Quark and lepton flavors with common modulus τ in A₄ modular symmetry

We study quark and lepton mass matrices with the common modulus τ in the A₄ modular symmetry. The viable quark mass matrices are composed of modular forms of weights 2, 4 and 6. It is remarked that the modulus τ is close to i, which is a fixed point in the fundamental region of SL (2, Z), and the CP symmetry is not violated. Indeed, the observed CP violation is reproduced at τ which is deviated a little bit from τ=i. The charged lepton mass matrix is also given by using modular forms of weights 2, 4 and 6, where five cases have been examined. The neutrino mass matrix is generated in terms of the modular forms of weight 4 through the Weinberg operator. Lepton mass matrices are also consistent with the observed mixing angles at τ close to i for NH of neutrino masses. Allowed regions of τ of quarks and leptons overlap each other for all cases of the charged lepton mass matrix. However, the sum of neutrino masses is crucial to test the common τ for quarks and leptons. The minimal sum of neutrino masses mᵢ is 140meV at the common τ. The inverted hierarchy of neutrino masses is unfavorable in our framework. It is emphasized that our result suggests the residual symmetry Z₂^S=\ I, S \ in the quark and lepton mass matrices.