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 Gamma(4N) and show that they can be decomposed into the irreducible multiplets of finite metaplectic group Gamma(4N). We construct concrete expressions of the half-integral/integral modular forms for Gamma(4) up to weight 6 and arrange them into the irreducible multiplets of (Gamma) over tilde (4). We present three typical models with (Gamma) over tilde (4) modular symmetry for neutrino masses and mixing, and the phenomenological predictions of each model are analyzed numerically.