Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves

IF 0.7 4区数学 Q2 MATHEMATICS Experimental Mathematics Pub Date : 2021-06-20 DOI:10.1080/10586458.2021.1926014

Hang Liu, H. Qin

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引用次数: 3

Abstract

Abstract In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K 2 of curves defined by these polynomials and applying Beilinson’s conjecture on K 2 of curves. We show that there are two and three explicit linearly independent elements in K 2 of certain families of genus 2 and genus 3 curves, respectively.

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定义2和3属曲线的多项式族的马勒测度

摘要本文研究了500多个互反多项式族的马勒测度，这些族定义了2属曲线和3属曲线。我们用数值方法找到了这些多项式的马勒测度与l函数的特殊值之间的关系。我们还在数值上发现了涉及不同多项式族的马勒测度之间的100多个恒等式，这些多项式族定义了格2和格3曲线。在此基础上，我们进一步研究了若干非互易多项式族的Mahler测度，并通过数值方法找到了这些族的Mahler测度与椭圆曲线l函数的特殊值之间的关系。我们还发现了这些非互易族的马勒测度与定义1属曲线的回火多项式之间的恒等式。我们将通过考虑由这些多项式定义的曲线K 2中某些元素的推进和后退，并将Beilinson猜想应用于曲线K 2来解释这些关系。我们证明了在某些2属和3属曲线族的K 2中分别有两个和三个显式线性无关的元素。

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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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