Rational Equivalences on Products of Elliptic Curves in a Family
Jonathan Love
备注：
12 pages, 1 figure, 2 tables. Organizational changes, expanded some comments into lemmas. Content is identical to the published version, aside from the organization of information in the Appendix
发表信息：
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 923-938
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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