Modular A4 invariance and neutrino mixing

We study the phenomenological implications of the modular symmetry (3) similar or equal to A(4) of lepton flavors facing recent experimental data of neutrino oscillations. The mass matrices of neutrinos and charged leptons are essentially given by fixing the expectation value of modulus , which is the only source of modular invariance breaking. We introduce no flavons in contrast with the conventional flavor models with A(4) symmetry. We classify our neutrino models along with the type I seesaw model, the Weinberg operator model and the Dirac neutrino model. In the normal hierarchy of neutrino masses, the seesaw model is available by taking account of recent experimental data of neutrino oscillations and the cosmological bound of sum of neutrino masses. The predicted sin(23)(2) is restricted to be larger than 0.54 and (CP) = +/-(50 degrees-180 degrees). Since the correlation of sin(23)(2) and (CP) is sharp, the prediction is testable in the future. It is remarkable that the effective mass m(ee) of the neutrinoless double beta decay is around 22 meV while the sum of neutrino masses is predicted to be 145 meV. On the other hand, for the inverted hierarchy of neutrino masses, only the Dirac neutrino model is consistent with the experimental data.