# Line Bundles on Moduli Space of Elliptic Curves and Modular Forms

I have several basic questions about modular forms I'm having trouble figuring out looking at the literature. Here are two:

It is pointed out in Milne's notes (pdf, bottom of page 44) that the holomorphic 1-form $dz$ transforms like a modular form of weight -2, in that under a modular transformation becomes $(cz + d)^{-2} dz$. Therefore, modular forms $f(z)$ of weight 2 are in correspondence with modular invariant 1-forms $f(z)dz$ and so we can think of them as sections of the line bundle $L$ of such forms over the quotient $\mathbb{H}^2//SL(2,Z)$, where $\mathbb{H}^2$ is the upper half-plane. Likewise, modular forms of weight $2k$ are sections of $L^k$.

It's known (and easy to show) that $H^2(\mathbb{H}^2//SL(2,Z),Z) = Z_{12}$. Does the Chern class of $L$ generate this group?

Or is there a square-root of this bundle whose sections are modular forms of weight 1? I think it cannot be because such forms cannot be invariant under the central element $-1 \in SL(2,Z)$. Probably instead they must be equivariant sections of some rank 2.

It's interesting that this Picard group of the moduli space is torsion. Is this what allows the discriminant modular form of weight 12 to have no zeros in $\mathbb{H}^2$?

Thanks.

• First, some minor confusions with this question: to be able to identify modular forms with sections of line bundles, you have to take compactifications of modular curves. Second, the behavior of moduli spaces of elliptic curves is (in some senses) technically easier at higher level than level $1$. Are you interested in level $1$ in particular or higher level? There are no weight $2$ forms of level $1$. Finally, and most seriously, you need to be very careful talking about "$\mathbb{H}//SL(2,Z)$." One could interpret this as an orbifold, a variety (the $j$-line), or (an open) stack. Mar 12, 2018 at 22:59
• The stack does admit a line bundle $\omega$ such that $\omega^{\otimes 12}$ is trivial and a trivialization is provided by $\Delta$. It's true that odd powers of this bundle do not admit any global sections over any ring $R$ of characteristic different from $2$, but so what? Mar 12, 2018 at 23:03
• Thanks. What you're saying about compactification is the same as some growth condition on the sections right? Also, I would rather work with twisted bundles or gerbes than increase the level. Is that possible? I've seen some string theory literature that seems to point in this direction, but I'd like to hear from a mathematician.
– wzzx
Mar 13, 2018 at 1:48
• Perhaps you should say what you are actually interested in (modular forms over general rings or over $\mathbf{C}$; modular forms of weight two or of higher weight, etc.) Mar 13, 2018 at 1:57
• (Yes, in this case, growth conditions on sections is related to extending over the cusps in some way.) Mar 13, 2018 at 1:58