Minimal Modular Flavor Symmetry and Lepton Textures Near Fixed Points

An extension of the Standard Model with ${\mathrm{\Gamma }}_2\simeq S_3$ modular flavor symmetry is presented. We consider the construction of the lepton sector, augmented by two right-handed neutrino states, in the vicinity of the fixed points $\tau =i\mathrm{\infty }$, $\tau =i$ and $\tau \text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}\omega \text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}-\frac{1}{2}\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}i\frac{\sqrt{3}}{2}$. Due to the residual symmetries at these points, and with the aid of nonholomorphic modular forms (which constitute representations of $S_3$) and by assigning specific transformation properties to the fermion fields, highly economical models (without flavon fields) are constructed with interesting Yukawa textures. All presented models strongly prefer the inverted ordering for the neutrino masses.