Lepton sector in modular $A_4$ and gauged $U(1)_R$ symmetry

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.