Texture-zero patterns of lepton mass matrices from modular symmetry

Texture zeros in fermion mass matrices have been widely considered in tackling the Standard Model flavour puzzle. In this work, we perform a systematic analysis of texture zeros in lepton mass matrices in the framework of $\Gamma_{3}'\cong T'$ modular symmetry. Assuming that the lepton fields transform as irreducible representations of $T'$, we obtain all possible texture-zero patterns for both charged-lepton and neutrino mass matrices which can be achieved from $T'$ modular symmetry. We provide representative models for the phenomenologically-viable textures which can accommodate the experimental data. The predictions for lepton mixing angles, CP-violating phases, light neutrino masses and effective neutrino mass relevant for neutrinoless doble beta decay, are discussed. We find that the minimal viable lepton model depends on only $7$ real free parameters including the modulus $\tau$ (the corresponding charged-lepton mass matrix contains $4$ vanishing entries, and the neutrino mass matrix has $1$ texture zero). Finally, we study in detail three benchmark models, one for each neutrino mass generation mechanism considered (Dirac, Majorana via Weinberg operator and Majorana via minimal type-I seesaw mechanism).