Hilbert’s 12th Problem, Complex Multiplication and Shimura Reciprocity

Abstract

We indicate the place of Shimura’s reciprocity law in class field theory and give a formulation of the law that reduces the technical prerequisites to a minimum. We then illustrate its practical use by dealing with a number of classical problems from the theory of complex multiplication that have been the subject of recent research. Among them are the construction of class invariants and the explicit generation of ring class fields.