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

通过Stark或Birch-Swinnerton-Dyer的猜想，l函数的特殊值与全局点之间的联系是现代数学的一大奥秘。拟议的项目旨在阐明这种联系。此外，提出的项目的一个方面是提供生成类字段的算法，并在实践中实现这些算法，从而涉及计算理论。此外，Gross-Stark单位的研究与模形式的p进族相关的伽罗瓦表示的快速增长的工作体之间的关系是一个令人兴奋的新联系。克罗内克的“青春之梦”是明确建构数场的类场；希尔伯特把这个问题排在他著名的问题列表的第12位。这个问题推动了数论的许多重大进展，包括克罗内克-韦伯定理和复乘法理论。数域的单位与阿基米德函数和p进ζ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。为数域提供明确的类场论或理解斯塔克和格罗斯的猜想方面的任何进展都应被视为对数学的重大贡献。

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.