Arithmetic Geometry and Applications

算术几何及其应用

• 批准号: 1902158
• 负责人: Mark Kisin
• 金额: $ 39万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902158&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Applications

The award supports the principal investigator's research in arithmetic geometry. Arithmetic geometry is a branch of mathematics that studies integer solutions of polynomial equations and has played a central role in solving many outstanding problems in number theory, such as Fermat's Last Theorem and the Mordell conjecture concerning the number of rational points on a curve. The main objects of study in this research project are called "Shimura varieties," the study of which is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of far-reaching and influential conjecturesThe project concerns problems in and applications of the arithmetic of Shimura varieties, which are generalizations of the moduli space of abelian varieties. In the first part of the project the PI plans to study the structure of the cohomology of Shimura varieties, and the structure of their mod p points. Specifically, there is a conjecture, proved by the PI in some cases, that the isogeny class of every mod p point contains the reduction of a special point. These results can be used to study the Hasse-Weil zeta function of a Shimura variety, following a program of Langlands. Another part of the project aims to study an analogue of the Mumford-Tate conjecture in crystalline cohomology; this amounts to an algebraisation problem for certain formal cycles on a Shimura variety. Finally, the PI plans to apply techniques from the deformation theory of finite flag group schemes to study Hilbert's 13th problem, which asks for the minimal m for which the solution of a general polynomial of degree n can be written as a composite of functions of m variables.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.

该奖项支持首席研究员在算术几何方面的研究。算术几何是研究多项式方程整数解的数学分支，在解决数论中的许多突出问题，如费马大定理和关于曲线上有理点数目的莫德尔猜想等方面发挥了核心作用。本课题的主要研究对象是志村变数，它是代数几何、数论和表示理论的交叉领域，具有广泛的应用前景。本课题涉及志村变数算法中的问题和应用，志村变数是对阿贝尔变数模空间的推广。在项目的第一部分，PI计划研究志村变种的上同调结构及其模p点的结构。具体地说，有一个猜想，在某些情况下由PI证明，即每个mod p点的等同系类包含一个特殊点的约化。这些结果可用于研究志村品种的Hasse-Weil zeta函数，遵循朗兰兹程序。该项目的另一部分旨在研究晶体上同调中的Mumford-Tate猜想的类似物；这相当于志村变上某些形式环的代数化问题。最后，PI计划应用有限标志群格式的变形理论技术来研究希尔伯特第13问题，该问题要求最小的m，其中n次一般多项式的解可以写成m变量函数的复合。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。