Minimal lepton models with non-holomorphic modular $A_{4}$ symmetry

We present a comprehensive bottom-up analysis of lepton mass and mixing based on the non-holomorphic $A_{4}$ modular symmetry. Neutrinos are assumed to be Majorana particles and the light neutrino masses are generated through the Weinberg operator. In this framework, we construct all phenomenologically viable models with minimal number of free parameters, where the Yukawa couplings are expressed in terms of polyharmonic Maa{\ss} forms of weights $\pm4$, $\pm2$ and $0$ at level $N=3$. Without imposing generalized CP (gCP) symmetry, we identify 147 (6) viable models with seven real free parameters that successfully reproduce the current experimental data of lepton sector for the normal (inverted) mass ordering. When gCP symmetry consistent with $A_{4}$ modular symmetry is included, the number of free parameters is reduced by one, yielding 47 (5) phenomenologically viable models in the normal (inverted) mass ordering. Finally, we present detailed numerical analyses of a representative model for both mass orderings to illustrate these results.