The q-unit circle: The unit circle in prime characteristics and its properties
| Autor: | Jacob Ward |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Root of unity Applied Mathematics Hyperbolic geometry 010102 general mathematics General Engineering 0102 computer and information sciences Reciprocity law Curvature 01 natural sciences Theoretical Computer Science symbols.namesake Unit circle 010201 computation theory & mathematics Eisenstein series symbols 0101 mathematics Real line Mathematics Vector space |
| Zdroj: | Finite Fields and Their Applications. 58:222-256 |
| ISSN: | 1071-5797 |
| DOI: | 10.1016/j.ffa.2019.04.006 |
| Popis: | 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 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, Mobius 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect