The space of lattices and modular forms of weight 1/2 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T01:42:47Z http://mathoverflow.net/feeds/question/99886 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99886/the-space-of-lattices-and-modular-forms-of-weight-1-2 The space of lattices and modular forms of weight 1/2 Freddie Manners 2012-06-18T11:05:50Z 2012-06-20T18:07:37Z <p>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.</p> <p>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":</p> <blockquote> <p>$f(a\ R_\theta\ T) = a^{-k} e^{- i k \theta} f(T)$</p> </blockquote> <p>where $R_\theta$ denotes the appropriate rotation matrix, and $a$ a positive real.</p> <p>[More precisely $f$ only has to be defined on one connected component of the space.]</p> <p>(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.</p> <hr> <p>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?</p> <p>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:</p> <ul> <li>What group plays the role of $SL_2(\mathbb{Z})$?</li> <li>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:<br> $G_4(\Lambda) = \sum_{w \in \Lambda \setminus 0} w^{-4}$<br> ?</li> </ul> <hr> <p>Apologies in advance if this is standard -- I've been unable to locate a satisfactory answer in the literature.</p> <p>Thanks,<br> Freddie</p> http://mathoverflow.net/questions/99886/the-space-of-lattices-and-modular-forms-of-weight-1-2/99892#99892 Answer by David Loeffler for The space of lattices and modular forms of weight 1/2 David Loeffler 2012-06-18T12:12:17Z 2012-06-18T12:12:17Z <p>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 $Mp_2$. </p>