# Do L-functions exist for Half-integral weight modular forms?

Classically, we can attach $L$-functions (with properties like, analytic continuation, functional equation) to Dirichlet characters, Hecke eigenforms, etc...

My question is: can one attach $L$-functions (properties similar to the above) to Half-Integral weight modular eigenforms as well? If they do exist, can someone provide a reference for this?

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Hi Ganesh, welcome to MO!!! –  Filippo Alberto Edoardo Nov 20 '12 at 1:50
By the way, are you aware of the paper by J. Park in Manuscripta Math, (2010)? He generalizes Stevens' construction of distribution-valued modular symbols to half-integral weight forms. From that, a "definition" of a $p$-adic $L$-function follows almost formally (although controlling interpolation properties is of course the hard core stuff....). –  Filippo Alberto Edoardo Nov 20 '12 at 1:56
Thanks for your reference Filippo. I am not aware of this paper. –  N. Kumar Nov 21 '12 at 14:23

Since the original question asked only for analytic continuation, functional equation, I'd like to add that the Mellin transform $\sum a_n n^{-s}$ of the half integral weight modular form $\sum a_n \exp(2 \pi i n z)$ has these two properties, but it lacks an Euler product decomposition even if the modular form is a Hecke eigenform, since the Fourier coefficients at square free indices are not multiplicatively related. A similar phenomenon occurs for Siegel modular forms with the so called Koecher-Maaß series; here the Fourier coefficients at matrices representing maximal lattices (Kern- resp. Stammformen in Brandt's terminology) have no multiplicative relation.

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@Rainer: Thank you very much. This was answer I really expected. Later, I could figure out that it was in Shimura's Annals of Math article on Half-integral weight modular forms. –  N. Kumar Jan 5 '13 at 6:16

You can certainly attach $L$-functions to half-integer weight eigenforms, but you don't get anything really new by doing so: they turn out be versions of $L$-functions of integer weight modular forms. More specifically, there is the "Shimura lifting" map from weight $k + 1/2$ to weight $2k$, which sends eigenforms to eigenforms; and the L-function of a half-integer weight eigenform will be closely related to that of its image under the Shimura lift. See e.g. here:

www.math.wisc.edu/~ono/reu06shimura.pdf

In fact this turns out to be a very powerful way of studying the L-functions of integer weight forms (used, for instance, in Tunnell's work on the congruent number problem, which uses modular forms of weight 3/2 to understand the values at $s=1$ of the $L$-functions of twists of elliptic curves.

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Just to emphasize something touched on in the last sentence here: I've always thought of the natural "$L$-function type object" of a half-integral weight form as the entire family of quadratic twists of its Shimura image form. I don't know enough about the metaplectic/automorphic picture to understand if there's something intrinsic about this point of view, but I've been idly curious about this. For that matter: is there a Langlands correspondence for metaplectic groups? Is the Galois side just the family of quadratic twists of the Galois representation of the Shimura lifting in this case? –  Ramsey Nov 19 '12 at 13:51
@David: Thanks for your reply. I agree that there is a relation between half-integral and integral weight modular forms by Shimura. However, the way the $L$-functions are related in the "Shimura lifting" it only involves $n^2$ terms of the half-integral weight modular forms. What about the non-square free terms? They dont matter much in defining the $L$-function, unlike the classical case? –  N. Kumar Nov 19 '12 at 14:40
Following up on Nick's question -- the Langlands correspondence for metaplectic groups is something I've thought about a lot in the past few years. I have a paper on L-groups for Metaplectic Groups on the ArXiv, for example. Since that paper is hard to read (I'm working hard to get rid of all the Hopf algebra machinery used there), I'll summarize here. An L-function should come from two pieces of data: an automorphic representation of $G$ (for half-integral weight eigenforms, $G$ is a metaplectic group) and an algebraic (finite-dimensional) representation of the L-group ${}^L G$. –  Marty Nov 19 '12 at 16:56
When $G$ is the simplest metaplectic group $\widetilde{SL}_2$, I argue that the L-group ${}^L G$ is a group scheme over $\mathbb{Z}$, which is isomorphic to $SL_2 \times \Gamma$ ($\Gamma$ the Galois group), but noncanonically and not until base change to $\mathbb{ZZ}[i]$. However, each additive character of ${\mathbb A} / {\mathbb Q}$ gives such an isomorphism, and this realizes a bijection between quadratic twists and (L-)isomorphisms of the L-group to $SL_2 \times \Gamma$. Taking the standard representation of $SL_2$, you get the L-function of a quadratic twist. –  Marty Nov 19 '12 at 17:01
So -- long story short -- I share Nick's intuition that the L-function is the entire family of quadratic twists, and I went so far as to cook up a theory of L-groups for (split, for now) metaplectic groups in order to understand this better. –  Marty Nov 19 '12 at 17:02
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Upon David Loeffler's request, here is a more fleshed out version of my former comments:

