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

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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.