Automorphy Factors and Bundles

Question

The question I'm considering is the following: given an 1-cocycle in of the modular group in Hom$(H;\textrm{GL}_{r}(C))$ call it $f$ when does it induce a vector bundle structure on the corresponding curve $X(1)$ (or more genrally $X(\Gamma)$)?

The reason why I'm confused over this is because I have two contradictory views on why it should hold in general/not: This first is baased around a similar intuition to filling in punctured Riemann surface - namely we can clearly define a vector bundle over $X(\Gamma)$ without the cusps & points of non-trivial stabilizers. As these points form a finite set we can choose suitable charts for them such that the cocycle doesn't need to be defined at these points so we can just take it to be as for the simpler quotient space.

The competing reasoning is by example - if we look at a non-trivial 1-dim rep of the modular group say $S \mapsto i$ and $T \mapsto -\overline{\rho}$ then this gives us a 1-cocycle (that is independant of $\tau$) and thus would induce a line bundle on $X(1)$. But as this line bundle would have it's 6th power isomorphic to the canonical line bundle this isn't possible.

I don't really feel which of these statements is wrong - any ideas would be musch appreciated.

Answer 1

A cocycle for a modular group is precisely the same as a descent datum for the quotient map $\mathbf{H} \to [\mathbf{H}/\Gamma]$, where the target is the quotient orbifold. This is a special case of the fact that pullback induces an equivalence of categories between vector bundles on the quotient, and vector bundles $V$ on the upper half-plane equipped with a cocycle with coefficients in $\operatorname{Aut}(V)$.

If you want to make a vector bundle on $X(1)$ or $X(\Gamma)$, your cocycle has to satisfy some properties, and you need to specify some additional data. In particular, you need triviality at elliptic points to descend from $[\mathbf{H}/\Gamma]$ to the affine coarse space $\mathbf{H}/\Gamma$ (which is often written $Y(\Gamma)$), and you need a gluing datum to describe the behavior at the cusps in order to define a vector bundle on the compact curve $X(\Gamma)$.

We have a standard example in level 1. The stack $Ell$ has Picard group $\mathbb{Z}/12\mathbb{Z}$ (I think this may be a theorem of Fulton). The trivial bundles descend to the coarse space $Y(1)$, which is an affine line, and all vector bundles on $Y(1)$ are trivial. Adding a cusp yields $X(1)$, which is a projective line, and vector bundles on $X(1)$ are just sums of line bundles parametrized by degree. The upshot is that if you don't specify gluing data at infinity, your cocycle will not tell you anything about the weight of a modular form.

Regarding your comment about level 1 forms of weight 6: they all vanish at $i$.

Answer 2

The way you would want to produce that vector bundle is gluing the trivial vector bundle $\mathbb H \times \mathbb C^r$ together along the maps:

for each $g \in \Gamma$, the map sending $(x,y)$ to $(g(x),\rho_g(y))$

First, note that to be well-defined at the elliptic points, the stabilizers of these elliptic points, meaning the torsion elements of the modular group, must act trivially. This is already enough to force the representation to be trivial, as the modular group is generated by its torsion elements.

One could solve this problem by considering the modular curve $X(1)$ as a stack or an orbifold.

Second, note that even if one could make a nontrivial vector bundle on $\mathbb H/\Gamma$, it is not clear at all how to extend this to the cusps.