The wide range of choices in the definition of an automorphic form is particularly annoying. Depending on the purposes, it could be a meromorphic function fully invariant by a certain discrete group of transformation, a holomorphic function almost-invariant, a differential form, a subrepresentation of an $L^2$ space, a classical or an adelic object, a solution to a partial differential equation, etc.

These are sometimes related, sometimes definitely different, and the lack of vocabulary consistency in such an active field requires some care when dealing with object (leading authors to usually restate a precise definition of the objects they are dealing with, or at least quoting the literature associated with the paradigm they have chosen.