Aspects of the Modular Symmetry Approach to Lepton Flavour

A new bottom-up approach to the flavour problem based on modular invariance has been recently proposed and has gained considerable attention in the literature. In the present thesis we develop basic aspects of the requisite modular symmetry formalism and explore its application to the lepton flavour problem. After introducing the relevant notions (the modular group, the modulus field and modular forms), we concentrate on the theoretical tools required for model-building such as explicit construction of the modular forms, the interplay of modular and CP transformations and of the related symmetries, classification of residual symmetries and their possible relation to the observed hierarchical flavour patterns. Armed with these tools, we construct and discuss three examples of viable models of lepton flavour: a simple predictive model depending on a small number of parameters, a model with an unbroken residual symmetry which leads to trimaximal neutrino mixing, and a model with a slightly broken residual symmetry which explains the observed pattern of charged-lepton masses without fine-tuning.