CAREER: Explicit class field theory, Stark's conjectures, and families of modular forms

职业：显式类场论、斯塔克猜想和模块化形式族

• 批准号: 0952251
• 负责人: Samit Dasgupta
• 金额: $ 47.13万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0952251&HistoricalAwards=false
• 关键词: CAREER; Explicit; class; field; theory

This project proposes to study the explicit construction of class fields by proving explicit formulas for Gross-Stark units. This will build on recent work of the principal investigator in collaboration with Henri Darmon and Robert Pollack, in which the weak Gross-Stark conjecture was proven under certain assumptions. It will also incorporate previous work of the principal investigator in which an exact formula for Gross-Stark units was conjectured. Furthermore, this project proposes to develop a unified theory of locally constructed "Darmon style" cohomology classes by linking Darmon's integration theory with the p-adic Langlands program. Connections to other outstanding conjectures concerning trivial zeroes of p-adic L-functions will be studied.This project aims to help foster an increased community of interaction and collaboration between the University of California, Santa Cruz, Stanford University, and the American Institute of Mathematics. This increased interaction will be highlighted by regularly held mini-conferences on the topics of Number Theory and Arithmetic Algebraic Geometry. Furthermore, a postdoc will be hired at UCSC to help foster the research environment, and in particular to interact with students on research topics. As part of this proposal, the PI plans to continue expository writing aimed at communicating high level mathematics to a student audience. Kronecker's "dream of youth" was to explicitly construct all the abelian extensions of quadratic imaginary fields. Hilbert presented the problem for general number fields as the 12th problem in his famous list. The search for an explicit class field theory has motivated many great advances in number theory. Its prime successes include the Kronecker-Weber theorem and the theory of complex multiplication. This project hopes to extend the understanding of explicit class field theory beyond the setting of complex multiplication. The main technique is to study the connection between units in number fields and special values of zeta-functions. This connection is a central motivating theme in number theory.

该项目提议通过证明针对总体形式的明确公式来研究班级领域的明确结构。这将建立在首席研究人员与亨利·达蒙（Henri Darmon）和罗伯特·波拉克（Robert Pollack）合作的最新工作的基础上，在某些假设下证明了弱总的猜想。它还将纳入主要研究人员的先前工作，其中猜想了一个确切的总体公式。此外，该项目提议通过将达尔蒙的集成理论与P-Adic Langlands计划联系起来，开发出本地建构的“ Darmon Style”的统一理论。 将研究与有关P-ADIC L功能琐碎零的其他杰出猜想的联系。该项目旨在帮助促进加利福尼亚大学，圣克鲁斯大学，斯坦福大学和美国数学学院之间增加互动和合作社区。 关于数字理论和算术代数几何学主题的定期小型会议将突出这种增加的相互作用。 此外，将在UCSC雇用博士后，以帮助培养研究环境，尤其是与学生就研究主题进行互动。 作为该提案的一部分，PI计划继续进行说明性写作，旨在将高级数学传达给学生受众。克罗内克（Kronecker）的“青年梦”（Dream of Youth）是要明确构建所有二次假想领域的亚伯式扩展。希尔伯特（Hilbert）将一般数字字段的问题作为他著名名单中的第12个问题。寻找明确的阶级领域理论促使了数量理论的许多巨大进步。它的主要成功包括Kronecker-Weber定理和复杂乘法理论。该项目希望将对显式阶级字段理论的理解扩展到复杂乘法的设置之外。主要技术是研究数字字段中单元与Zeta功能的特殊值之间的连接。这种联系是数字理论中的主要动机主题。