Sergei Vostokov (S.-Petersburg State University)
sergei@vostokov.usr.pu.ru Originally the problem of the reciprocity law was stated by Hilbert as his 9-th problem. After creation of class field theory the problem boils down to the finding of an explicit form of the Hilbert pairing in a local field. In the talk I will give a short review of the history of the explicit reciprocity laws, explain the state of art and describe several application to number theory (explicit class field theory, representation theory, etc.).