Shimura Varieties, p-Adic Shtukas, and Local Systems

志村品种、p-Adic Shtukas 和本地系统

基本信息

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

项目摘要

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。这是一门融合了两个最古老的数学领域的学科：可以用最简单的方程（即多项式）描述的形状几何学和数字研究。事实证明，这种学科的结合取得了非凡的成果——解决了几代人都无法解决的问题（例如“费马大定理”）。一般领域与物理学有联系，并且在纠错码和密码学的构造中找到了重要的应用。 PI 的工作主要集中于研究描述具有多种对称性的形状的特定方程以及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一些项目。PI 正在努力描述 Shimura 簇在非平滑还原素数处的积分模型并研究相关空间。特别是，他将继续研究此类素数的阿贝尔型志村簇的奇点。他计划使用 p-adic shtukas 的新颖理论来表征这些积分模型，并在正交 Shimura 簇的情况下，明确研究其还原的局部结构。他还想将 Shimura 簇解释为“算术 shtukas”的更一般模空间的特例，并将 Shimura 簇的特殊点概念推广到此类模空间。最后，通过与数学物理学中出现的黎曼曲面束模理论的类比，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 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。