most recent 30 from http://mathoverflow.net 2013-05-22T11:35:28Z http://mathoverflow.net/feeds/question/99575 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99575/is-there-a-theory-of-maass-forms-over-finite-fields Chandan Singh Dalawat 2012-06-14T09:08:10Z 2012-06-14T09:08:10Z

<p>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 :</p> <p>Is there a theory of Maaß forms over $\overline{\mathbf{F}}_p$ (where $p$ is a prime) which would be related to <em>even</em> 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 <em>odd</em> representations $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\overline{\mathbf{F}}_p)$ ? </p> <p>Somehow, for $p=2$, the two theories (Maaß forms and modular forms over $\overline{\mathbf{F}}_2$) will have to coincide.</p>