We study a continued fraction U(τ) of order twelve using the modular function theory. We obtain the modular equations of U(τ) by computing the affine models of modular curves X(Γ) with Γ = Γ1(12) ∩ Γ0(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(τ). 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(τ) and the value U(τ) at an imaginary quadratic argument is a unit. In addition, if U(τ) is expressed in terms of radicals, then we can express U(rτ) in terms of radicals for a positive rational number r.
Bibliographical noteFunding Information:
The first-named author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the Korea government (MEST) (NRF-2017R1A2B2004574). The second-named author was supported by RP-Grant 2016 of Ewha Womans University and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03029519).
© 2017 American Mathematical Society.
- Modular function
- Ramanujan continued fraction