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.