Rational Equivalences on Products of Elliptic Curves in a Family

Jonathan Love

doi:10.5802/jtnb.1148

Jonathan Love ¹

¹ Stanford University, Dept. of Mathematics Building 380 Stanford, California, USA 94305

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

Abstract (VO)
Résumé

Given a pair of elliptic curves $E_{1}, E_{2}$ over a field $k$ , we have a natural map ${CH}^{1} {(E_{1})}_{0} \otimes {CH}^{1} {(E_{2})}_{0} \to {CH}^{2} (E_{1} \times E_{2})$ , 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 $E_{1} \times E_{2}$ .

Si $E_{1}$ et $E_{2}$ sont deux courbes elliptiques sur un corps $k$ , nous avons une application naturelle ${CH}^{1} {(E_{1})}_{0} \otimes {CH}^{1} {(E_{2})}_{0} \to {CH}^{2} (E_{1} \times E_{2})$ . 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 $E_{1} \times E_{2}$ .

Reçu le : 2020-05-11
Révisé le : 2020-09-03
Accepté le : 2020-09-18
Publié le : 2021-01-08

DOI : 10.5802/jtnb.1148

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

Affiliations des auteurs :

Jonathan Love ¹

¹ Stanford University, Dept. of Mathematics Building 380 Stanford, California, USA 94305

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Rational {Equivalences} on {Products} of {Elliptic} {Curves} in a {Family}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Jonathan Love. Rational Equivalences on Products of Elliptic Curves in a Family. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 923-938. doi : 10.5802/jtnb.1148. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1148/

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