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

本计画提出通过证明Gross-Stark单位的显式公式来研究类域的显式构造。 这将建立在首席研究员与Henri Darmon和Robert Pollack合作的最近工作的基础上，其中弱Gross-Stark猜想在某些假设下得到了证明。 它还将包括主要研究者以前的工作，其中确定了Gross-Stark单位的精确公式。 此外，这个项目希望发展一个统一的理论公式斯塔克单位在一阶阿贝尔设置。 该项目将通过群上同调的形式主义在概念上统一一些作者的“模块化方法”与其他作者的“Shintani方法”。 该项目的一个目标是将达蒙的积分理论与朗兰兹纲领联系起来。克罗内克的“青春梦想”是明确构造二次虚域的所有阿贝尔扩张。 希尔伯特提出的问题一般数领域的第12个问题在他著名的名单。 寻找一个明确的类域理论激发了数论的许多重大进展。 它的主要成就包括克罗内克-韦伯定理和复数乘法理论。这个项目希望扩展对显式类域理论的理解，超越复数乘法的设置。 主要技术是研究数域中的单位与zeta函数的特殊值之间的联系。 这种联系是数论中的一个中心动机主题。

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.