We study the modularity of Ramanujan's function k (τ) = r (τ) r2 (2 τ), where r (τ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k (τ) of "an"level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ in an imaginary quadratic field, the value k (τ) k generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 (10). Furthermore, we suggest a rather optimal way of evaluating the singular values of k (τ) using the modular equations in the following two ways: one is that if j (τ) is the elliptic modular function, then one can explicitly evaluate the value k (τ) and the other is that once the value k (τ) is given, we can obtain the value k (r τ) for any positive rational number r immediately.
|Number of pages||15|
|State||Published - 1 Jan 2020|
- Ramanujan's function k
- class field theory
- modular function