Gross-Stark units, Stark-Heegner points, and explicit class field theory for totally real fields

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Abstract for the award DMS-0653023 of DasguptaThe connection between special values of L-functions and global points,through the conjectures of Stark or Birch-Swinnerton-Dyer, is one of the great mysteries of modern mathematics. The proposed project aims to shed light on this connection. Furthermore, one aspect of the proposed project is to provide algorithms to generate class fields, and implement these algorithms in practice, thereby involving the theory of computing. Also, the relationship between the study of Gross-Stark units and the rapidly growing body of work on Galois representations associated to p-adic families of modular forms is an exciting new link to be explored.Kronecker's "dream of youth" was to explicitly construct class fieldsof number fields; Hilbert presented the problem as the 12th in his famous list. This question has motivated many great advances in number theory, including the Kronecker-Weber theorem and the theory of complex multiplication. The connection between units in number fields and special values of archimedean and p-adic zeta-functions is also a central motivating problem in number theory. Any advances towards providing an explicit class field theory for number fields or understanding the conjectures of Stark and Gross should be considered significant contributions to mathematics.