The $q$-unit circle

TL;DR

This paper introduces the $q$-unit circle in global function fields, demonstrating its properties analogous to the classical circle, and develops related structures like hyperbolic plane and modular forms.

Contribution

It defines the $q$-unit circle and explores its properties, extending classical concepts to the setting of global function fields with new geometric and algebraic structures.

Findings

The $q$-unit circle has properties similar to the classical circle.

Mutually tangent horoballs satisfy a Descartes-type relation.

The hyperbolic plane is uniquely determined by the $q$-unit circle.

Abstract

We define the unit circle for global function fields. We demonstrate that this unit circle (endearingly termed the \emph{$q$-unit circle}, after the finite field $\mathbb{F}_q$ of $q$ elements) enjoys all of the properties akin to the classical unit circle: center, curvature, roots of unity in completions, integrality conditions, embedding into a finite-dimensional vector space over the real line, a partition of the ambient space into concentric circles, M\"{o}bius transformations, a Dirichlet approximation theorem, a reciprocity law, and much more. We extend the exponential action of Carlitz by polynomials to an action by the real line. We show that mutually tangent horoballs solve a Descartes-type relation arising from reciprocity. We define the hyperbolic plane, which we prove is uniquely determined by the $q$-unit circle. We give the associated modular forms and Eisenstein series.