On the p-adic Geometry of Modular Curves

摘要 罗伯特科尔曼加州大学伯克利分校98 01389 教授罗伯特科尔曼计划继续他的研究古典模形式，特别注意Mazur-Gouvea猜想。 该项目围绕着最近发现的研究人员的一个新的几何结构的某些模块形式。 科尔曼的“特征曲线”提供了一个新的观点，一些最研究的数学对象在现代数论。如果计划中的项目只完成了一部分，它可能会改变数论150年历史的分支的面貌。 专家们正急切地等待这项工作的第一份报告。 当数学家们第一次使用微积分技术来研究模形式时，他们对模形式有趣的分析性质很感兴趣。模形式最初被视为具有令人难以置信的强对称性的数学函数。尽管模形式的这些方面仍然引起人们的强烈兴趣，但现代数论家主要对模形式所包含的算术信息感兴趣。 模形式编码了关于其他数学结构的大小的信息，这些结构乍一看（甚至第二眼）是完全不相关的。 你从模形式中得到的数，其意义与模形式本身是完全不同的。 基本上，模块化形式可以计数。 世纪后半叶的《数论》一书中，有很大一部分都致力于理解这种奇怪的计数方法的原理。 科尔曼教授对模形式有一些非常新的想法，应该会增加我们对一个重大数学谜团的理解。

ABSTRACT Robert Coleman University of California, Berkeley 98 01389 Professor Robert Coleman plans to continue his study of classical modular forms with particular attention to the Mazur-Gouvea conjecture. The project revolves around a recent discovery of the investigator's of a new geometric structure on certain modular forms. Coleman's "eigencurve" provides a new view of some of the most studied mathematical objects in modern number theory. If only a part of the planned project is completed, it could change the face of 150 year old branch of Number Theory. The experts are eagerly awaiting the first reports from this work. When mathematicians first used the techniques of calculus to study modular forms, they were interested in their interesting analytical properties. Modular forms were first seen as mathematical functions with incredibly strong symmetries. While these aspects of modular forms are still of intense interest, modern number theorists are mainly interested in the arithmetical information that modular forms contain. Modular forms encode information about the size of other mathematical constructions that at first (and even second) glance are totally unrelated. Numbers you can obtain from modular forms have a meaning quite separate from the forms themselves. Basically, modular forms count things. A large part of Number Theory in the second half of the twentieth century has been devoted to understanding why and how this strange counting works. Professor Coleman has some very new ideas about modular forms that should increase our understanding of a major mathematical mystery.