Automorphic functions on the upper half plane, especially modular functions.- Elliptic curves and the fundamental theorems of the classical theory of complex multiplication.- Relation between the points of finite order on an elliptic curve and the modular functions of higher level.- Abelian varieties and siegel modular functions.- The endomorphism-ring of an abelian variety; the field of moduli of an abelian variety with many complex multiplications.- The class-field-theoretical characterization of K¿ (?(z)).- A further method of constructing class fields.- The hasse zeta function of an algebraic curve.- Infinite galois extensions with l-adic representations.- Further generalization and concluding remarks.