Abstract
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.
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