论文标题
跨越先验鸿沟：从翻译表面到代数曲线
Crossing the transcendental divide: from translation surfaces to algebraic curves
论文作者
论文摘要
我们研究了通过翻译表面给出的riemann表面构建代数曲线的，该曲线是平面中有限多个多边形的集合，侧面通过翻译识别。我们使用离散的Riemann表面的理论给出了一种算法，用于近似雅各布的翻译表面，其多边形可以分解为正方形。在$ l $形的多边形的情况下，我们首先实施该算法，而代数曲线已经知道。该算法在任何属中也实施了詹金斯 - 斯特雷贝尔代表的特定例子，这是一个密集的翻译表面家族，到目前为止，该算法一直生活在riemann表面和代数曲线之间的超越鸿沟的分析方面。使用Riemann theta函数，我们提供了数值实验，并产生的猜想为5属。
We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of $L$ shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins-Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.
