~~I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of trying to internalize some of the basic definitions of the theory.~~

It's commonly said that one way to think about the definition of a modular form $f(z)$ of weight $k$ is that the $k$-fold differential $f(z) (dz)^k$ is invariant under $\Gamma(N)$, i.e. it defines a $k$-fold differential on $Y(N)$. According to Milne, these two definitions are only equivalent for meromorphic modular and differentials respectively, and the analogous relationship between modular forms and holomorphic $k$-fold differentials is more complicated.

**Question 1:** What is the nature of the conceptual relationship between modular forms and holomorphic differentials? In other words, to what extent is the construction of modular forms a special case of a more general construction for an arbitrary Riemann surface, and to what extent does it depend on special properties of $Y(N)$ (and what are those properties)?

Now that I've read more carefully, this question is more or less resolved by Lemma 4.11 in Milne. The point seems to be that the two definitions of the order of a pole coming from modular forms and from differentials disagree at the elliptic points and cusps because fixed points of the group action count with different multiplicity, or something. So now I'm only interested in the second question. (The original title of this question was, somewhat tongue-in-cheek, "what is a modular form?")

Another way to think about modular forms is that they are particularly well-behaved functions on the set of lattices in $\mathbb{C}$. Now, the set of lattices in $\mathbb{C}$ forms a locally finite poset under inclusion, and Gian-Carlo Rota has taught me to think about incidence algebras whenever I see functions on a locally finite poset. This perspective seems relevant to the combinatorial definition of the Hecke operators so I want to know if it can be developed more thoroughly.

**Question 2:** What is the relationship, if any, between modular forms and the incidence algebra of the poset of lattices in $\mathbb{C}$ under inclusion? In particular, does Mobius inversion have any significance?

(I'm not really sure how to tag this.)