Finite Modular Groups and Lepton Mixing

We study lepton mixing patterns which are derived from finite modular groups ΓN, requiring subgroups Gν and Ge to be preserved in the neutrino and charged lepton sectors, respectively. We show that only six groups ΓN with N=3, 4, 5, 7, 8, 16 are relevant. A comprehensive analysis is presented for Ge arbitrary and Gν=Z2×Z2, as demanded if neutrinos are Majorana particles. We discuss interesting patterns arising from both groups Ge and Gν being arbitrary. Several of the most promising patterns are specific deviations from tri-bimaximal mixing, all predicting θ13 non-zero as favoured by the latest experimental data. We also comment on prospects to extend this idea to the quark sector.