关于某些代数曲线K2群的研究

The algebraic K-theory of algebraic varieties over number fields reflects important arithmetic properties of algebraic varieties and it is of great significance in both K-theory and number theory. The celebrated Beilinson's conjecture gives the structure of non-torsion part of K-groups of algebraic varieties over number field, but even for the K2 group of algebraic curves, we only have very limited understanding of the structure of its non-torsion part. This project will study the K2 of certain algebraic curves to enhance the understanding its structure. The content of this project includes two aspects. Firstly, for certain quartic curves of genus 3 over rational field, we will construct three elements in the integral K2 group and prove that they are linearly independent by calculating the limit of the regulator. Secondly, we construct elements in the integral K2 group of certain hyperelliptic curves over number field and prove that they are linearly independent by calculating the limit of the regulator. Specifically, we intend to prove the lower bound of the rank of integral K2 group of certain elliptic curves over quadratic field is the same as the rank predicted by Beilinson's conjecture.

数域上代数簇的代数K理论反映了代数簇重要的算术性质，在K理论和数论中都有重要意义。著名的Beilinson猜想给出了一般数域上代数簇K群的无挠部分的结构，但是即使对于代数曲线的K2群，我们对其无挠部分具体结构的了解也极为有限。本项目将通过对某些代数曲线K2群的研究，增进对这些曲线K2群的认识，具体研究内容包括如下两方面。第一，拟对某些有理数域上亏格3的四次曲线构造出其整K2群中的三个元素，通过计算导子的极限证明这些元素线性无关。第二，拟对某些数域上的超椭圆曲线具体构造出其整K2群中的一些元素，通过计算导子证明这些元素线性无关。特别地，证明其中某些二次域上的椭圆曲线的整K2群秩的下界与Beilinson猜想中对K2群秩的预测相同。

著名的Beilinson猜想是代数K理论中最重要的猜想之一。本项目研究了某些代数曲线的K2群和Beilinson猜想以及典型群等相关问题。本项目圆满地完成了预定目标，在Proceedings of the American Mathematical Society，Communications in Algebra，中国科学等国内外著名杂志上接收和发表论文3篇，其中SCI论文两篇。主要研究成果包括三方面：第一，对亏格3的四次曲线族构造了K2群中的三个元素，当参数满足一定条件时证明了这些元素是整元素，通过计算正则子的极限证明了这些元素一般线性无关，从而验证了这些曲线关于K2群秩的Beilinson猜想的下界。第二，构造了四族任意亏格的 (超) 椭圆曲线 K2 群中的元素，证明了在某些条件下这些元素是整元素，当曲线参数满足一定条件时，证明其中一些元素线性无关。 同时给出了分别在两个实二次域上有两个具体的线性无关整元素的一些椭圆曲线族。第三，在稳定秩条件下，研究了奇酉群的分类并给出了一个三明治型定理。