  We consider the modular symmetry associated with the compactification of
extra dimensions as the flavor symmetry on lepton sector. Especially, we
propose a model based on the modular $A_4$ symmetry with three right-handed
neutrinos and a gauge singlet Higgs, which works well in the so-called large
volume limit of the extra dimensions, i.e., $\mbox{Im}\tau \to \infty$ for a
modulus $\tau$. The right-handed neutrinos are introduced to realize the seesaw
mechanism for tiny neutrino masses observed in oscillation experiments. The
vacuum expectation value of the singlet Higgs gives the Majorana masses for
right-handed neutrinos and the $\mu$-term for Higgs fields. The model can
explain the observed masses and mixing angles of neutrinos successfully. We
find that one of the mixing angles should be in the range $\sin^2 \theta_{23}
\ge 0.58$. Importantly, the CP violating phases of neutrinos are predicted in
the two restricted regions. One is that the Dirac phase is $\delta_{\rm CP}
\simeq - 0.5 \pi$ and the Majorana phases are $\alpha_{21} \simeq 0$ and
$\alpha_{31} \simeq \pi$. The other is $\delta_{\rm CP} \simeq + 0.5 \pi$,
$\alpha_{21} \simeq 2 \pi$ and $\alpha_{31} \simeq \pi$. The effective neutrino
mass in the neutrinoless double beta decay is found to be $m_{\rm eff} =
0.037$--0.047 eV. These predictions will be tested in the future neutrino
experiments.
