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Construction of generators of modular function fields. (English) Zbl 0536.10016

The purpose of this paper is to give generators for the field of meromorphic functions on \(\Gamma\) (N)\(\backslash {\mathbb{H}}\) where \(\Gamma\) (N) is the usual principal congruence subgroup of the modular group. These are found using Kubert-Lang’s theory of ”Klein-forms”, Galois- theoretic and function-theoretic considerations. For primes \(N>5\) systems of two functions are given which generate the field.
As an example for composite N the author considers the case \(N=6\) in considerable detail. He introduces three further groups lying between \(\Gamma\) (6) and the modular group and describes the function field in each case. It turns out that these represent the four isomorphism classes of elliptic curves of conductor 36.
Reviewer: S.J.Patterson

MSC:

11F03 Modular and automorphic functions
11F06 Structure of modular groups and generalizations; arithmetic groups
11R58 Arithmetic theory of algebraic function fields
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