On unramified automorphic forms over the projective line

Roberto Alvarenga; Valdir Pereira Junior

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arXiv · 2301.11369

On unramified automorphic forms over the projective line

Roberto Alvarenga·Valdir Pereira Junior

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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.

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Roberto Alvarenga, Valdir Pereira Junior. 2023-01-26. On unramified automorphic forms over the projective line. https://arxiv.org/abs/2301.11369

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