Arithmetic Volumes of Shimura Varieties

志村品种算术卷

• 批准号: 1801905
• 负责人: Benjamin Howard
• 金额: $ 24万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801905&HistoricalAwards=false
• 关键词: Arithmetic; Volumes; Shimura; Varieties

The Principal Investigator will study aspects of the theory of Shimura varieties. These are particular kinds of higher-dimensional surfaces whose rich geometry and arithmetic puts them among the central objects of study in modern mathematics. The goal is to find new relations between geometrically defined quantities such as volumes and intersection multiplicities, and quantities that come from other branches of mathematics, like representation theory, that have no a priori geometric meaning. In addition to their importance in pure mathematics, Shimura varieties play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are important tools for understanding elliptic curves and abelian varieties, which now play a major role in cryptography. More broadly, the project concerns the theory of numbers and arithmetic geometry. This is an area of research which has applications to cyber security, through cryptography, and to some aspects of coding theory. The primary goal of the Principal Investigator's research on Shimura varieties is to prove new relations between their geometric invariants and L-functions. For example, the Principal Investigator will use recent advances in the theory of Borcherds products and integral models to express the arithmetic volumes of Shimura varieties of unitary and orthogonal type as special values of L-functions. The Principal Investigator will also prove new examples of generalized Gross-Zagier formulas, for example by expressing the intersection multiplicities of Shimura curves embedded into the Siegel threefold in terms of the central derivative of a Langlands L-function attached to a cuspidal representation of GSp(4). Similar higher dimensional formulas are also expected.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

主要研究者将研究Shimura品种理论的各个方面。这些是高维表面的特殊种类，它们的丰富几何形状和算术使它们成为现代数学研究的中心对象。 目的是在几何定义的数量（例如体积和相交的多重性）中找到新的关系，以及来自其他数学分支（例如代表理论）的数量，它们没有先验的几何含义。 除了它们在纯数学中的重要性外，Shimura品种在Langlands计划中发挥了至关重要的作用，该计划对理论物理学引起了人们的兴趣，并且是理解椭圆曲线和Abelian品种的重要工具，该工具现在在密码学中起着重要作用。 更广泛地说，该项目涉及数字和算术几何理论。这是一个研究领域，通过密码学以及编码理论的某些方面适用于网络安全。 首席研究者对Shimura品种的研究的主要目标是证明其几何不变和L功能之间的新关系。 例如，主要研究者将使用Borcherds产品和积分模型理论的最新进展来表达单一和正交类型的Shimura品种的算术体积作为L功能的特殊值。 主要研究者还将证明普遍的Gross-Zagier公式的新示例，例如，通过表达嵌入Siegel曲线的相交曲线相交的多重性，该曲线嵌入了Siegel三倍的langlands l函数的中心衍生物，该范围附着在GSSP的cuscpidal代表上（4）。 该奖项也反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准来评估的。

项目成果

On the Ekedahl–Oort stratification of Shimuracurves

关于 Shimura 曲线的 EkedahlâOort 分层

• DOI: 10.2140/ant.2020.14.961
• 发表时间: 2020
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Howard, Benjamin

Linear invariance of intersections on unitary Rapoport–Zink spaces

酉 Rapoport-Zink 空间上交集的线性不变性

• DOI: 10.1515/forum-2019-0023
• 发表时间: 2019
• 期刊: Forum Mathematicum
• 影响因子: 0.8
• 作者: Howard, Benjamin

Arithmetic of Borcherds products

Borcherds 产品的算术

• DOI: 10.24033/ast.1128
• 发表时间: 2020
• 期刊: Asterisque
• 影响因子: 1.1
• 作者: Howard, Benjamin;Madapusi Pera, Keerthi
• 通讯作者: Madapusi Pera, Keerthi

Modularity if generating series of divisors on unitary Shimura varieties II: arithmetic applications

如果在酉 Shimura 簇上生成一系列除数，则模块化 II：算术应用

• DOI: 10.24033/ast.1127
• 发表时间: 2020
• 期刊: Asterisque
• 影响因子: 1.1
• 作者: Bruinier, Jan H.;Howard, Benjamin;Kudla, Stephen S.;Rapoport, Michael;Yang, Tonghai
• 通讯作者: Yang, Tonghai

Modularity of generating series of divisors on unitary Shimura varieties

酉志村簇上因数生成级数的模块化

• 发表时间: 2021
• 期刊: Astérisque
• 作者: J. Bruinier, B. Howard