Fermion masses and mixing from the double cover and metaplectic cover of the A5 modular group

We perform a comprehensive study of the homogeneous finite modular group A₅^' which is the double covering of A₅. The integral weight and level 5 modular forms have been constructed up to weight 6 and they are decomposed into the irreducible representations of A₅^'. Then we perform a systematical analysis of the A₅^' modular models for lepton masses and mixing. The phenomenologically viable models with minimal number of free parameters and the results of fit are presented. We find out 15 models with 9 real free parameters which can accommodate the experimental data of lepton sector. After including generalized $CP$ symmetry, 9 viable models with 7 free parameters are found out. We apply A₅^' modular symmetry to the quark sector, and a quark-lepton unification model is given. The framework of modular invariance is extended to include the rational weight modular forms of level 5. The ring of modular forms at level 5 can be generated by two algebraically independent weight $1/5$ modular forms denoted by F₁(τ) and F₂(τ). We give the expressions of the rational weight modular forms of level 5 up to weight 3 and arrange them into the irreducible multiplets of finite metaplectic group Γ₅A₅^'×Z₅. A neutrino mass model with Γ₅ modular symmetry is presented, and the phenomenological predictions of the model are analyzed numerically.