In his comment, Nick Ramsey mentioned that the natural L-function for a half-integral weight modular form is really the entire family of quadratic twists of an L-function coming from a Shimura-type lift. I agree with Nick's perspective and it motivated me to work towards a more general framework. I don't mean this post as shameless self-promotion, but here's my preprint on Split Metaplectic Groups and their L-groups. I think this will probably not be widely read due to the excessive use of Hopf algebras and reliance on Lusztig's canonical bases. Fortunately, I've recently worked out ways to avoid these completely, and I will hopefully have another preprint up soon.

Back to the question at hand: the general non-metaplectic perspective is that an L-function can be produced from a pair $(\pi, \rho)$ where $\pi$ is an automorphic representation of $G_{\mathbb A}$ and $\rho$ is an algebraic representation of the L-group ${}^L G$. For classical modular forms, one might take $G = PGL_2$ and ${}^L G = SL_2(C) \times \Gamma$ where $\Gamma$ is the absolute Galois group of ${\mathbb Q}$.

My perspective on the metaplectic groups is that again, an L-function should be associated to a pair $(\pi, \rho)$ where $\pi$ is a genuine automorphic representation of $\tilde G_{\mathbb A}$ (this makes sense in a framework of Brylinski-Deligne, for example) and $\rho$ is an algebraic representation of a putative L-group ${}^L \tilde G$. In the simplest case, $\tilde G = Mp_2$ is the metaplectic group. My preprint is devoted to the construction of such an L-group.

The key subtlety, observed by Nick and others, is the ambiguity if one uses the Shimura correspondence as guidance. Indeed, work of Shimura and Waldspurger requires the choice of an additive character, and thus for the definition of an L-function. In my construction, this choice gets wrapped up in the choice of algebriac representation $\rho$ of the L-group.

Roughly speaking, the L-group ${}^L Mp_2$ of $Mp_2$ is noncanonically isomorphic to $SL_2(C) \times \Gamma$. It arises in my preprint from a somewhat magical/contrived twisting of both multiplication and comultiplication in the Hopf algebra of the direct product. A less contrived non-Hopfy approach will appear in a new paper sometime soon (I hope). I hope to tackle global issues as well.

It turns out that every nontrivial additive character $\psi$ of ${\mathbb A} / {\mathbb Q}$ can be used to generate an isomorphism from the L-group ${}^L Mp_2$ to the direct product $SL_2(C) \times \Gamma$, whence a natural 2-dimensional representation $\rho_\psi$. The L-functions produced by various choices of additive character (various isomorphisms of the L-group to the L-group of $PGL_2$) should comprise an orbit, under quadratic twisting, of a single L-function.

To summarize, the L-functions could be written $L(\pi, \rho_\psi)$ for $\pi$ a genuine automorphic representation of $Mp_{2n}$ and $\rho_\psi$ a two-dimensional representation of ${}^L Mp_{2n}$ coming from an additive character $\psi$.

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Thanks Prof. Marty, for throwing some light. –  N. Kumar Nov 21 '12 at 14:24