Multiple modular symmetries as the origin of flavor

We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries S₄^A, S₄^B, and S₄^C, associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bitriplet scalars to the diagonal S₄ subgroup. The low energy effective theory involves the three independent moduli fields ₀, ₁, and ₂, which preserve the residual modular subgroups Z₃^A, Z₂^B, and Z₃^C, in their respective sectors, leading to trimaximal TM₁ lepton mixing, consistent with current data, without flavons.