On unramified automorphic forms over the projective line

TL;DR

This paper studies unramified automorphic forms over the projective line, proving the triviality of cusp forms and the one-dimensionality of eigenforms for PGL_n, with specific results for PGL_3, and conjectures about toroidal forms.

Contribution

It establishes new results on the structure of unramified automorphic forms over function fields, including triviality of cusp forms and dimensionality of eigenforms, and proposes conjectures on toroidal forms.

Findings

The space of unramified cusp forms is trivial.

For n=3, the space of eigenforms is one-dimensional.

No nontrivial unramified toroidal forms for PGL_3 over P^1.

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.