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.

Si E 1 et E 2 sont deux courbes elliptiques sur un corps k, nous avons une application naturelle CH 1 (E 1 ) 0 CH 1 (E 2 ) 0 CH 2 (E 1 ×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 ×E 2 .

Given a pair of elliptic curves E 1 ,E 2 over a field k, we have a natural map CH 1 (E 1 ) 0 CH 1 (E 2 ) 0 CH 2 (E 1 ×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 ×E 2 .

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DOI : https://doi.org/10.5802/jtnb.1148
Classification : 14C15,  14J27,  11G05
Mots clés : Chow Group, Kummer surface, clean, pencil, cubic curve, zero-cycle
@article{JTNB_2020__32_3_923_0,
     author = {Jonathan Love},
     title = {Rational {Equivalences} on {Products} of {Elliptic} {Curves} in a {Family}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {923--938},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {3},
     year = {2020},
     doi = {10.5802/jtnb.1148},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1148/}
}
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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