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

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation 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 $\tau = i\infty$, $\tau$ being the vacuum expectation value of the modulus. We consider first a model in which the up-type and down-type quark mass matrices $M_u$ and $M_d$ involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. We also consider the case of $M_u$ and $M_d$ depending on the same $\tau$ and involving modular forms of weights 8, 4, 2 and 6, 4, 2, respectively, with $M_u$ receiving a tiny SUSY breaking or higher dimensional operator contribution. Both the mass hierarchies of up-type and down-type quarks as well and the CKM mixing angles and CP violating phase are reproduced successfully with one complex parameter and all parameters being in magnitude of the same order. The relatively large value of ${\rm Im}\,\tau$, needed for describing the down-type quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies.