Modular $S_3$-invariant flavor model in SU(5) grand unified theory

We present a flavor model with $S_3$ modular invariance in the framework of SU(5) grand unified theory (GUT). The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $\tau$. The GUT relation of down-type quarks and charged leptons is imposed by the vacuum expectation value (VEV) of the adjoint 24-dimensional Higgs multiplet in addition to the VEVs of $5$ and $\overline{5}$ Higgs multiplets of SU(5). The observed Cabibbo-Kobayashi-Maskawa and-vo-Maki-Nakagawa-Sakata mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic charge-parity phase and the effective mass of the neutrinoless double beta decay with the sum of neutrino masses.