On the p-adic Geometry of Modular Curves

加州大学伯克利分校罗伯特·科尔曼教授98 01389计划继续他对经典模形式的研究，特别关注Mazur-Gouvea猜想。该项目围绕着研究人员最近的一项发现，即在某些模块形式上发现一种新的几何结构。科尔曼的“本征曲线”为现代数论中一些最受研究的数学对象提供了一个新的视角。如果只完成计划项目的一部分，它可能会改变数论150年历史的分支的面貌。专家们正在热切地等待这项工作的第一批报告。当数学家第一次使用微积分技术来研究模形式时，他们对其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。虽然模形式的这些方面仍然引起人们的浓厚兴趣，但现代数学家主要对模形式所包含的算术信息感兴趣。模块化形式编码了有关其他数学结构的大小的信息，这些结构乍一看(甚至是第二眼)是完全无关的。您可以从模数形式获得的数字具有与形式本身完全不同的含义。基本上，模块化形式很重要。二十世纪下半叶的《数论》有很大一部分致力于理解这种奇怪的计数为什么以及如何工作。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.