关于代数曲线K2群的Beilinson猜想和Deligne猜想的研究

The celebrated Beilinson’s conjecture and Deligne's conjecture are deep and difficult conjectures in algebraic K-theory. They give very detailed descriptions of the structure of K-groups of projective algebraic varieties over number fields and the relations between these K-groups and the value of L-functions of these varieties at integers. This project will study Beilinson's conjecture and Deligne's conjecture about K2 of certain algebraic curves, including the following three aspects. Firstly, we will construct several non-integer elements in the K2 group of families of elliptic curve over rational field. By studying the properties of these elements mod p, we intend to prove that they are linearly independent so as to verify Deligne's conjecture on the lower bound of the rank of K2 groups of these curves. Secondly, we will construct integral elements in K2 groups of non-hyperelliptic curves of arbitrary genus over rational field and prove that they are linearly independent by calculating the regulator so as to verify Beilinson's conjecture for the lower bound of the integral K2 groups of these curves. Thirdly, we will construct infinitely many elements in the K2 groups of certain hyperelliptic curves over number fields and investigate their integrality and linearly independence. Specifically, we will study the relation between the number of linearly independent integral elements and the rank of integral K2 groups predicted by Beilinson's conjecture when the curves are defined over low degree number fields and have genus not greater than 2.

著名的Beilinson猜想和Deligne猜想是代数K理论中极为深刻而困难的猜想，它们对数域上射影代数簇的K群的结构以及K群与代数簇的L函数在整点处的值之间的关系给出了具体的刻画。本项目将研究代数曲线K2群的Beilinson猜想和Deligne猜想，具体包括以下三方面。第一，构造有理数域上的椭圆曲线族K2群中的多个（非整）元素，通过研究这些元素在模p约化处的性质证明它们线性无关，从而对这些曲线K2群秩的下界验证Deligne猜想。第二，构造有理数域上的任意亏格非超椭圆曲线K2群中的整元素，通过计算正则子证明它们线性无关，从而对这些曲线整K2群秩的下界验证Beilinson猜想。第三，构造数域上的超椭圆曲线K2群中的无穷多个元素，研究这些元素的整性和线性无关性。特别地，当数域次数较低且曲线的亏格不大于2时探讨线性无关的整元素个数与Beilinson猜想所预测整K2群的自由秩之间的关系。

Beilinson猜想是关于代数簇K理论最重要的猜想，特别地，它将多项式的Mahler测度与L函数的特殊值联系起来。本项目研究了某些高亏格代数曲线K2群的Beilinson猜想，同时本项目还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。本项目完成了预定目标，已经在SCI期刊Journal of Mathematical Analysis and Applications发表论文一篇，在SCI期刊Journal of Pure and Applied Algebra和Experimental Mathematics各有一篇论文已接收并上网。主要研究成果包括以下三方面：第一，我们构造了几族亏格g=1，2，4，7的代数曲线上g个K2群中相互独立的整元素。此外，我们证明了这些元素的除子的支撑集上两点的差有非挠除子，这与以往的构造不同。第二，我们研究了亏格2和3的自反多项式族的Mahler测度，数值验证了Mahler测度和L函数特殊值之间的关系，并在假设Beilinson猜想成立的条件下证明了这些关系。建立了多个不同多项式族Mahler测度之间的关系。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的Mahler测度与定义出亏格1曲线的多项式族Mahler测度之间的关系。第三，我们证明了任意亏格多项式族Mahler测度之间的恒等式，作为这些恒等式的应用，证明了多个非tempered多项式的Mahler测度与椭圆曲线L函数在0处导数的关系。