Since an algebraic function w(z) is defined implicitly by an equation of the form f(z,w) = 0, where f is a polynomial, it is understandable that the study of such functions should be possible by algebraic methods. Such methods also have the advantage that the theory can be developed in the most general setting, viz. over an arbitrary field, annd not only over the field of complex numbers (the classical case).
... H. Hasse perceived the connection between complex multiplication and the Riemann hypothesis for congruence zeta functions, which was later proved by A. Weil in a fully general form. This observation led M. Deuring to establish a purely algebraic treatmene of complex multiplication of elliptic curves. He could, moreover, along the same line of ideas, determine the zeta functions of elliptic curves with complex multiplication. The definition of zeta function of an algebraic curve defined over an algebraic number field is originally due to Hasse; and Weil is the first contributor to this subject.