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; 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将在算术代数几何学领域进行研究。这是一个融合了两个最古老的数学领域的主题：形状的几何形状可以用最简单的方程式（即多项式）和数字研究来描述。事实证明，这种学科的结合非常富有成果 - 解决了经受住几代人的问题（例如“ Fermat的最后一个定理”）。通用领域与物理有联系，并发现了纠正误差代码和密码学的构建重要应用。 PI的工作主要集中于研究特定方程的研究，这些方程描述了许多对称性的形状以及对象与数学物理学中某些构造的联系。 PI计划将研究生参与一些项目。PI正在努力描述Shimura品种不可或缺的模型，这是在非平滑降低和与研究相关的空间的数量中。特别是，他将继续研究这种素数的阿贝利亚类型的shimura品种的奇异性。他计划通过使用P-Adic Shtukas的新理论来表征这些整体模型，并且在正交Shimura品种的情况下，明确研究了其减少的局部结构。他还想将Shimura品种解释为“算术shtukas”更通用的模量空间的特殊情况，并将Shimura品种特殊点的特殊点概念推广到此类Moduli空间。最后，以与莱曼表面相比的模量理论的类比为动机，在数学物理学中出现时，PI将研究本地系统和Galois表示的变形空间的符合性属性。该奖项反映了NSF的法定任务，并通过使用基金会和广泛的范围来评估了NSF的法定任务，并通过评估范围进行了评估。