Suppose my favorite way of thinking about modular forms is as functions on the space of (real, 2D) lattices. One can identify this space with $SL\_2(\mathbb{Z}) \backslash GL\_2(\mathbb{R})$, i.e. bases for the lattice up to reparameterization.


A function $f : SL\_2(\mathbb{Z}) \backslash GL\_2(\mathbb{R}) \rightarrow \mathbb{C}$ is a modular form of weight $k$ if it satisfies the "scaling relation":



> 
> $f(a\ R\_\theta\ T) = a^{-k} e^{- i k \theta} f(T)$
> 
> 
> 


where $R\_\theta$ denotes the appropriate rotation matrix, and $a$ a positive real.


[More precisely $f$ only has to be defined on one connected component of the space.]


(I think) the standard definition is equivalent to this, by considering the value of $f$ on canonical lattices $\langle 1,\ z \rangle$ for $z$ in the complex upper half-plane. Note it's entirely clear in this language that e.g. $G\_4$ is a modular form of weight 4.




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So, my question is: is it possible to make sense of modular forms of half-integral weight (for concreteness, say $\vartheta$) in a similar way?


I'm aware that a necessary step is to pass to some kind of double cover such as $Mp\_2$, to make the scaling relation make sense for $k=1/2$; but I am having trouble making this sufficient. In particular, as sub-questions:


* What group plays the role of $SL\_2(\mathbb{Z})$?
* While I could extend $\vartheta$ by "brute force" to a function on e.g. $Mp\_2$ (by callously applying the scaling relation), is there a natural way to define $\vartheta$ on the larger space, similar to the "obvious" definition:  

$G\_4(\Lambda) = \sum\_{w \in \Lambda \setminus 0} w^{-4}$  

?




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Apologies in advance if this is standard -- I've been unable to locate a satisfactory answer in the literature.


Thanks,  

Freddie



The space of lattices and modular forms of weight 1/2
You need to replace $SL\_2(\mathbb{Z})$ with a discrete subgroup of $Mp\_2$, and you want this subgroup not to contain the kernel of $Mp\_2 \to SL\_2$, since otherwise there are trivially no half-integer-weight forms. If you take a small enough finite-index subgroup of $SL\_2(\mathbb{Z})$, then it will admit a lifting to $Mp\_2$. 


