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 $\tau$ we find that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of $\mathbb{Z}_3^{ST}$ residual symmetry, $\tau \simeq e^{2\pi i/3}$. 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.