Gross-Stark units and p-adic families of Hilbert modular forms

本课题拟通过证明Gross-Stark单位的显式公式来研究类域的显式构造。这将建立在最近首席研究员与亨利·达蒙和罗伯特·波拉克合作的工作基础上，其中在某些假设下证明了格罗斯-斯塔克的弱猜想。它也将纳入之前的工作，其中一个格罗斯-斯塔克单位的精确公式被推测。此外，本项目希望发展一种斯塔克单位在秩一阿贝尔设置公式的统一理论。该项目将通过群上同源的形式将一些作者的“模块化方法”与其他作者的“Shintani方法”在概念上统一起来。该项目的一个目标是将达蒙的整合理论与朗兰兹计划联系起来。克罗内克的“青春梦想”是明确地构造二次虚域的所有阿贝尔扩展。希尔伯特提出了一般数域的问题，作为他著名的列表中的第12个问题。对显式类场论的探索推动了数论的许多重大进展。它的主要成就包括克罗内克-韦伯定理和复乘法理论。本项目希望将显式类场理论的理解扩展到复数乘法的设定之外。主要技术是研究数域单位与ζ函数的特殊值之间的联系。这种联系是数论中一个核心的激励主题。

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 hopes to develop a unified theory of formulas for Stark units in rank one abelian settings. The project would conceptually unify the "modular methods" of some authors with the "Shintani methods" of others through the formalism of group cohomology. One goal of the project is to connect Darmon's integration theory to the Langlands program.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.