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.