Shimura Varieties, p-Adic Shtukas, and Local Systems

志村品种、p-Adic Shtukas 和本地系统

• 批准号: 2100743
• 负责人: Georgios Pappas
• 金额: $ 25万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-05-01 至 2025-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100743&HistoricalAwards=false
• 关键词: Shimura; Varieties; Adic; Shtukas; Local

The PI will conduct research in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest areas of mathematics: The geometry of shapes that can be described by the simplest equations, namely polynomials, and the study of numbers. This combination of disciplines has proved extraordinarily fruitful - having solved problems that withstood generations (such as "Fermat's last theorem"). The general field has connections with physics, and has found important applications to the construction of error correcting codes and cryptography. The PI's work mainly concentrates on the study of specific equations which describe shapes with many symmetries and on connections of the subject with certain constructions in mathematical physics. The PI plans to involve graduate students in some of the projects.The PI is working to describe integral models for Shimura varieties at primes of non-smooth reduction and study related spaces. In particular, he will continue to investigate the singularities of Shimura varieties of abelian type at such primes. He plans to characterize these integral models by using the novel theory of p-adic shtukas and, in the case of orthogonal Shimura varieties, explicitly study the local structure of their reductions. He would also like to interpret Shimura varieties as special cases of more general moduli spaces of "arithmetic shtukas" and to generalize the concept of special points of Shimura varieties to such moduli spaces. Finally, motivated by an analogy with the theory of moduli of bundles over Riemann surfaces as it appears in mathematical physics, the PI will investigate symplectic properties of deformation spaces of local systems and Galois representations.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.

PI将在算术代数几何领域进行研究。这门学科融合了数学中两个最古老的领域：可以用最简单的方程（即多项式）来描述形状的几何，以及对数字的研究。这些学科的结合被证明是非常富有成效的——解决了几代人经受住考验的问题（比如“费马大定理”）。一般领域与物理学有联系，并在纠错码和密码学的构建中发现了重要的应用。PI的工作主要集中在研究描述具有许多对称性的形状的特定方程，以及研究该学科与数学物理中某些结构的联系。PI计划让研究生参与其中的一些项目。PI的工作是描述非光滑约简素数下Shimura变量的积分模型并研究相关空间。特别是，他将继续研究阿贝尔型的志村变体在这些素数处的奇异性。他计划用新颖的p进shtukas理论来描述这些积分模型，并且在正交Shimura变量的情况下，明确地研究它们约简的局部结构。他还想将志村变数解释为“算术shtukas”的更一般模空间的特例，并将志村变数的特殊点的概念推广到这样的模空间。最后，在与数学物理中出现的黎曼曲面上束的模理论的类比的激励下，PI将研究局部系统和伽罗瓦表示的变形空间的辛性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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

Regular integral models for Shimura varieties of orthogonal type

正交型Shimura品种的正则积分模型

• DOI: 10.1112/s0010437x22007370
• 发表时间: 2022
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Pappas, G.;Zachos, I.
• 通讯作者: Zachos, I.

Volume and symplectic structure for ℓ-adic local systems

α-adic 局部系统的体积和辛结构

• DOI: 10.1016/j.aim.2021.107836
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Pappas, Georgios

On integral models of Shimura varieties

志村品种的积分模型

• DOI: 10.1007/s00208-022-02387-8
• 发表时间: 2022
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Pappas, Georgios