$q$ -单位圆
The $q$-unit circle
摘要：
我们定义了全局函数域的单位圆。我们证明了这个单位圆（在$q$元素的有限域$\mathbb{F}_q$之后，亲切地称为\emph{$q$-unit circle}）享有类似于经典单位圆的所有性质：中心，曲率，完备化中的单位根，完整性条件，嵌入到实直线上的有限维向量空间中，将周围空间划分为同心圆，莫比乌斯变换，狄利克雷近似定理，互易定律，等等。我们将Carlitz的多项式指数作用推广到实数线的作用。我们发现，相互相切的horoball解决了笛卡尔型的关系所产生的互惠。定义了双曲平面，证明了它是由$q$ -单位圆唯一确定的。给出了相应的模形式和Eisenstein级数。
英文摘要：
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, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
