Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction☆

By Bumkyu Cho, J. Koo, Y. Park. 2009

47 citations recorded by the source corpus.

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–Gordon continued fraction v(τ) by computing the affine models of modular curves X(Γ) with Γ=Γ1(8)∩Γ0(16p). 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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/d942c3269c34ed57bbfd76b1a5a397c8fa0c56ce

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In Algebra and Number Theory

1. Frank G. Garvan
2. Jie Frye
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4. Karl Mahlburg
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In Applied Mathematics

1. Frank G. Garvan
2. Kathrin Bringmann

Citations in the corpus

• Automatic Proof of Theta-Function Identities (2018, background): “EXAMPLE: > with(qseries): > with(thetaids): > with(ramarobinsids): > G:=j->1/GetaL([1,3,4],13,j): H:=j->1/GetaL([2,5,6],13,j): > GM:=j->1/MGetaL([1,3,4],13,j): HM:=j->1/MGetaL([2,5,6],13,j): > GE:=j->-GetaLEXP([1,3,4],13,j): HE:=j->-GetaLEXP([2,5,6],13,j): > GHID:=( G(1)* G(2)+ H(1)* H(2))/( G(2)* H”