CAREER: Explicit class field theory, Stark's conjectures, and families of 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，本计画建议将Darmon的积分理论与p-adic Langlands程式连结起来，发展出一个局部建构的“Darmon风格”上同调类的统一理论。 该项目旨在帮助促进加州大学、圣克鲁斯大学、斯坦福大学和美国数学研究所之间的互动和合作。 这种增加的互动将突出定期举行的小型会议上的数论和算术代数几何的主题。 此外，博士后将在UCSC聘请，以帮助促进研究环境，特别是与学生的研究课题互动。 作为该提案的一部分，PI计划继续进行旨在向学生观众传达高水平数学的临时写作。克罗内克的“青春梦想”是明确地构造所有的二次虚域的阿贝尔扩张。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 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.