Arithmetic Algebraic Geometry

DMS-0400482“算术代数几何”奖摘要本项目属于算术代数几何领域，包含两个部分：Shimura簇的几何和半交换簇的Nelon模型。这个项目的第一部分是围绕由Oort提出的关于Shimura簇的良好约化的Hecke轨道猜想，该猜想指出素数到PHecke轨道的Zariski闭包等于叶的Zariski闭包。部分是与他的合作者F.Oort和C.F.Yu合作，Chai发展了几种针对Hecke轨道猜想的技术，并制定了一个计划来证明Abel簇的模空间的Hecke轨道猜想。计划的最后一步是由余正福完成的，给出了关于交换变种的模空间的Hecke轨道猜想的一个证明提纲。这项建议的一个主要目标是详细阐述这一证据，并为今后的应用进一步发展这一方法。此外，还将探讨隶属于酉群的Shimura变种的Hecke轨道问题。第二部分的重点是局域上半阿贝尔变元的一个数值不变量，称为基变导体，它是用农隆模型定义的。本文的目的是了解当残余场不完美时基变导体的行为，并探索刚性解析空间的形式Nelon模型的基本性质。本项目的第一部分研究了一类非常特殊的多项式方程的对称性。这类多项式方程，称为下村变种，在数论中占有重要地位。据推测，这些对称性表征了下村品种上的一种结构，称为“叶化”。这项提议是开发并记录最近构思的关于这一猜想的证据。这个项目的第二部分涉及多项式方程组的另一个方面，即当允许使用更一般的数字来求解时，降低方程组的复杂程度。这个项目有望增加我们在数论方面的知识，这是一个尽管在过去被认为是柏拉图式的纯粹的学科，但在数字时代发现了过多的应用。

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.