If modular forms are (global?) sections of the structural sheaf of the moduli space of elliptic curves, the differential forms view will just be the standard construction of sheaf of 1-differentials. Similarly, since elliptic curves are easily defined over arithmetic fields, arithmetic modular forms will just be same thing, but over C_p or something like that.
I actually might be totally off in the above, but I doubt I am: that’s the power of Grothendieck approach, where everything just falls into its natural place in the framework.
There is rich structure in this area of maths that goes well beyond just sections of some sheaf, or at least this is what Serre, Deligne, Langlands, Mazur, Katz, Hida, Taylor, Wiles and many others seem to think.
First, they can be differential forms, not only functions. Second, there's an important note that we don't look only at things over C. For example, specifically in the context of Fermat's Last Theorem, we need Hida's theory of p-adic families of modular forms. Much of the arithmetic of modular forms comes from the modular curves being algebraic and (almost) defined over the integers.