# Is there a theory of Maaß forms over finite fields ?

Here is a somewhat naïve question which must have occurred to many people, so it would be nice to record here the attempts at an answer :

Is there a theory of Maaß forms over $\overline{\mathbf{F}}_p$ (where $p$ is a prime) which would be related to even representations $$\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\overline{\mathbf{F}}_p)$$ in the same way as the theory of modular forms over $\overline{\mathbf{F}}_p$ is related via Serre's conjecture (now a theorem of Khare-Wintenberger) to odd representations $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\overline{\mathbf{F}}_p)$ ?

Somehow, for $p=2$, the two theories (Maaß forms and modular forms over $\overline{\mathbf{F}}_2$) will have to coincide.

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I think your title is misleading as it suggests that all Maass forms are attached to Galois representations (while only very special ones are). –  GH from MO Jun 16 '12 at 8:54
Maybe I should have said "Maaß forms of Galois type". What I'm looking for is a thingummy $f_\rho$ which would do for even representations $\rho$ what modular forms (of a certain kind) do for odd representations, namely to parametrise them in such a way that 1) the $f_\rho$ are easier to compute and understand, and 2) all invariants of $\rho$ can be recovered from $f_\rho$. So basically the question amounts to : Do we understand --- at least conjecturally --- even representations as well as we understand odd representations ? –  Chandan Singh Dalawat Jun 16 '12 at 12:32