2018/01/27 by Kenneth Ward, Ward, Kenneth
Mathematics · #11B57 #11E12 #11E76 #11F03 #11J61 #11R18 #11R60 #11T22 #52C26 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B57 #msc:11E12 #msc:11E76 #msc:11F03 #msc:11J61 #msc:11R18 #msc:11R60 #msc:11T22 #msc:52C26
paper · pdf · doi:10.48550/arxiv.1801.09147
32 pages
arxiv created 2018/01/27 · arxiv updated 2018/01/30
We define the unit circle for global function fields. We demonstrate that this unit circle (endearingly termed the q-unit circle, after the finite field \mathbbFq 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. 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.