Fermion Hierarchies from Fricke Modular Symmetry

We show how Fricke modular symmetry can generate realistic fermion hierarchies and quark CP violation from a single modulus in a supersymmetric effective theory. Matter fields are labelled by the two one-dimensional characters of ${\mathrm{\Gamma }}_0^{+}(N)$ that are trivial on ${\mathrm{\Gamma }}_0(N)$, and interactions use complete spaces of cusp-regular holomorphic modular forms. At level three, the Fricke extension enlarges an interior stabilizer from order three to order six, allowing its parity to enforce three additional powers of the displacement $t$. An explicit assignment gives quark mass powers $(7,3,0)$ and $(5,4,2)$ and CKM powers $(1,2,3)$. The leading textures conserve CP, but an exact relation among the modular generators produces an appreciable CKM phase at finite displacement. A fit at $\mathrm{\mid }t\mathrm{\mid }\simeq 0.306$ reproduces the quark and lepton reference values with order one coefficients in a fixed normalization. We also relate the Fricke double-coset algebra to coset non-invertible symmetry; on the coefficient spaces used here its action reduces to ordinary Fricke parity.