Gross-Stark units, Stark-Heegner points, and explicit class field theory for totally real fields

摘要奖DMS-0653023的Dasgupta特殊值之间的联系L-功能和全球点，通过斯塔克或Birch-Swinnerton-Dyer的图解，是一个伟大的奥秘现代数学。拟议的项目旨在阐明这一联系。此外，所提出的项目的一个方面是提供算法来生成类字段，并在实践中实现这些算法，从而涉及到计算的理论。此外，Gross-Stark单位的研究和与模形式的p-adic族相关的伽罗瓦表示的快速增长的工作之间的关系是一个令人兴奋的新的联系有待探索。克罗内克的“青年梦想”是明确地构造数域的类域;希尔伯特提出了这个问题作为他著名的列表中的第12个。这个问题激发了数论的许多重大进展，包括克罗内克-韦伯定理和复数乘法理论。数域中的单位与阿基米德和p-adic zeta函数的特殊值之间的联系也是数论中的一个中心激励问题。任何进展提供了一个明确的类领域理论的数字领域或理解的abstractures斯塔克和格罗斯应被视为重大贡献的数学。

Abstract for the award DMS-0653023 of DasguptaThe connection between special values of L-functions and global points,through the conjectures of Stark or Birch-Swinnerton-Dyer, is one of the great mysteries of modern mathematics. The proposed project aims to shed light on this connection. Furthermore, one aspect of the proposed project is to provide algorithms to generate class fields, and implement these algorithms in practice, thereby involving the theory of computing. Also, the relationship between the study of Gross-Stark units and the rapidly growing body of work on Galois representations associated to p-adic families of modular forms is an exciting new link to be explored.Kronecker's "dream of youth" was to explicitly construct class fieldsof number fields; Hilbert presented the problem as the 12th in his famous list. This question has motivated many great advances in number theory, including the Kronecker-Weber theorem and the theory of complex multiplication. The connection between units in number fields and special values of archimedean and p-adic zeta-functions is also a central motivating problem in number theory. Any advances towards providing an explicit class field theory for number fields or understanding the conjectures of Stark and Gross should be considered significant contributions to mathematics.