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.

首席研究员将研究志村品种理论的各个方面。这些是特殊类型的高维表面，其丰富的几何和算术使它们成为现代数学研究的中心对象。 目标是找到几何定义的量（例如体积和交叉重数）与来自其他数学分支（例如表示论）的没有先验几何意义的量之间的新关系。 除了在纯数学中的重要性之外，志村簇在朗兰兹纲领中也发挥着重要作用，该纲领在理论物理学中越来越受到人们的关注，并且是理解椭圆曲线和阿贝尔簇的重要工具，而椭圆曲线和阿贝尔簇现在在密码学中发挥着重要作用。 更广泛地说，该项目涉及数论和算术几何。这是一个通过密码学以及编码理论的某些方面应用于网络安全的研究领域。 首席研究员对 Shimura 簇的研究的主要目标是证明它们的几何不变量和 L 函数之间的新关系。 例如，首席研究员将利用 Borcherds 乘积理论和积分模型的最新进展，将酉和正交型 Shimura 簇的算术体积表示为 L 函数的特殊值。 首席研究员还将证明广义 Gross-Zagier 公式的新示例，例如通过附加到 GSp(4) 尖峰表示的 Langlands L 函数的中心导数来表达嵌入到 Siegel 三重中的 Shimura 曲线的交集重数。 类似的更高维度的公式也在预期之中。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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

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

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 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