The space of lattices and modular forms of weight 1/2

2012-06-18T11:05:50Z

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.

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

Answer by David Loeffler for The space of lattices and modular forms of weight 1/2

2012-06-18T12:12:17Z

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 $Mp_2$.

The space of lattices and modular forms of weight 1/2 - MathOverflow

The space of lattices and modular forms of weight 1/2

Answer by David Loeffler for The space of lattices and modular forms of weight 1/2