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.