TY - JOUR
T1 - Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction
AU - Cho, Bumkyu
AU - Koo, Ja Kyung
AU - Park, Yoon Kyung
PY - 2009/4
Y1 - 2009/4
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=60649090795&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2008.09.018
DO - 10.1016/j.jnt.2008.09.018
M3 - Article
AN - SCOPUS:60649090795
SN - 0022-314X
VL - 129
SP - 922
EP - 947
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 4
ER -