论文标题
在数字字段上的曲线产物上的零循环的扭转现象
Torsion phenomena for zero-cycles on a product of curves over a number field
论文作者
论文摘要
对于平滑的投影品种，$ k $ k $ k $ a bloch和贝林森预测，$ x $的阿尔巴尼人地图的内核是一个扭转组。在本文中，我们考虑了一个产品$ x = C_1 \ times \ cdots \ times \ times c_d c_d c_d $ of光滑的投射曲线，并证明，如果猜想对于两条曲线的任何子产品都是正确的，那么对于$ x $来说是正确的。此外，我们制作了许多非异构椭圆曲线的新示例$ e_1，e_2 $，其正等级超过$ \ mathbb {q} $，而自然映射的图像$ e_1（\ mathbb {q}）\ outimes e_2 e_2（\ mathbb {q}） \ text {ch} _0（e_1 \ times e_2）$是有限的，包括第一个已知的等级示例，大于$ 1 $。结合两个结果，我们获得了无限的许多非平凡产品$ x = c_1 \ times \ cdots \ times c_d $，类似地图$ \ varepsilon $具有有限的图像。
For a smooth projective variety $X$ over a number field $k$ a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of $X$ is a torsion group. In this article we consider a product $X=C_1\times\cdots\times C_d$ of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for $X$. Additionally, we produce many new examples of non-isogenous elliptic curves $E_1, E_2$ with positive rank over $\mathbb{Q}$ for which the image of the natural map $E_1(\mathbb{Q})\otimes E_2(\mathbb{Q})\xrightarrow{\varepsilon} \text{CH}_0(E_1\times E_2)$ is finite, including the first known examples of rank greater than $1$. Combining the two results, we obtain infinitely many nontrivial products $X=C_1\times\cdots\times C_d$ for which the analogous map $\varepsilon$ has finite image.
