Singular values of generalized $\lambda$ functions

TL;DR

This paper investigates the algebraic properties of special values of a generalized lambda function at imaginary quadratic points, establishing their integrality and their role in generating modular function fields.

Contribution

It proves that special values of the generalized lambda function are algebraic integers and that these functions generate the modular function field for mma_1(N).

Findings

Special values at imaginary quadratic points are algebraic integers.

Lambda and the modular invariant generate the modular function field.

Results contribute to understanding the algebraic structure of modular functions.

Abstract

We study special values of a modular function $\Lambda$ which is one of generalized $\lambda$ functions. We show special values of $\Lambda$ at imaginary quadratic points are algebraic integers. Further we prove that $\Lambda$ and the modular invariant function generate the modular function field with respect to the modular subgroup $\Gamma_1(N)$.