Generalized Lambda Functions and Modular Function Fields of Principal Congruence Subgroups

Abstract

Let $N$ be a positive integer greater than 1. We define a modular function of level $N$ which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function $j$ generate the modular function field with respect to the principal congruence subgroup of level $N$. Further we study its values at imaginary quadratic points.