Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis (Theorem 2.4), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}(\sqrt{-1})}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}(\sqrt{-3})}$$\end{document} (Theorem 4.2). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer (≥2) (Remark 4.3).