Research Proposal on Arithmetic Geometry

算术几何研究计划

• 批准号: 0100441
• 负责人: Ching-Li Chai
• 金额: $ 11.33万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100441&HistoricalAwards=false
• 关键词: Research; Proposal; Arithmetic; Geometry

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.

该项目属于算术代数几何领域，包含两部分：Neron模型和志村变异几何。第一部分的重点是使用Neron模型定义的称为基极变化导体的数值不变量。这个数值不变量度量了半abel变量在有限基扩展前后的Neron模型之间的差异，从而使半abel变量获得半稳定约简。对于数场上的阿贝尔变化，基极变化导体等于稳定状态下法尔廷高度的减小。最近E. de Shalit， J.-K.。Yu和Chai利用他们发现的Neron模型的同余性质，证明了环面的基变换导体等于Artin导体的一半。柴将这一同余性质推广到阿贝尔变体。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耳变量的基变导体等于伽罗瓦群上的两个中心函数之间的配对：由Neron模型得到的形式环面特征群上的伽罗瓦表示的特征，以及为每一个有限伽罗瓦扩展定义的特定中心函数。这种特殊的中心函数在p进数域的环切扩展中有值；它可以被认为是“阿廷导体的平分线”因为这个函数和它的复共轭的和等于阿廷特征。(b)证明基变导体的可加性。(c)研究基变导体的初等因数。第二个项目以Hecke轨道问题和Oort的“叶状结构”为中心，以更好地减少志村品种。研究了具有常点的Shimura变约的“状态-线性”子变的概念。(d)验证在低秩情况下，关于每一个状态线性子变种都等于一个Shimura子变种的约简的猜想。(e)证明了在叶状结构中，一个素数到p Hecke轨道的Zariski闭包等于叶的Zariski闭包的Oort猜想的一些情形。这是一个在数学领域被称为“算术几何”的提议。在这门学科中，代数几何和数论的问题和技术相互融合，对这两个领域都有好处。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。对于多项式方程的系统，如果允许使用更一般的数字，其复杂程度就会下降，因为它变得更容易找到解。这个项目的第一部分研究了一个数值不变量，它测量了复杂性下降了多少。本课题的第二部分研究了一类非常特殊的多项式方程的对称性，称为志村变量，它在数论中非常重要。