On the representation of the Picard modular function by $θ$ constants I-II

Abstract

In this paper the author shows the representation of the Picard modular function by 0 constants, and characterizes this function as modular forms on the domain D= {(M, z;) eC : 2Re v + \u <0} = relative to a certain arithmetic discontinuous group, 0 1 0 where H= 1 0 0 . 0 0 1 We divide the paper in two parts. In Part I we discuss the former subject and in Part II we study the latter. This modular function was constructed originally by [P], and was investigated by several mathematicians recently [D-M], [F], [H], [Sh] and [T]. This modular function is defined as the inverse mapping of the period mapping 0 for the family of the complex algebraic curves in (z,w) -space