The q-unit circle: The unit circle in prime characteristics and its properties

Abstract

We define the unit circle for global function fields. We demonstrate that this unit circle, endearingly termed the q-unit circle (pronounced “cue-nit”), after the finite field Fq 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, Möbius transformations, a Dirichlet approximation theorem, a reciprocity law, and much more. In addition, we extend the polynomial exponential action of Carlitz to an action by all points on 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; and we give the associated modular forms and Eisenstein series.