Abstract
We study the modularity of Ramanujan’s function
k
(
τ
)
=
r
(
τ
)
r
2
(
2
τ
)
k(\tau )=r(\tau ){r}^{2}(2\tau )
, where
r
(
τ
)
r(\tau )
is the Rogers-Ramanujan continued fraction. We first find the modular equation of
k
(
τ
)
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
τ
\tau
in an imaginary quadratic field, the value
k
(
τ
)
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
Γ
1
(
10
)
{{\mathrm{\Gamma}}}_{1}(10)
. Furthermore, we suggest a rather optimal way of evaluating the singular values of
k
(
τ
)
k(\tau )
using the modular equations in the following two ways: one is that if
j
(
τ
)
j(\tau )
is the elliptic modular function, then one can explicitly evaluate the value
k
(
τ
)
k(\tau )
, and the other is that once the value
k
(
τ
)
k(\tau )
is given, we can obtain the value
k
(
r
τ
)
k(r\tau )
for any positive rational number r immediately.
On Representation of an Integer by X 2 + Y 2 + Z 2 and the Modular Equations of Degree 3 and 5
