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.

本项目是关于志村品种的研究。这些几何对象被定义为系数为有理数的多项式方程的解。志村变种在解决几个长期存在的问题中发挥了至关重要的作用，包括莫德尔猜想。 PI和他的合作者建议工作的问题找到多项式方程的整数系数定义志村品种。这个问题是数论和算术几何研究的基础，并在几个重要而著名的数学中有广泛的应用。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讲习班，旨在帮助这些学生获得背景知识，开始研究这一领域的研究问题。该项目还为本科生提供了研究问题的机会，以及研究生的论文问题。PI将致力于研究积分模型和志村簇的p-adic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簇的积分模型，并研究与Shimura簇上的p-adic超越有关的问题（包括Borel代数性定理的p-adic模拟，和与Shimura簇上的p-adic双代数性有关的问题）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。