Flavor symmetries from modular subgroups in magnetized compactifications

We study the flavor structures of zero-modes, which are originated from the modular symmetry on $T_1^2\times T_2^2$ and its orbifold with magnetic fluxes. We introduce the constraint on the moduli parameters by ${\tau }_2=N{\tau }_1$, where ${\tau }_i$ denotes the complex structure moduli on $T_i^2$. Such a constraint can be derived from the moduli stabilization. The modular symmetry of $T_1^2\times T_2^2$ is $SL(2,\mathbb{Z}{)}_{{\tau }_1}\times SL(2,\mathbb{Z}{)}_{{\tau }_2}\subset Sp(4,\mathbb{Z})$ and it is broken to ${\mathrm{\Gamma }}_0(N)\times {\mathrm{\Gamma }}^0(N)$ by the moduli constraint. The wave functions represent their covering groups. We obtain various flavor groups in these models.