Modular symmetry at level 6 and a new route towards finite modular groups

A bstract We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ( N′ )/Γ( N″ ), and the modular group SL(2 , ℤ) is ex- tended to a principal congruence subgroup Γ( N′ ). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of Γ₆ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with Γ₆ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′ , and a benchmark model is constructed.