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)$$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.
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:
$G_4(\Lambda) = \sum_{w \in \Lambda \setminus 0} w^{-4}$
?
Apologies in advance if this is standard -- I've been unable to locate a satisfactory answer in the literature.
Thanks,
Freddie