Number Theory and Allied Topics

本研究的主要目的是在设定函数域中建立Rohrlich和Shimura关于某些阿贝尔变量周期坐标的代数依赖关系的主要猜想的类似物。目前的步骤包括分析某些类型的t模的结构，它们大致对应于阿贝尔变体。有了这些工具，人们期望在相关问题上取得进一步的进展。特别是，PI和他的同事将为某些一维t模（称为德林菲尔德模）的周期独立性建立精确的条件。对Nesterenko最近关于拉马努金函数的研究的另一种方法也将通过确定某微分方程的有理解来尝试。这将允许应用PI的Lojasiewicz不等式，而不是最初的基本的，但由于Philippon而相当复杂的独立性准则。此外，PI将提供基本lojasiewicz不等式的相当一般的算术版本，这在其重要方面是最好的。本建议以数和多项式的性质为中心。一个核心问题是，在数学（分析或几何）的各种背景下出现的那些数字是否有未知的联系。100多年前，这种类型的一些首次研究解决了困扰数学家两千年的问题。本研究的一个主要目的是表明，在类似于今天某些中心难题的广泛背景下，这种联系并不存在。在这种情况下，多项式对应于整数，幂级数对应于实数。这项工作的核心将由PI与M.A. Papanikolas和G.W. Anderson一起进行。要在广泛的分析、几何和数论领域之间建立这些结果，协调相当不同的元素是至关重要的。在经典情况下，PI将试图发展对某些微分方程行为的理解，以简化我们对Nesterenko最近某些工作的看法。正如人们试图证明值之间没有“隐藏的”联系一样，人们也努力建立多项式值不能“不自然地”小。PI有这样一个结果的一个很好的版本，他将进一步扩展其适用性。

The main thrust of this research is to establish in the setting offunction fields analogues of major conjectures by Rohrlich and,Shimura on the algebraic (in)dependence of coordinates of periods of certain abelian varieties. The current step involves the analysis of the structure of certain types of t-modules, which correspond roughly to abelian varieties. With these tools, one expects further progress on related questions. In particular, the PI and his coworkers will establish precise conditions for the independence of the periods of certain one dimensional t-modules, called Drinfeld modules. Another approach to Nesterenko's recent work on the Ramanujan functions will also be attempted by means of determining the rational solutions to acertain differential equation. This would allow the application of the PI's Lojasiewicz inequality rather than the originalfundamental, but considerably more complicated, independencecriterion due to Philippon. Moreover the PI will provide a quite general arithmetic version of the fundamental Lojasiewiczinequality which is best possible in its important respects.The present proposal centers about the properties of numbers andpolynomials. One central question is whether those numbers whicharise in various contexts in mathematics (analysis or geometry) have unkown linkages. Some of the first research of this type over a hundred years ago resolved problems which had puzzledmathematicians for two millenia. One main objective of the present research is to show that such linkages do not exist in a broad setting analogous to certain central puzzles of today. In that setting, polynomials correspond to integers and power series correspond to real numbers. The core of this work will be carried out by the PI together with M.A. Papanikolas and G.W. Anderson. A coordination of quite diverse elements is crucial to establishing these results on the boundary between the broad fields of analysis, geometry, and number theory. In the classical case, the PI will try to develop an understanding of the behaviour of certain differential equations to simplify our view of certain recent work of Nesterenko. Just as one tries to show that there are no "hidden" linkages between values, one also strives toestablish that polynomial values cannot be "unnaturally" small. The PI has one good version of such a result, and he will extend its applicability even further.