A continued fraction of order twelve as a modular function

Abstract

We study a continued fraction U ( τ ) U(\tau ) of order twelve using the modular function theory. We obtain the modular equations of U ( τ ) U(\tau ) by computing the affine models of modular curves X ( Γ ) X(\Gamma ) with Γ = Γ 1 ( 12 ) ∩ Γ 0 ( 12 n ) \Gamma = \Gamma _1 (12) \cap \Gamma _0(12n) for any positive integer n n ; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer n n . We point out that we provide an explicit construction method for finding the modular equations of U ( τ ) 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 12 over imaginary quadratic fields by using U ( τ ) U(\tau ) and the value U ( τ ) U(\tau ) at an imaginary quadratic argument is a unit. In addition, if U ( τ ) U(\tau ) is expressed in terms of radicals, then we can express U ( r τ ) U(r \tau ) in terms of radicals for a positive rational number r r .