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Shimura品种的研究。这些是几何对象，被定义为具有有理数系数的多项式方程的解决方案。 Shimura品种在解决包括Mordel猜想在内的几个长期猜想中发挥了至关重要的作用。 PI及其合作者提议研究以整数系数来定义Shimura品种的多项式方程的问题。这个问题是数字理论和算术几何形状的研究至关重要的，并且在几种重要且众所周知的猜想中具有广泛的应用。该项目的教育组成部分包括一个针对希望从事算术几何学工作的早期研究生的研讨会，旨在帮助这些学生获得背景，以开始在该领域的研究问题。该项目还为本科生提供了解决研究问题的机会，以及研究生的论文问题。 PI将致力于研究积分模型和Shimura品种的P-Adic几何形状的基本问题。具体而言，PI及其合作者将努力研究异常的Shimura品种的整体模型，并研究与Shimura品种的P-AdiC超越性有关的问题（包括Borel代数定理的P-AdiC类似物，以及与P-Adic Biadic Bi Bi Bi Bi的问题 - 关于Shimura品种的代数）。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估的。