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

摘要

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