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We study the modular symmetry of zero-modes on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.

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We study orbifolding by the ℤ2(per) permutaion of T12 × T22 with magnetic fluxes and its twisted orbifolds. We classify the possible three generation models which lead to nonvanishing Yukawa couplings on the magnetized T12 × T22 and orbifolds including the ℤ2(per) permutation and ℤ2(t) twist. We also study the modular symmetry on such orbifold models. As an illustrating model, we examine the realization of quark masses and mixing angles.

We study Majorana neutrino masses induced by D-brane instanton effects in magnetized orbifold models. We classify possible cases, where neutrino masses can be induced. Three and four generations are favored in order to generate neutrino masses by D-brane instantons. Explicit mass matrices have specific features. Their diagonalizing matrices correspond to the bimaximal mixing matrix in the case with even magnetic fluxes, independently of the modulus value τ . On the other hand, for odd magnetic fluxes, diagonalizing matrices correspond nearly to the tri-bimaximal mixing matrix near τ = i, while they become the bimaximal mixing matrix for larger Imτ . For even fluxes, neutrino masses are modular forms of the weight 1 on T 2 /Z 2 , and they have symmetries such as S 4 and ∆ (96) × Z 3 .