Number Theory and Allied Topics

数论及相关主题

• 批准号: 0355390
• 负责人: W. Dale Brownawell
• 金额: --
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355390&HistoricalAwards=false
• 关键词: Number; Theory; Allied; Topics

Abstract for award DMS-0355390 of BrownawellThis project focuses primarily on establishing function fieldanalogues of major conjectures in transcendence theory which arecurrently far out of the reach of methods in the classical setting ofthe complex numbers. The first step (joint with Papanikolas) will bethe analogue of the converse of the Shimura-Deligne relations on theperiods of abelian varieties. This will extend joint work withAnderson and Papanikolas precisely determining the linear relations onmonomials in special Gamma values. Other topics will involveindependence of divided derivatives of periods, transcendence andvariation of characteristic through powers of a fixed prime, andinvestigation of applications of interpolation matrices. Anotherdirection will be to approach Nesterenko's theorem for Ramanujan'sfunctions based on the investigator's independence criterion. Lastly,a ``final'' version of the arithmetic Liouville-Lojasiewicz inequalityis planned.As has been understood since the middle of the nineteenth century,rational functions in one variable behave amazingly like the rationalnumbers. Questions in this setting are fascinating in their ownright, and often they are easier to resolve than their classicalanalogues. This project pursues an extension in several well-definedsettings in function fields of the basic premise of all transcendenceinvestigations since Hermite: There are no surprising algebraicrelationships. By that we mean that all algebraic relationshipsbetween numbers that one might care are consequences of some extremelybasic underlying structure.

Brownawell的DMS-0355390奖项摘要该项目主要侧重于建立超越理论中主要结构的函数场模拟，这些理论通常远远超出了经典复数设置中的方法。 第一步（与Papanikolas联合）将是志村-德利涅关系在阿贝尔品种周期上的匡威的模拟。 这将扩展与安德森和Papanikolas的联合工作，精确地确定特殊Gamma值中单项式的线性关系。 其他的主题将包括周期的除导数的独立性，特征通过固定素数的幂的超越和变化，以及插值矩阵的应用研究。 另一个方向是在研究者独立性准则的基础上，进一步研究Ramanujan函数的Nesterenko定理。 最后，一个算术Liouville-Lojasiewicz不等式的“最终”版本被计划了。正如自19世纪中期以来所理解的那样，一个变量的有理函数的行为与有理数惊人地相似。 这种情况下的问题本身就很吸引人，而且往往比经典的类似问题更容易解决。 这个项目追求在几个定义良好的设置在函数领域的基本前提的所有超越调查，因为厄米：有没有令人惊讶的代数关系的扩展。 我们的意思是，人们可能关心的数字之间的所有代数关系都是一些非常基本的潜在结构的结果。