Retrieving texture zeros in 3+1 active-sterile neutrino framework under the action of $A_4$ modular-invariants

The flavour problem and the viability of texture zeros of the Majorana mass matrix in the 3+1 active-sterile neutrino mixing scenario are investigated in this work using a novel bottom-up technique where we leverage the pertinent concepts of full modular group, modular invariants, and $A_4$ flavour symmetry as theoretical tools for the explicit construction of neutrino models. In this approach we treat each chiral field as modular forms in the 3+1 neutrino mixing which are constrained by the $A_4$ modular symmetry. Using these techniques, we create straightforward predictive models that only depends on a few parameters and simultaneously explains the observed pattern of neutrino mixing without the need for fine-tuning, allowing us to perceive the feasible zero textures of the Majorana mass matrix under the 3+1 framework.We discuss the implications of the allowed 3+1 zero textures by analyzing the sterile neutrino parameters, providing insight into the flavor problem and the viability of the Majorana mass matrix. The values of the active-sterile mixing (ASM) parameters predicted from our models are highly consistent with the $3\sigma$ values of the ASM parameters.