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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 thesemodels, all matter fields are assigned to three inequivalent singlets, ${\bf1}$, ${\bf 1^\prime}$ and ${\bf 1^{\prime\prime}}$, of the finite modular group$\Gamma_3 \simeq A_4$. Considering tensor products of the $A_4$ group, it iseasy to make the charged lepton mass matrix $M_\ell$ diagonal. Since not allmodular forms of a specific weight and level 3 can be arranged into threeinequivalent singlets of $A_4$ simultaneously, we can always make some entriesin $M_\nu$ vanish by properly assigning the representations and modular weightsfor the matter fields. We consider two cases where neutrino masses originatefrom the Weinberg operator and the type-I seesaw mechanism, respectively. Forthe former case, all seven viable two-zero textures of $M_\nu$ (${\bfA_{1,2}}$, ${\bf B_{1,2,3,4}}$ and ${\bf C}$) can be realized successfully. Forthe latter case, only five of them (namely ${\bf A_{1,2}}$, ${\bf B_{3,4}}$ and${\bf C}$) can be achieved due to the intrinsic structure of the right-handedneutrino mass matrix $M_{\rm R}$ in our assumption for simplicity.

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