A modular $A_4$ symmetry realization of two-zero textures of the Majorana neutrino mass matrix

We show how to realize two-zero textures of the Majorana neutrino mass matrix $M_{\nu }$ based on modular $A_4$ invariant models without flavons. In these models, all matter fields are assigned to three inequivalent singlets, ${\bf 1}$,${\bf 1^\prime}$and${\bf 1^{\prime\prime}}$,ofthefinitemodulargroup$\Gamma_3 \simeq A_4$\mathrm{.}Consideringtensorproductsofthe$A_4$ group, it is easy to make the charged lepton mass matrix $M_{\mathrm{\ell }}$ diagonal. Since not all modular forms of a specific weight and level 3 can be arranged into three inequivalent singlets of $A_4$ simultaneously, we can always make some entries in $M_{\nu }$ vanish by properly assigning the representations and modular weights for the matter fields. We consider two cases where neutrino masses originate from the Weinberg operator and the type-I seesaw mechanism, respectively. For the former case, all seven viable two-zero textures of $M_{\nu }$ (${\bf A_{1,2}}$,${\bf B_{1,2,3,4}}$and${\bf C}$) can be realized successfully. For the latter case, only five of them (namely ${\mathbf{A}}_{\mathbf{1},\mathbf{2}}$, ${\mathbf{B}}_{\mathbf{3},\mathbf{4}}$ and $\mathbf{C}$) can be achieved due to the intrinsic structure of the right-handed neutrino mass matrix $M_{\mathrm{R}}$ in our assumption for simplicity.