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We study $A_4 \times A_4 \times A_4$ modular symmetric flavor models torealize quark mass hierarchies and mixing angles without fine-tuning. Massmatrices are written in terms of modular forms. At modular fixed points $\tau =i\infty$ and $\omega$, $A_4$ is broken to $Z_3$ residual symmetry. When themodulus $\tau$ is deviated from the fixed points, modular forms showhierarchies depending on their residual charges. Thus, we obtain hierarchicalstructures in mass matrices. Since we begin with $A_4\times A_4 \times A_4$,the residual symmetry is $Z_3 \times Z_3 \times Z_3$ which can generatesufficient hierarchies to realize quark mass ratios and absolute values of theCKM matrix $|V_{\textrm{CKM}}|$ without fine-tuning. Furthermore, CP violationis studied. We present necessary conditions for CP violation caused by thevalue of $\tau$. We also show possibilities to realize observed values of theJarlskog invariant $J_{\textrm{CP}}$, quark mass ratios and CKM matrix$|V_{\textrm{CKM}}|$ simultaneously, if $\mathcal{O}(10)$ adjustments incoefficients of Yukawa couplings are allowed.

Quark mass hierarchies and CP violation in $A_4\times A_4\times A_4$ modular symmetric flavor models