Abelian symmetries of the N-Higgs-doublet model with Yukawa interactions

We investigate finite abelian groups which can represent symmetries of the N-Higgs-doublet models with quarks. We build a general formalism based on the powerful method of the Smith normal form and obtain an analytic upper bound on the order of abelian symmetry groups for any N. We investigate in detail the case N=2 and rederive known results in a more compact and intuitive fashion. We also study the NHDM with the maximal finite abelian symmetry for all small N cases up to N=5, and show that in each case all Yukawa textures compatible with such symmetry originate from a unique basic structure. This work opens the way to a systematic exploration of phenomenology of the NHDM with a desired symmetry, and illustrates the power of the Smith normal form technique.