代数曲线的K2群与二元多项式的Mahler测度

There are deep relations between the Mahler measure of some polynomials and the special value of L-functions of elliptic curves. In many cases, the Beilinson's conjecture about the K2 group of algebraic curves could explain these relations very well. This project will establish new relations between the Mahler measure of different families of polynomials and new relations between the Mahler measure of polynomials and the special value of L-functions of elliptic curves by studying the K2 group of algebraic curves defined by certain bivariate polynomials. The content of this project includes three aspects. Firstly, we will give relations of Mahler measure between certain birational families of polynomials. In particular, we will establish relations between the Mahler measure of certain families of reciprocal and non-reciprocal polynomials which define elliptic curves. Secondly, for certain families of hyperelliptic curves of genus g equal to 2 or 3 defined by tempered polynomials, we plan to construct g linearly independent elements in the K2 group. If the Jacobian of the curves split into the product of elliptic curves, we will study the relation between these elements and the K2 group of corresponding elliptic curves. Thirdly, we plan to establish relations between the Mahler measure of certain families of hyperelliptic curves of genus equal to 3, special value of L-functions of elliptic curves and the Mahler measure of corresponding families of elliptic curves.

多项式的Mahler测度与椭圆曲线L函数的特殊值之间存在深刻的联系。在许多情况下，关于代数曲线K2群的Beilinson猜想可以很好的解释这些联系。本项目将通过研究某些二元多项式定义出代数曲线的K2群，建立不同多项式族Mahler测度之间以及多项式Mahler测度与椭圆曲线L函数特殊值之间新的联系。本课题的研究内容包括以下三方面：第一，对某些双有理等价的多项式族给出其Mahler测度之间的联系。特别地，建立某些定义出椭圆曲线的自反多项式与非自反多项式之间Mahler测度的关系。第二，对某些由tempered多项式定义的亏格为2和3的超椭圆曲线族构造K2群中的亏格个线性无关元素。如果曲线的雅克比簇分裂为椭圆曲线的乘积，研究这些元素与相应椭圆曲线K2群之间的关系。第三，对某些亏格为3的超椭圆曲线族建立Mahler测度与椭圆曲线L函数特殊值以及相应椭圆曲线族Mahler测度之间的联系。

多项式的Mahler测度与椭圆曲线L函数的特殊值之间存在深刻的联系。在许多情况下，关于代数曲线K2群的Beilinson猜想可以很好的解释这些联系。本项目研究了不同多项式族Mahler测度之间以及多项式Mahler测度与椭圆曲线L函数特殊值之间新的联系，同时本项目还研究了有限阿贝尔群代数的K2群和K3群的结构。本项目圆满完成了预定目标，已经在SCI期刊Journal of Algebra and its Applications发表论文一篇，另有两篇论文在撰写中。主要研究成果包括以下三方面：第一，对某些由tempered多项式定义的亏格为2的超椭圆曲线族构造K2群中的两个线性无关元素，由这些元素出发给出了定义亏格2曲线的非自反多项式族与定义出亏格1曲线的多项式族Mahler测度之间的关系。著名数论学家Wadim Zudilin评价这一结果为"spectacular and exceptional"。第二，建立了某些定义亏格3超椭圆曲线的自反多项式族Mahler测度与椭圆曲线L函数特殊值以及相应定义出亏格1和亏格2曲线的多项式族Mahler测度之间的联系。第三，具体计算出了一些有限阿贝尔群代数的K2群和K3群的结构。