The theory of automorphic functions in one complex variable was created during the second half of the nineteenth and the beginning of the twentieth centuries. Important contributions are due to such illustrious mathematicians as F. Klein, P. Koebe and H. Poincaré. Two sources may be traced: the uniformization theory of algebraic functions, and certain topics in number theory. Automorphic functions with respect to groups with compact quotient space on the one hand and elliptic modular functions on the other are examples of these two aspects. In several complex variables there is no analogue of uniformization theory; the class of automorphic functions which can be considered becomes much narrower, and the underlying groups are, in general, arithmetically defined.

In the mid-1930s C.L. Siegel discovered a new type of automorphic forms and functions in connection with his famous investigations on the analytic theory of quadratic forms. He denoted these functions as ‘modular functions of degree n’; nowadays they are called ‘Siegel modular functions’. Next to Abelian functions they are the most important example of automorphic functions in several complex variables, and they very soon became a touchstone to test the efficiency of general methods in several complex variables and other fields. Only recently, Hilbert modular functions have achieved a similar position due to the progress made in that area by K. Doi, F. Hirzebruch, F. W. Knöller, H. Naganuma and D. Zagier, amongst others.