On unramified automorphic forms over the projective line

Abstract

Let $q$ be a prime power and ${\mathbb{F}}_q$ be the finite field with $q$ elements. In this article we investigate the space of unramified automorphic forms for ${\mathrm{PGL}}_n$ over the rational function field defined over ${\mathbb{F}}_q$ (i.e.\ for ${\mathbb{P}}^1$ defined over ${\mathbb{F}}_q$). In particular, we prove that the space of unramified cusp form is trivial and (for $n=3$) that the space of eigenforms is one dimensional. Moreover, we show that there are no nontrivial unramified toroidal forms for ${\mathrm{PGL}}_3$ over ${\mathbb{P}}^1$ and conjecture that the space of all toroidal automorphic forms is trivial.

arXiv comment15 pages