Modular flavour symmetries and modulus stabilisation

We study the problem of modulus stabilisation in the framework of the modular symmetry approach to the flavour problem. By analysing simple UV-motivated CP-invariant potentials for the modulus τ we find that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of $$ {\mathrm{\mathbb{Z}}}_3^{ST} $$ residual symmetry, τ ≃ e2πi/3. Stabilising the modulus at these novel minima breaks spontaneously the CP symmetry and can naturally explain the mass hierarchies of charged leptons and possibly of quarks.