CAREER: Algebraicity and Integral Models of Shimura Varieties

本项目涉及下村品种的研究。这些几何对象被定义为系数为有理数的多项式方程的解。下村品种在解决几个长期存在的猜想方面发挥了关键作用，包括莫德尔猜想。PI和他的合作者提出了寻找定义下村变种的整系数多项式方程的问题。这个问题是数论和算术几何研究的基础，并广泛应用于几个重要的和众所周知的猜想。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背景知识，开始研究该领域的研究问题。该项目还为本科生提供了研究问题的机会，也为研究生提供了论文问题的机会。PI将致力于研究积分模型和Shimura簇的p-进几何的基本问题。具体地说，PI和他的合作者将致力于研究特殊的Shimura变种的积分模型，并研究与Shimura变种上的p-ar超越有关的问题(包括Borel代数性定理的p-ar类似，以及与Shimura变种上的p-ar双代数性有关的问题)。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.