Abstract
We begin this chapter by proving some general theorems about fields of modular functions. We then turn to the investigation of special fields. Here we will not make use of existence theorems of function theory. We start with the absolute invariant J for which we have a representation as the quotient of modular forms. This representation may also be considered as its definition. By various processes we shall derive new functions from functions of level 1. We shall then determine their behavior under modular substitutions. The same will hold for forms of level 1. The so-called division fields, generated from the Weierstrass ℘-function of the theory of elliptic functions, should in a certain sense be discussed here, but they will be more conveniently treated in the next chapter.
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