Special Fibers of Modular Varieties

模块化品种特种纤维

• 批准号: 1502147
• 负责人: Liang Xiao
• 金额: $ 13.5万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502147&HistoricalAwards=false
• 关键词: Special; Fibers; Modular; Varieties

The Langlands program aims at establishing deep relations between number theory (arithmetic information about integers, for example their factorization into products of primes) and harmonic analysis (for example, harmonic functions on certain space with additional symmetry). Such relations are often established through the understanding of certain spaces, called the modular varieties. In particular, the geometry of these spaces encodes fruitful arithmetic data that will help solve problems in both number theory and analysis. In this project, one main objective is to give an explicit characterization of certain Newton strata of the special fiber of the modular varieties, including the basic stratum (in some cases). For one application, the Principal Investigator will seek to verify the Tate conjecture for the special fiber of modular varieties under some genericity condition. The explicit description is tied to the Taylor-Wiles modularity lifting technique for coherent cohomology of Hilbert modular varieties of low weight. It is also tied to the Euler system method that gives rise to results relating Selmer groups and the vanishing of p-adic L-functions.

朗兰兹计划旨在建立数论（关于整数的算术信息，例如它们被分解成素数的乘积）和调和分析（例如，具有额外对称性的特定空间上的调和函数）之间的深层关系。这种关系通常是通过对某些空间的理解而建立起来的，这些空间被称为模变体。特别是，这些空间的几何编码了富有成效的算术数据，这将有助于解决数论和分析中的问题。在这个项目中，一个主要目标是给出特定牛顿层的特殊纤维的模块化品种的明确特征，包括基本层（在某些情况下）。对于一个应用程序，首席研究员将寻求在某些一般条件下验证模变体特殊纤维的Tate猜想。明确描述了低权重Hilbert模变体的相干上同调的Taylor-Wiles模提升技术。它还与欧拉系统方法联系在一起，欧拉系统方法产生了与Selmer群和p进l函数消失有关的结果。

项目成果

Tate cycles on some quaternionic Shimura varieties mod $p$

一些四元 Shimura 品种的泰特循环 mod $p$

• DOI: 10.1215/00127094-2018-0068
• 发表时间: 2019
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Tian, Yichao;Xiao, Liang
• 通讯作者: Xiao, Liang