  We propose a radiative seesaw model with a light dark matter candidate (DM)
under modular $A_4$ and gauged $U(1)_{B-L}$ symmetries, in which neutrino
masses are generated via one-loop level, and we have a bosonic DM candidate
$\sim 0.01-50$ GeV. DM is stabilized by nonzero modular weight and a remnant
$Z_2$ symmetry of $U(1)_{B-L}$ symmetry. The naturalness of tiny DM mass is
indirectly realized by the radiatively induced mass of neutral fermions that
interact with our DM candidate. Thus, DM mass does not have to exceed the
neutral fermion masses, whose scale is assumed to be 50 GeV. Taking several
benchmark points of DM mass in DM analysis, we show the numerical analysis for
the neutrino oscillation data in the normal and inverted hierarchy cases and
find several features for both cases.
