关于某些代数曲线K2群的研究

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。

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.