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Let $\Gamma \leq SL_2(\mathbb{Z})$ be a congruence subgroup, $X$ the corresponding compact modular curve. I often see the statement (for example, in many posts here on SE) that modular forms (of weight $k$) $M_k(\Gamma)$ can be interpreted as being global sections of a line bundle $\mathcal{L}$ on $X$. However I am getting confused between two meanings of this:

  1. For each $k \in \mathbb{N}$, there is a line bundle $\mathcal{L}_k$ such that $M_k = H^0(X,\mathcal{L}_k)$

  2. There exists a line bundle $\mathcal{L}$ such that for each $k \in \mathbb{N}$, we have $M_k = H^0(X, \mathcal{L}^{\otimes k})$

Clearly 2 implies 1. But when are these statements true? Is 1 always true, for instance? Or do we need further conditions on $\Gamma$, concerning for example, elliptic elements and irregular cusps?

I've seen it also stated in some places (for example, Eyal Goren's lectures, also Milne's notes construct the line bundle assuming free action of $\Gamma$) that the existence of a line bundle in the sense of 1 requires $\Gamma$ not to have elliptic elements or irregular cusps. But in other places I just see the assertion that this line bundle exists, with no conditions on $\Gamma$. Some people use the canonical bundle on $X$ and assert that this gives the modular forms as global sections in the sense of 2. But for some reason I find it hard to believe that this is always true; in any case, I couldn't find any clear reference about it.

I would greatly appreciate any help to clear away my confusion. Most references I was able to find about this are very sketchy and confusing. Does anyone know a good place to look it up?

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1 Answer 1

The statements are both true, but not because they are equivalent. The line bundle $L$ you consider is probably the dualizing sheaf $\omega$ (or just canonical sheaf or just sheaf of relative differentials depending on your base scheme). Then modular forms of weight $k$ correspond to global sections of $\omega$.

Maybe Peter Bruin's thesis could help here.

See http://user.math.uzh.ch/bruin/thesis.pdf

Especially

Equation I.2.1 (The language is a bit heavy in this section but it becomes more down to earth later).

page 76 might also be helpful.

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Thanks. The line bundle considered in the equation you pointed to is a line bundle on a moduli stack, and not on a scheme. This stack is representable for level structure $\Gamma_1(n)$, $n\geq 5$ (says so on page 8). I know that $\Gamma_1(n)$ has no elliptic points. Does that have to do maybe with whether this line bundle can be moved to the modular curve $H^*/\Gamma$ (which is a Riemann surface)? –  Nadim Rustom May 19 '12 at 10:37
    
@Hamburger: Are you sure? I agree that statement (1) is true for any $\Gamma$, but I am sure that statement (2) is false -- if it were true, then by Riemann-Roch the dimension of $M_k$ would be a linear function of $k$ for $k \gg 0$, which is false for general $\Gamma$. –  David Loeffler May 19 '12 at 11:55
    
@David. Hmm, I think I was confused. You're right that statement 2 is false in this generality. –  Hamburger May 19 '12 at 11:57
    
A necessary condition for $\mathcal{L}^{\otimes k}$ to descend to $\Gamma$ is that $k$ should kill the torsion in $\Gamma$, and that $k$ should also be even if $\Gamma$ has irregular cusps. I think this condition is also sufficient but I'm not sure. –  David Loeffler May 19 '12 at 12:13
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When I said "descend", I meant: choose some auxilliary level structure, say full $\Gamma(N)$ level structure for some $N \gg 0$, and consider $X(\Gamma \cap \Gamma(N))$; this certainly has a sheaf $\mathcal{L}$ on it, and also there is a natural Galois cover $X(\Gamma \cap \Gamma(N)) \to X(\Gamma)$ and one can ask if $\mathcal{L}$ descends. This is an alternative and more explicit approach that avoids using stacks. –  David Loeffler May 19 '12 at 16:04

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