# Real weight modular forms as sections of a line bundle

Background: I've been trying to read

Baily, Walter L., Jr. The decomposition theorem for V-manifolds. Amer. J. Math. 78 1956

and I have problems with the language used in the paper. Firstly I am not familiar with V-manifolds (or orbifolds). Secondly I want to apply some of the theorems in the paper to modular forms of real weight and I don't know much about the 'modular forms as sections of a line bundle on $\Gamma/\mathbb{H}$ point of view, for $\Gamma$ a Fuchsian group of first kind and $\mathbb{H}$ the upper halfplane.

My main question is: Can we view modular forms of arbitrary weight $r$ and multiplier system $v$ as sections of a line bundle $L_{r,v}$ on $\Gamma/\mathbb{H}^*$ viewed as an orbifold?

My main reference for this are Milne's lecture notes on modular forms (Proposition 4.25) where he explains briefly a factor of automorphy (includes real weight) corresponds uniquely to a line bundle $L$ on $\Gamma/\mathbb{H}$ and an isomorphism $\mathbb{H}\times \mathbb{C}\cong p^*(L)=\{(h,l)\in \mathbb{H}\times L|p(h)=\pi(L)\}$, where $p$ is the quotient map $\mathbb{H}\rightarrow \Gamma/\mathbb{H}$ and $\pi$ is the map $L\rightarrow\Gamma/\mathbb{H}$. He then says that for even integer weight $2k$ one can extend this line bundle to a line bundle $L^*$ on $\Gamma/\mathbb{H}^*$ and sections of $L^*$ are modular forms of weight $2k$. Is there any reason for restricting the weight here? To me it seems that one can do this with any factor of automorphy.

The other restriction he makes is that $\Gamma$ should act freely on $\mathbb{H}$. Can one ommit this condition by talking about line bundles on orbifolds the orbifold $\Gamma/\mathbb{H}$ or is there a caveat that I am missing?

## 1 Answer

The map $f(z) \mapsto f(z)\cdot(dz)^k$ establishes a bijection between modular forms of weight $2k$ on $\Gamma \backslash \mathbb{H}^*$ and sections of $\Omega^{\otimes k}(\log \text{cusps})$, the $k$th pluricanonical bundle with at most log poles at cusps. Because we are working with a compact curve, line bundles have a well-defined degree that takes discrete values, and when $k$ is positive, the line bundle $\Omega^{\otimes k}(\log \text{cusps})$ has nonzero degree. To sum up, we can't interpolate indefinitely between weights, because the line bundles of the corresponding degrees may not exist.

You can remove the freeness condition by considering orbifolds. Also, you may talk about forms of arbitrary real or complex weight on compactified curves by replacing the curves with suitable analytic gerbes. Unfortunately, I do not know a good reference for such constructions.

• Thanks! I think the construction you mention is explained very well in Diamond and Shurman's book, they use it to calculate dimension formulas for modular forms. However I'm not sure if that answers my question. I am not looking to write general weight $r$ modular forms as sections of line bundles but only forms with a particular multiplier system. I.e. forms that transform like this $$f(\gamma z) = v(\gamma)j(\gamma,z)^r f(z),$$ where $v$ is a multiplier system on $\Gamma$ and $j(\gamma,z)^r=exp(\log(j(\gamma,z))\cdot r)$ for a previously chosen branch of log. – MichalisN May 5 '14 at 9:51