Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction

Abstract

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 = Γ 1 ( 8 ) ∩ Γ 0 ( 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.