Modular invariant dynamics and fermion mass hierarchies around τ = i

A bstract We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point τ = i , where modular invariant theories possess a residual Z 4 invariance. In this region the breaking of Z 4 can be fully described by the spurion ϵ ≈ τ − i , that flips its sign under Z 4 . Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the Z 4 symmetry at τ = i , are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of |ϵ| . We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal Kähler potential.