Arithmetic of the Ramanujan-Gollnitz-Gordon continued fraction

Abstract

Text. We extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Gollnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Gollnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Gollnitz-Gord on continued fraction v(tau) by computing the affine models of modular curves X(Gamma) with Gamma = Gamma(1)(8) boolean AND Gamma(0)(16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(tau) is a modular unit over Z we give a new proof of the fact that the singular values of v(tau) 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. (C) 2008 Elsevier Inc. All rights reserved.