The formalism of combined finite modular and generalised CP (gCP) symmetries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular S 4 model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus τ being the only source of CP violation.

We investigate models of charged lepton and neutrino masses and lepton mixing based on broken modular symmetry. The matter fields in these models are assumed to transform in irreducible representations of the finite modular group Γ 4 S 4 . We analyse the minimal scenario in which the only source of symmetry breaking is the vacuum expectation value of the modulus field. In this scenario there is no need to introduce flavon fields. Using the basis for the lowest weight modular forms found earlier, we build minimal phenomenologically viable models in which the neutrino masses are generated via the type I seesaw mechanism. While successfully accommodating charged lepton masses, neutrino mixing angles and mass-squared differences, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as specific correlations between the values of the atmospheric neutrino mixing parameter sin 2 θ 23 and i) the Dirac CPV phase δ, ii) the sum of the neutrino masses, and iii) the effective Majorana mass in neutrinoless double beta decay. We consider also the case of residual symmetries Z ST 3 and Z S 2 respectively in the charged lepton and neutrino sectors, corresponding to specific vacuum expectation values of the modulus.

In the framework of the modular symmetry approach to lepton flavour, we consider a class of theories where matter superfields transform in representations of the finite modular group Γ 5 A 5 . We explicitly construct a basis for the 11 modular forms of weight 2 and level 5. We show how these forms arrange themselves into two triplets and a quintet of A 5 . We also present multiplets of modular forms of higher weight. Finally, we provide an example of application of our results, constructing two models of neutrino masses and mixing based on the supersymmetric Weinberg operator.

We construct phenomenologically viable models of lepton masses and mixing based on modular A 4 invariance broken to residual symmetries Z T 3 or Z ST 3 and Z S 2 respectively in the charged lepton and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfully describing the charged lepton masses, neutrino mass-squared differences and the atmospheric and reactor neutrino mixing angles θ 23 and θ 13 , these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as the existence of specific correlations between i) the values of the solar neutrino mixing angle θ 12 and the angle θ 13 (which determines θ 12 ), ii) the values of the Dirac CPV phase δ and of the angles θ 23 and θ 13 , iii) the sum of the neutrino masses and θ 23 , iv) the neutrinoless double beta decay effective Majorana mass and θ 23 , and v) between the two Majorana phases. *