On the p-adic Geometry of Modular Curves

模曲线的 p 进几何

• 批准号: 9801389
• 负责人: Robert Coleman
• 金额: $ 20.71万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801389&HistoricalAwards=false
• 关键词: adic; Geometry; Modular; Curves

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.

罗伯特·科尔曼（Robert Coleman）教授计划继续他对经典模形式的研究，并将重点放在Mazur-Gouvea猜想上。该项目围绕着研究人员最近在某些模块形式上发现的新几何结构展开。科尔曼的“特征曲线”为现代数论中一些最受研究的数学对象提供了一种新的观点。如果计划项目的一部分完成，它可能会改变拥有150年历史的数论分支的面貌。专家们急切地等待着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极强对称性的数学函数。虽然模形式的这些方面仍然引起了人们的强烈兴趣，但现代数论学家主要对模形式包含的算术信息感兴趣。模块形式编码的信息与其他数学结构的大小有关，这些结构乍一看（甚至第二看）是完全不相关的。从模形式中得到的数字，其意义与模形式本身是完全不同的。基本上，模形式可以计数。20世纪下半叶，数论的很大一部分都致力于理解这种奇怪的计数为何以及如何工作。科尔曼教授对模形式有一些非常新的想法，这些想法应该会增加我们对一个主要数学奥秘的理解。