Mathematical Sciences: Diophantine Approximations of Algebraic Points

数学科学：代数点的丢番图近似

• 批准号: 9532018
• 负责人: Paul Vojta
• 金额: $ 13.89万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9532018&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Diophantine; Approximations; Algebraic

This project deals with diophantine approximation and value distribution theory of holomorphic functions. Work by P. Vojta, C. Osgood, and others has shown that these two fields are in fact connected by a formal analogy. This observation has led to new conjectures, shorter proofs of old theorems, and proofs of new theorems. This project will continue this trend, primarily working in the field of diophantine approximation. Results whose proofs may be simplified include finiteness theorems for rational points on abelian varieties and Faltings' theorem on the finiteness of principally polarized abelian varieties over a given field with finite reduction. New results which may be proved include "moving targets" type results; i.e., diophantine approximation statements which involve a slowly varying family of equations. A distant goal is the "abc conjecture." Technical improvements in proofs using Thue's method will also be pursued. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

本项目主要研究丢番图逼近和全纯函数的值分布理论。作者：P. Vojta，C.奥斯古德和其他人已经表明，这两个领域实际上是由一个正式的类比。这一观察导致了新的证明，旧定理的较短证明，以及新定理的证明。这个项目将继续这一趋势，主要工作在丢番图近似领域。其证明可以简化的结果包括阿贝尔簇上有理点的有限性定理和关于给定域上有限约化的主极化阿贝尔簇的有限性的Faltings定理。可以证明的新结果包括“移动目标”类型的结果;即，丢番图近似语句，涉及一个缓慢变化的方程族。一个遥远的目标是“abc猜想”。“使用Thue方法的证明的技术改进也将被追求。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在各种 在诸如数据传输和处理以及通信系统的领域中的应用。