Diophantine approximation, Nevanlinna theory, and jet differentials

丢番图近似、Nevanlinna 理论和射流微分

• 批准号: 0500512
• 负责人: Paul Vojta
• 金额: $ 10.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500512&HistoricalAwards=false
• 关键词: Diophantine; approximation; Nevanlinna; theory; jet

The investigator studies diophantine problems from the point of view ofthe analogy with the value distribution theory of holomorphic curves(Nevanlinna theory), and vice versa. Specifically, a substantialcomponent will consist of general work in algebraic geometry on thetheory of jet spaces. This work will then be applied to one or moreof the following topics: (i) work on finding a suitable generalization of Semple-Demailly quotient jet spaces to the general context of an arbitrary morphism of schemes, (ii) formulation of a theory of log jet spaces over log schemes, (iii) investigation of Yamanoi's proof of the "1+epsilon" conjecture, and (iv) a search for a number-theoretic counterpart to the lemma on the logarithmic derivative.This project involves number theory (specifically, solutions of polynomialequations in integers or rational numbers, as well as related inequalities)and value distribution theory (inequalities satisfied by power seriesfunctions of one or several complex variables). These two seeminglydisparate areas of mathematics have many phenomena in common. This hasbeen partly explained by a formal analogy between the two fields, whichhas led to new conjectures, shorter proofs of existing theorems, and newtheorems. The present project uses this analogy to further elucidatethe underlying structure of number theory, value distribution theory,and the analogy between the two fields.

从与全纯曲线的值分布理论(nevanlinna理论)类比的角度研究丢番图问题，反之亦然。具体地说，一个实质性的组成部分将包括关于喷流空间理论的代数几何的一般工作。这项工作随后将应用于下列一个或多个主题：(I)寻找半数商JET空间到任意方案态射的一般上下文的适当推广；(Ii)在对数方案上建立对数JET空间的理论；(Iii)研究Yamanoi对“1+epsilon”猜想的证明；以及(Iv)寻找关于对数导数的引理的数论对应物。本项目涉及数论(具体地说，整数或有理数中多项式方程的解，以及相关的不等式)和值分布理论(由一个或多个复变量的幂函数所满足的不等式)。这两个看似完全不同的数学领域有许多共同的现象。这部分是由两个领域之间的形式类比来解释的，这导致了新的猜想，现有定理的更短的证明，以及新的定理。本项目使用这个类比来进一步阐明数论、值分布理论的基本结构，以及这两个领域之间的类比。