Flavor symmetries from modular subgroups in magnetized compactifications

We study the flavor structures of zero-modes, which are originated from the modular symmetry on T²₁× T²₂ and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by τ₂=Nτ₁, where τᵢ denotes the complex structure moduli on T²ᵢ. Such a constraint can be derived from the moduli stabilization. The modular symmetry of T²₁ × T²₂ is SL(2,Z)τ₁ × SL(2,Z)τ₂ ⊂ Sp(4,Z) and it is broken to Γ₀(N) × Γ⁰(N) by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.