Modular Flavor Symmetries and Fermion Mass Hierarchies

We investigate fermion mass hierarchies in models with modular flavor symmetries. Several key conclusions arise from the observation that the determinants of mass matrices transform as 1-dimensional vector-valued modular forms. We demonstrate that, under some fairly general assumptions, achieving hierarchical fermion masses requires the vacuum expectation value of the modulus $\tau$ to be located near one of the critical points, $i$, $i\infty$, or $\omega$. We also revisit the universal near-critical behavior around these points and classify the resulting mass hierarchies for the critical points $i$ and $\omega$. We compare the traditional Froggatt--Nielsen mechanism with its modular variant. The knowledge and boundedness of Fourier and Taylor coefficients are crucial to the predictive power of modular flavor symmetries.