We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Frohlich.

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[87][88][89][90][91][92][93][94][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 (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.Video. For a video summary of this paper, please visit http:// www.youtube.com/watch?v=FWdmYvdf5Jg.

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ 1 (N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ 1 (13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).