CAREER: Algebraicity and Integral Models of Shimura Varieties

职业：志村品种的代数性和积分模型

• 批准号: 2338942
• 负责人: Ananth Shankar
• 金额: $ 49.43万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2338942&HistoricalAwards=false
• 关键词: CAREER; Algebraicity; Integral; Models; Shimura

This project concerns the study of Shimura varieties. These are geometric objects that are defined as solutions to polynomial equations with coefficients that are rational numbers. Shimura varieties have played a crucial role in settling several long standing conjectures, including the Mordel Conjecture. The PI and his collaborators propose to work on the question of finding polynomial equations with integer coefficients which define Shimura varieties. This question is fundamental to the study of Number Theory and Arithmetic Geometry and has broad applications to several important and well-known conjectures. The educational component of the project includes a workshop targeted at early-stage graduate students looking to work in Arithmetic Geometry, aimed at helping these students acquire background to start working on research problems in this field. The project also provides opportunities for undergraduate students to work on research problems, as well as thesis-problems for graduate students. The PI will work on the fundamental problems of studying integral models and the p-adic geometry of Shimura varieties. Specifically, the PI and his collaborators will work on studying integral models of exceptional Shimura varieties, and studying questions pertaining to p-adic transcendence on Shimura varieties (including a p-adic analogue of Borel's algebraicity theorem, and questions pertaining to p-adic bi-algebraicity on Shimura varieties).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.

该项目涉及志村品种的研究。这些几何对象被定义为系数为有理数的多项式方程的解。志村簇在解决几个长期存在的猜想（包括莫德尔猜想）方面发挥了至关重要的作用。 PI 和他的合作者提议研究寻找具有定义 Shimura 簇的整数系数的多项式方程的问题。这个问题是数论和算术几何研究的基础，并且在几个重要且众所周知的猜想中具有广泛的应用。该项目的教育部分包括一个针对希望从事算术几何工作的早期研究生的研讨会，旨在帮助这些学生获得开始解决该领域研究问题的背景。该项目还为本科生提供研究问题的机会，并为研究生提供论文问题的机会。 PI 将致力于研究 Shimura 簇的积分模型和 p 进几何的基本问题。具体来说，PI 和他的合作者将致力于研究特殊 Shimura 品种的积分模型，并研究与 Shimura 品种的 p 进超越有关的问题（包括 Borel 代数性定理的 p 进模拟，以及与 Shimura 品种的 p 进双代数有关的问题）。该奖项反映了 NSF 的法定 使命，并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。