Number Theory and Allied Topics

本项目主要致力于建立超越理论中的主要猜想的函数域的类比，这些猜想在复数的经典背景下经常是方法所不能及的。第一步(与Papanikolas联合)将是关于阿贝尔变种周期的Shimura-Deligne关系的逆的类比。这将扩展与安德森和帕帕尼科拉斯的联合工作，精确地确定特殊伽马值中的单项式的线性关系。其他主题将涉及周期的除导数的独立性，通过固定素数的幂的特征的超越和变化，以及内插矩阵的应用的研究。另一个方向是根据研究者的独立性标准来逼近内斯特连科关于拉马努詹函数的定理。最后，设计了算术Liouville-Lojasiewicz不等式的“最终”版本。自19世纪中叶以来，人们一直在理解，一元有理函数的行为与有理数惊人地相似。这种背景下的问题本身就很吸引人，而且往往比它们的经典类似问题更容易解决。这个项目追求在自Hermite以来所有超越研究的基本前提的函数域中的几个定义良好的设置中的扩展：没有令人惊讶的代数关系。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极端基本的底层结构的结果。

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.