一类椭圆曲线乘积的有理等价

Pub Date : 2020-03-05 DOI:10.5802/JTNB.1148

Jonathan R. Love

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引用次数: 1

摘要

给定域$k$上的一对椭圆曲线$E_1，E_2$，我们有一个自然映射$\text{CH}^1（E_1）_0\otimes\text{CH}^1（E2）_0\to\text{CH}^2（E_1\timesE_2）$，并且由Beilinson提出的猜想预测当$k$是一个数域时，该映射的图像是有限的。我们构造了一个$2$参数的椭圆曲线族，该族可用于生成对$E_1，E_2$的例子，其中该图像是有限的。构造该族是为了保证通过$E_1\times E_2$的Kummer曲面中指定点的有理曲线的存在。

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Rational Equivalences on Products of Elliptic Curves in a Family

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1\times E_2$.

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