Crossing the Transcendental Divide: From Translation Surfaces to Algebraic Curves

IF 0.7 4区数学 Q2 MATHEMATICS Experimental Mathematics Pub Date : 2022-11-01 DOI:10.1080/10586458.2023.2203413

Turku Ozlum cCelik, S. Fairchild, Yelena Mandelshtam

引用次数: 1

Abstract

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.

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跨越超越鸿沟：从平移曲面到代数曲线

我们研究从通过平移曲面给出的黎曼曲面构造代数曲线，平移曲面是平面中有限多个多边形的集合，边通过平移确定。我们利用离散黎曼曲面理论，给出了一种近似平移曲面的雅可比变换的算法，该平移曲面的多边形可以分解为正方形。我们首先在代数曲线已知的$L$形多边形的情况下实现该算法。对于Jenkins-Strebel代表的具体例子，该算法也可以在任何亏格中实现，这是一个密集的平移曲面族，直到现在，它一直生活在黎曼曲面和代数曲线之间超越划分的分析侧。利用Riemann-theta函数，我们给出了数值实验和由此产生的亏格为5的猜想。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.

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