Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction

Text: We extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Göllnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v (τ) by computing the affine models of modular curves X (Γ) with Γ = Γ₁ (8) ∩ Γ₀ (16 p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v (τ) is a modular unit over Z we give a new proof of the fact that the singular values of v (τ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=FWdmYvdf5Jg.