Given a pair of elliptic curves over a field , we have a natural map , and a conjecture due to Bloch and Beilinson predicts that the image of this map is finite when is a number field. We construct a -parameter family of elliptic curves that can be used to produce examples of pairs 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 .
Si et sont deux courbes elliptiques sur un corps , nous avons une application naturelle . Quand 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 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 .
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Accepted:
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Keywords: Chow Group, Kummer surface, clean, pencil, cubic curve, zero-cycle
@article{JTNB_2020__32_3_923_0, author = {Love, Jonathan}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.1148/} }
TY - JOUR AU - Love, Jonathan TI - Rational Equivalences on Products of Elliptic Curves in a Family JO - Journal de théorie des nombres de Bordeaux PY - 2020 SP - 923 EP - 938 VL - 32 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.1148/ DO - 10.5802/jtnb.1148 LA - en ID - JTNB_2020__32_3_923_0 ER -
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Love, Jonathan. Rational Equivalences on Products of Elliptic Curves in a Family. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 923-938. doi : 10.5802/jtnb.1148. http://archive.numdam.org/articles/10.5802/jtnb.1148/
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