Diophantine approximation and Nevanlinna theory

丢番图近似和 Nevanlinna 理论

• 批准号: 0901149
• 负责人: Paul Vojta
• 金额: $ 28.78万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901149&HistoricalAwards=false
• 关键词: Diophantine; approximation; Nevanlinna; theory

The analogy between number theory and Nevanlinna theory has led to much interplay between the two fields in both directions, but at its most fundamental level is not well understood. In particular, the use of the derivative of a holomorphic function in Nevanlinna theory stands out as something with no known counterpart in number theory. The recent "tautological inequality" of M. McQuillan, however, is of a form that can be translated into number theory, leading to a conjecture that the present project will investigate. The project includes work on trying to prove this conjecture, starting with special cases stemming from the Subspace Theorem of W. M. Schmidt, and from Faltings' work on closed subvarieties of abelian varieties. In addition, it will further study the extent to which McQuillan's inequality can serve as a gateway to the main theorems of Nevanlinna theory. As a separate but related project, the Principal Investigator will also continue work on completed invariant jet spaces in the arithmetical context.The Thue-Siegel-Roth method in number theory is a method for showing that certain types of diophantine equations have only finitely many solutions, or at least showing that their families of solutions obey additional equations. It made its debut 100 years ago this year, but it has been gaining momentum in the last 20 years, due in part to similarities with Nevanlinna theory. The latter is a part of complex analysis, encompassing methods for showing that meromorphic functions having certain properties do not exist, or more generally that in certain cases a non-constant holomorphic function from the complex plane to a complex algebraic manifold must satisfy additional equations.The similarities between these two fields have benefited both areas of mathematics, allowing ideas, conjectures, and methods to be carried over from one area to the other, in both directions. However, the fundamental reasons for these similarities are not at all understood at the present time. In particular, the derivative -- specifically the "lemma on the logarithmic derivative" -- plays a central role in Nevanlinna theory, but has no known counterpart in number theory. Based on a recent inequality of McQuillan in Nevanlinna theory, however, the proposer has a conjectural counterpart for this lemma in number theory. This grant will support work on this conjecture and its possible ramifications.

数论和内瓦林纳理论之间的类比导致了这两个领域在两个方向上的许多相互作用，但在其最基本的层面上还没有被很好地理解。特别是，在纳瓦林纳理论中使用全纯函数的导数是数论中没有已知的对应的东西。然而，M.McQuillan最近的“重言式不等式”是一种可以转化为数论的形式，导致了一个猜想，即本项目将进行调查。该项目包括试图证明这一猜想的工作，从源于W.M.施密特的子空间定理的特殊情况开始，以及法林斯关于阿贝尔变种的闭子变种的工作。此外，它还将进一步研究麦克奎兰不等式在多大程度上可以作为通向内瓦林纳理论主要定理的门户。作为一个独立但相关的项目，首席研究人员还将继续在算术上下文中研究完备的不变喷流空间。数论中的Thue-Siegel-Roth方法是一种方法，用于证明某些类型的丢番图方程只有有限多个解，或至少证明它们的解族服从附加方程。它在100年前的今年首次亮相，但在过去20年里，它的势头一直在增强，部分原因是它与纳瓦林纳理论的相似之处。后者是复分析的一部分，包括证明具有某些性质的亚纯函数不存在的方法，或者更广泛地说，在某些情况下，从复平面到复代数流形的非常数全纯函数一定满足附加方程。这两个领域之间的相似之处使数学的两个领域都受益，允许思想、猜想和方法从一个领域双向传播到另一个领域。然而，这些相似之处的根本原因目前还不清楚。特别是，导数--特别是“关于对数导数的引理”--在内瓦林纳理论中起着核心作用，但在数论中没有已知的对应物。然而，基于内瓦林纳理论中麦克奎兰最近的一个不等式，提出者对数论中的这个引理有一个猜想的对应物。这笔赠款将支持关于这一猜想及其可能的后果的工作。