Universal predictions of Siegel modular invariant theories near the fixed points

We analyze a general class of locally supersymmetric, CP and modular invariant models of lepton masses depending on two complex moduli taking values in the vicinity of a fixed point, where the theory enjoys a residual symmetry under a finite group. Like in models that depend on a single modulus, we find that all physical quantities exhibit a universal scaling with the distance from the fixed point. There is no dependence on the level of the construction, the weights of matter multiplets and their representations, with the only restriction that electroweak lepton doublets transform as irreducible triplets of the finite modular group. Also the form of the kinetic terms, which here are assumed to be neither minimal nor flavor blind, is irrelevant to the outcome. The result is remarkably simple and the whole class of examined theories gives rise to five independent patterns of neutrino mass matrices. Only in one of them, the predicted scaling agrees with the observed neutrino mass ratios and lepton mixing angles, exactly as in single modulus theories living close to $\tau=i$.