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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 themodular $A_4$ 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 andphenomenologically interesting.

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