Generators of modular function fields obtained from generalized lambda functions

TL;DR

This paper introduces a generalized modular function extending the elliptic lambda function, demonstrating it generates the modular function field and analyzing its values at imaginary quadratic points.

Contribution

It defines a new generalized lambda function and proves it, along with the modular invariant, generate the modular function field for principal congruence subgroups.

Findings

The generalized lambda function, together with the modular invariant, generates the modular function field.

The paper studies the values of this function at imaginary quadratic points.

It extends classical results on modular functions and lambda functions.

Abstract

We define a modular function which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function generate the modular function field with respect to the principal congruence subgroup. Further we study its values at imaginary quadratic points.