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.