On some arithmetic properties of Siegel functions

We deal with several arithmetic properties of the Siegel functions which are modular units. By modifying the ideas in Kubert and Lang (Modular Units. Grundlehren der mathematischen Wissenschaften, vol 244. Spinger, Heidelberg, 1981), we establish certain criterion for determining a product of Siegel functions to be integral over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}[j]}$$\end{document} . We also find generators of the function fields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{K}(X_1(N))}$$\end{document} by examining the orders of Siegel functions at the cusps and apply them to evaluate the Ramanujan’s cubic continued fraction systematically. Furthermore we construct ray class invariants over imaginary quadratic fields in terms of singular values of j and Siegel functions.