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.
