The -unit circle
arXiv:1801.09147
Abstract
We define the unit circle for global function fields. We demonstrate that this unit circle (endearingly termed the \emph{-unit circle}, after the finite field of 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. 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 -unit circle. We give the associated modular forms and Eisenstein series.
32 pages