Arithmetic Algebraic Geometry

摘要奖DMS-0400482“算术代数几何“的蔡庆立这个项目是在算术代数几何领域，包括两个部分：几何志村簇和Neron模型的半阿贝尔簇。 这个项目的第一部分是围绕着Hecke轨道猜想的Shimura品种，由Oort制定，其中规定，一个素数到p的Hecke轨道的Zebraki闭包等于一个叶子的Zebraki闭包。部分是与他的合作者，F。Oort和C.- F. Yu，Chai发展了几种方法来实现Hecke轨道猜想，并制定了一个计划来证明阿贝尔簇的模空间的Hecke轨道猜想。 计划的最后一步是由C完成的。F.余，和一个大纲的证明Hecke轨道猜想的模空间的交换品种是可用的。 本提案的一个主要目标是详细阐述该证明，以及为未来应用进一步开发该方法。还将探讨的是Hecke轨道问题志村品种重视酉群。 第二部分的重点是一个数值不变量的semiabelian品种在当地的领域，称为基地变化导体，使用Neron模型定义。这里的目标是了解基变化导体的行为时，剩余场是不完善的，并探讨刚性解析空间的形式Neron模型的基本性质。本项目的第一部分研究的是一类非常特殊的多项式方程的对称性。这类多项式方程，称为志村变种，在数论中具有核心重要性。推测，这些对称性特征的结构，称为“叶理”，志村品种。 该建议是发展和文件最近构思证明这一猜想。本项目的第二部分涉及多项式方程组的另一个方面，当允许使用更一般的数字来解决问题时，方程组的复杂程度会下降。 这个项目预计将提高我们在数论，一个主题，虽然在过去被认为是柏拉图式的和纯粹的知识，在数字时代已经发现了大量的应用程序。

Abstract for Award DMS-0400482 "Arithmetic algebraic Geometry"by Ching-Li ChaiThis project is in the field of arithmetic algebraic geometry and contains two parts: geometry of Shimura varieties and Neron models of semiabelian varieties. The first part of this project is centered around the Hecke orbit conjecture for good reductions of Shimura variety, formulated by Oort, which states that the Zariski closure of a prime-to-p Hecke orbit is equal to the Zariski closure of a leaf. Partly in collaboration with his collaborators, F. Oort and C.-F. Yu, Chai has developed several techniques toward the Hecke orbit conjecture, and also formulated a plan to prove the Hecke orbit conjecture for the moduli space of abelian varieties. The last step of the plan was finished by C.-F. Yu, and an outline of a proof of the Hecke orbit conjecture for the moduli space of abelian varieties is available. A main objective of this proposal is a detailed exposition of that proof, as well as further development of the method for future applications. Also will be explored is the Hecke orbit problem for Shimura varieties attached to unitary groups. The focus of the second part is a numerical invariant of semiabelian varieties over local fields, called the base change conductor, defined using the Neron models. The goal here is to understand the behavior of the base change conductor when the residue field is not perfect, and to explore the foundational properties of formal Neron models of rigid analytic spaces.The first part of this project studies the of symmetries on a very special class of polynomial equations. This class of polynomial equations, called Shimura varieties, are of central importance in Number Theory. Conjecturally, these symmetries characterize a structure, called "foliation", on Shimura varieties. The proposal is to develop and document a recently conceived proof of this conjecture. The second part of this project deals with another aspect of system of polynomial equations, on the drop of the level of complexity of a system of equation when more general numbers are allowed to be used for solutions. This project is expected to enhance our knowledge in number theory, a subject which, though considered platonic and pure in the past, has found a plethora of applications in the digital age.