A continued fraction of order twelve as a modular function

Abstract:

We study a continued fraction $U(\tau )$ of order twelve using the modular function theory. We obtain the modular equations of $U(\tau )$ by computing the affine models of modular curves $X(\Gamma )$ with $\Gamma = \Gamma _1 (12) \cap \Gamma _0(12n)$ for any positive integer $n$; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer $n$. We point out that we provide an explicit construction method for finding the modular equations of $U(\tau )$. We also prove that these modular equations satisfy the Kronecker congruence relations. Furthermore, we show that we can construct the ray class field modulo $12$ over imaginary quadratic fields by using $U(\tau )$ and the value $U(\tau )$ at an imaginary quadratic argument is a unit. In addition, if $U(\tau )$ is expressed in terms of radicals, then we can express $U(r \tau )$ in terms of radicals for a positive rational number $r$.