CAREER: Explicit class field theory, Stark's conjectures, and families of modular forms

这个项目建议通过证明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亨利·达蒙和罗伯特·波拉克合作的最近工作的基础上，在这些工作中，在某些假设下证明了弱的格罗斯-斯塔克猜想。它还将结合首席研究人员之前的工作，其中推测出了格罗斯-斯塔克单位的准确公式。此外，该项目还建议通过将Darmon的积分理论与p-adic朗兰兹程序相结合来发展本地构建的“Darmon式”上同调类的统一理论。这项计画旨在促进加州大学圣克鲁斯分校、斯坦福大学与美国数学学会之间加强互动与合作。这种增加的互动将通过定期举办的关于数论和算术代数几何主题的小型会议来突出。此外，加州大学洛杉矶分校将聘请一名博士后，帮助培养研究环境，特别是与学生就研究主题进行互动。作为这项提议的一部分，PI计划继续编写旨在向学生受众传达高级数学知识的说明性写作。Kronecker的“青春梦想”是明确地构造出二次虚场的所有阿贝尔延拓。希尔伯特在他的著名列表中将一般数字域的问题作为第12个问题提出。对显式类场理论的探索推动了数论的许多重大进步。它的主要成就包括Kronecker-Weber定理和复数乘法理论。这个项目希望将对显式类域理论的理解扩展到复数乘法的设置之外。其主要技巧是研究数域上的单位与zeta函数的特殊值之间的关系。这种联系是数论中一个重要的激励主题。

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 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.