Rational Equivalences on Products of Elliptic Curves in a Family

Love, Jonathan

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 923-938.

Si $E_1$ et $E_2$ sont deux courbes elliptiques sur un corps $k$, nous avons une application naturelle ${\mathrm{CH}}^1{\left(E_1\right)}_0\otimes {\mathrm{CH}}^1{\left(E_2\right)}_0\to {\mathrm{CH}}^2(E_1\times E_2)$. Quand $k$ est un corps de nombres, une conjecture due à Bloch et Beilinson prédit que l’image de cette application est finie. Nous construisons une famille de courbes elliptiques à deux paramètres qui peut être utilisée pour produire des exemples de couples $E_1,E_2$ pour lesquels cette image est finie. La famille est définie pour garantir l’existence d’une courbe rationnelle passant par un point spécifié de la surface de Kummer de $E_1\times E_2$.

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map ${\mathrm{CH}}^1{\left(E_1\right)}_0\otimes {\mathrm{CH}}^1{\left(E_2\right)}_0\to {\mathrm{CH}}^2(E_1\times E_2)$, and a conjecture due to Bloch and 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 $E_1\times E_2$.

Classification : 14C15, 14J27, 11G05
Mots clés : Chow Group, Kummer surface, clean, pencil, cubic curve, zero-cycle

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