Special values of generalized $\lambda$ functions at imaginary quadratic points

TL;DR

This paper investigates a generalized lambda function and demonstrates its algebraic properties and its role in generating class fields when evaluated at imaginary quadratic points.

Contribution

It proves that the generalized lambda function and the j-invariant generate the modular function field and that the lambda function's values at imaginary quadratic points are algebraic integers.

Findings

Lambda_{k,ell} and j generate the modular function field for (N)

Values of mbda_{k,ell} at imaginary quadratic points are algebraic integers

These values generate ray class fields over Hilbert class fields

Abstract

We study a modular function $\Lambda_{k,\ell}$ which is one of generalized $\lambda$ functions. We show $\Lambda_{k,\ell}$ and the modular invariant function $j$ generate the modular function field with respect to the modular subgroup $\Gamma_1(N)$. Further we prove that $\Lambda_{k,\ell}$ is integral over $\mathbf Z[j]$. From these results, we obtain that the value of $\Lambda_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and generates a ray class field over the Hilbert class field.