Here it is said that a Hauptmodul (a generator of a modular function field) is unique up to a Möbius transformation.
My impression is that it is really hard to find references on Hauptmoduln and their properties such as that uniqueness up to Möbius transformation.
Has anybody references on this?
So I'm not sure about references specific to Hauptmoduln (except specialized research articles), so perhaps you would be better served by reading up on modular functions, forms and curves. I suggest Milne's online notes which are excellent, or Diamond and Shurmon's book which is also excellent.
As far as your specific and interesting question regarding the uniqueness of the Hauptmodul up to Möbius transformation, it boils down what are the automorphisms of a function field. You can see at this MSE question detailed answer as to why they must be Möbius transformations.
