Research Proposal on Arithmetic Geometry

这个项目是在算术代数几何领域，包括两个部分：Neron模型和志村品种的几何。 第一部分的重点是一个数值不变，称为基地变化导体，使用Neron模型定义。 这个数值不变量测量之间的差异Neron模型的半阿贝尔品种之前和之后，使有限的基础扩展，使半阿贝尔品种获得半稳定的减少。 对于数域上的阿贝尔簇，基变导体等于稳定化后的Faltings高度的减少。 最近E. de Shalit，J. K.余和柴证明，基地变化导体的环面是等于一半的阿廷导体，使用一个全等性质的Neron模型，他们发现。 Chai将这一同余性质推广到阿贝尔簇。 第一个项目的明确目标包括：（a）证明具有潜在普通约化的阿贝尔簇的基变化导体等于伽罗瓦群上两个中心函数之间的配对：从Neron模型获得的形式环面的特征标群上的伽罗瓦表示的特征标，以及为每个有限伽罗瓦扩张定义的特定中心函数。 这个特定的中心函数在p进数域的某些分圆扩展中具有值;它可以被认为是“阿丁导体的二等分”，因为这个函数与其复共轭的和等于阿丁特征标。 (b)证明了基变导体的可加性。 (c)研究基变导体的初等因子。 第二个项目是围绕Hecke轨道问题和Oort的“叶状结构”，以良好的减少志村品种。 本文研究了Shimura簇的常点约化的“Tate-linear”子簇的概念。 (d)证明在低秩情形下，每个Tate-线性子簇都等于一个Shimura子簇的约化。 (e)证明Oort猜想的几个例子，即素数到p的Hecke轨道的Zebriki闭包等于叶理结构中叶子的Zebriki闭包。这是数学领域中的一个建议，被称为“算术几何”。“在这个主题的问题和技术都代数几何和数论混合，以造福于这两个领域。 数论是数学最古老的分支。近年来，它已成为通信系统，数据传输和密码学等领域不可或缺的工具。 算术几何中的一个典型问题涉及多项式方程。 对于一个多项式方程组，如果允许使用更一般的数字，复杂度就会下降，因为它更容易找到解决方案。 这个项目的第一部分研究了一个数值不变量，它可以衡量复杂度下降了多少。这个项目的第二部分研究一类非常特殊的多项式方程的对称性，称为志村簇，这在数论中具有核心重要性。

This project is in the field of arithmetic algebraic geometry and contains two parts: Neron models and the geometry of Shimura varieties. The focus of the first part is a numerical invariant, called the base change conductor, defined using the Neron models. This numerical invariant measures the difference between the Neron models of a semiabelian variety before and after making a finite base extension so that the semiabelian variety acquires semistable reduction. For an abelian variety over a number field, the base change conductor is equal to the decrease of Faltings height under stabilization. Recently E. de Shalit, J.-K. Yu and Chai proved that the base change conductor for a torus is equal to one half of the Artin conductor, using a congruence property for Neron models they discovered. This congruence property has been extended to abelian varieties by Chai. The explicit goals of the first project include: (a) Prove that the base change conductor for an abelian variety with potentially ordinary reduction is equal to the pairing between two central functions on the Galois group: the character of the Galois representation on the character group of a formal torus obtained from the Neron model, and a specific central function defined for every finite Galois extension. This specific central function has values in some cyclotomic extension of the field of p-adic numbers; it can be thought of as a "bisection of the Artin conductor" because the sum of this function with its complex conjugate is equal to the Artin character. (b) Prove an additivity property of the base change conductor. (c) Study the elementary divisors of the base change conductor. The second project is centered around the Hecke orbit problem and Oort's "foliation structure" for good reductions of Shimura variety. A notion of "Tate-linear" subvarieties of reduction of Shimura varieties with ordinary points will be investigated. (d) Verify in lower-rank cases the conjecture that every Tate-linear subvarieties is equal to the reduction of a Shimura subvariety. (e) Prove some cases of Oort's conjecture that the Zariski closure of a prime-to-p Hecke orbit is equal to the Zariski closure of a leaf in the foliation structure.This is a proposal in the area of mathematics known to as "Arithmetic Geometry." In this subject the problems and techniques of both Algebraic Geometry and Number Theory intermingle, to the benefit of both areas. Number theory is the oldest branch of mathematics. In recent years it has become an indispensable tools in areas such as communication systems, data transmission, and cryptology. A typical problem in Arithmetic Geometry concerns polynommial equations. For a system of polynomial equations, the degree of complexity of drops if one is allowed to use more general numbers, because it becomes easier to find solutions. The first part of this project studies a numerical invariant which measures how much the complexity drops. The second part of this project studies the of symmetries of a very special class of polynomial equations, called Shimura varieties, which are of central importance in Number Theory.