Number Theory and Allied Topics

本研究的主要目的是建立在设置offunction领域的类似物的主要成果由Rohrlich和志村的代数（不）依赖的坐标周期的某些交换品种。 目前的步骤涉及到分析某些类型的t-模的结构，它们大致对应于阿贝尔变种。 有了这些工具，人们期待在相关问题上取得进一步进展。 特别是，PI和他的同事将建立某些一维t-模（称为Drinfeld模）的周期独立性的精确条件。 另一种方法涅斯捷连科最近的工作拉马努金职能也将试图通过确定合理的解决方案，以acertain微分方程。 这将允许应用PI的Lojasiewicz不等式，而不是原始的基本的，但相当复杂的，独立性由于Philippon。 此外，PI将提供一个相当普遍的算术版本的基本Lojasiewicz不等式，这是最好的可能在其重要的方面。 一个中心问题是，那些在数学（分析或几何）的各种背景下出现的数字是否有未知的联系。 一百多年前，这种类型的一些第一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标是表明，这种联系并不存在于一个广泛的设置类似于今天的某些中心难题。 在这种情况下，多项式对应于整数，幂级数对应于真实的数。这项工作的核心将由PI与M.A.一起进行。Papanikolas和G.W.安德森。 协调相当不同的元素是至关重要的，以建立这些结果之间的边界广泛的领域分析，几何和数论。 在经典的情况下，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可能“不自然地”小。 PI对这样的结果有一个很好的版本，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.