$A_4$ Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point $\tau = \omega$

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau = \omega \equiv \exp({i\,2\pi/3})$ (the left cusp of the fundamental domain of the modular group), $\tau$ being the VEV of the modulus. The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a source of CP violation in addition to the VEV of the modulus, $\tau = \omega + \epsilon$, $(\epsilon)^* \neq \epsilon$, $|\epsilon|\ll 1$. We find that in the case of real (CP-conserving) $g_u$ and $g_d$ and common $\tau$ ($\epsilon$) in the down-quark and up-quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $|\epsilon| \cong 0.03$, all other constants being of ${\cal O}(1)$, and correspond approximately to $1 : |\epsilon| : |\epsilon|^2$. The up-type quark mass hierarchies can be achieved with the same $|\epsilon| \cong 0.03$ but allowing $g_u\sim {\cal O}(10)$ and correspond to $1 : |\epsilon|/|g_u| : |\epsilon|^2/|g_u|^2$. In this setting, we discuss the CKM quark mixing and CP violation.