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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.)

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The ordering on lattices has nothing to do with the definition of a modular form, but if as well as remembering the ordering you remember the index of the smaller lattice in the larger, then the resulting "labelled ordering" has got a lot to do with Hecke operators. – Kevin Buzzard Feb 2 '10 at 11:42
As for your question about differentials, I think the issue Milne is highlighting is simply that the standard definition of modular form includes some boundedness condition at infinity as well as the functional equations, and the boundedness condition then translates into a boundedness condition on the differentials, but if you were just to take a guess, you'd guess wrong. For example weight 2 modular forms are not holomorphic differentials on the compactified modular curve---that's weight 2 cusp forms. Weight 2 modular forms are differentials with at most simple poles at the cusps. – Kevin Buzzard Feb 2 '10 at 11:44
You a.s. know this, but by the Hecke-Petersson theorem, the Hecke operators (form a C*-algebra w/r/t the Petersson inner product and) have a basis of simultaneous eigenvectors in the space of cusp forms. Möbius inversion gives nontrivial relationships between the Hecke operators. – Steve Huntsman Feb 2 '10 at 12:03
For the record, it is only forms of even weight that can be interpreted in terms of holomorphic differentials. The forms of odd weight correspond to the cotangent bundle of the universal elliptic curve at the identity, and I feel like the connection between the tensor square of this and the cotangent bundle has nontrivial content. – Tyler Lawson Feb 2 '10 at 13:58
@Tyler, I think the connection is typically called the "Kodaira-Spencer style isomorphism" e.g., in Katz's p-adic properties paper. – S. Carnahan Feb 2 '10 at 16:40

Because $T_m T_n = \sum_{d|(m,n)} T_{mn/d^2}$ Möbius inversion (apparently) gives

$T_{mn} = \sum_{d|(m,n)} \mu(d)T_{m/d}T_{n/d}$

I have not bothered to work this out (it has been well over a decade since I messed around with modular forms at all) but am cribbing from a homework assignment PDF. But see also here.

It looks like Carol Hamer's article here also has some relevance to your question.

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Regarding the specific applications to modular forms, the book you linked mentions how the formula you wrote allows you to express some coefficients of eigenforms in terms of other coefficients. – S. Carnahan May 4 '10 at 4:29

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