Ramanujan's function k(tau) = r(tau) r(2)(2 tau) and its modularity

Abstract

We study the modularity of Ramanujan's function k(tau) = r(tau) r(2)(2 tau), where r(tau) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k(tau) 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 tau in an imaginary quadratic field, the value k(tau) 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 Gamma(1)(10). Furthermore, we suggest a rather optimal way of evaluating the singular values of k(tau) using the modular equations in the following two ways: one is that if j(tau) is the elliptic modular function, then one can explicitly evaluate the value k(tau), and the other is that once the value k(tau) is given, we can obtain the value k(r tau) for any positive rational number r immediately.