Flipped SU(5) with modular $A_4$ symmetry

We study Flipped $SU(5)\times U(1)$ Grand Unified Theories (GUTs) with $\Gamma_3\simeq A_4$ modular symmetry. We propose two models with different modular weights assignments, where the fermion mass hierarchy can arise from weighton fields. In order to relax the constraint on the Dirac neutrino Yukawa matrix we appeal to mechanisms which allow incomplete GUT representations, allowing a good fit to quark and charged lepton masses and quark mixing for a single modulus field $\tau$, with the neutrino masses and lepton mixing well determined by the type I seesaw mechanism, at the expense of some tuning. We also discuss the double seesaw possibility allowed by the extra singlets generically predicted in such string inspired theories.