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 K₂ of curves defined by these polynomials and applying Beilinson’s conjecture on K₂ of curves. We show that there are two and three explicit linearly independent elements in K₂ of certain families of genus 2 and genus 3 curves, respectively.

摘要

在本文中，我们研究了定义 2 格和 3 格曲线的 500 多个倒多项式族的马勒测度。我们以数值方式找到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定义 2 格和 3 格曲线的不同多项式族的马勒测度之间的 100 多个恒等式。此外，我们研究了定义 genus 2 曲线的几个非互易多项式族的马勒测度，并以数值方式找出这些族的马勒测度与L的特殊值之间的关系-椭圆曲线的函数。我们还发现这些不可逆族的马勒测度与定义 1 种曲线的调节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K ₂曲线中某些元素的前推和后拉以及将贝林森猜想应用于曲线K _{2来解释这些关系。}我们证明了 2 格曲线和 3 格曲线的某些族的K ₂中分别存在两个和三个显式线性独立元素。