  We propose a lepton model under modular $A_4$ and gauged $U(1)_R$ symmetries,
in which the neutrino masses are induced at one-loop level. Thanks to the
modular $A_4$ symmetry, we have several predictions on the lepton sector,
especially, on fixed points of $\tau=i,\omega\equiv e^{2\pi i/3},i\times
\infty$ each of which has remnant symmetry; $Z_2$ for $\tau=i$ and $Z_3$ for
$\tau=\omega, i\infty$. These points are favored by a string theory and
phenomenologically interesting.
