Half-integral weight modular forms and application to neutrino mass models

We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral weight modular forms for congruence subgroup $Γ(4N)$ and show that they can be decomposed into the irreducible multiplets of finite metaplectic group $\widetildeΓ_{4N}$. We construct concrete expressions of the half-integral/integral modular forms for $Γ(4)$ up to weight 6 and arrange them into the irreducible representations of $\widetildeΓ_4$. We present three typical models with $\widetildeΓ_4$ modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.