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 : 10.5802/jtnb.1148
Classification : 14C15, 14J27, 11G05
Mots clés : Chow Group, Kummer surface, clean, pencil, cubic curve, zero-cycle
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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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/

[1] Alexander A. Beĭlinson Higher regulators and values of L-functions, Current problems in mathematics, Vol. 24 (Itogi Nauki i Tekhniki), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1984, pp. 181-238 | MR

[2] Alexander A. Beĭlinson Height pairing between algebraic cycles, K-theory, arithmetic and geometry (Moscow, 1984–1986) (Lecture Notes in Mathematics), Volume 1289, Springer, 1987, pp. 1-25 | DOI | MR | Zbl

[3] Spencer Bloch Algebraic cycles and values of L-functions, J. Reine Angew. Math., Volume 350 (1984), pp. 94-108 | DOI | MR | Zbl

[4] Spencer Bloch Lectures on algebraic cycles, New Mathematical Monographs, 16, Cambridge University Press, 2010, xxiv+130 pages | DOI | MR | Zbl

[5] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 Computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[6] Brian Conrad; Keith Conrad; Harald Helfgott Root numbers and ranks in positive characteristic, Adv. Math., Volume 198 (2005) no. 2, pp. 684-731 | DOI | MR | Zbl

[7] John Cremona The elliptic curve database for conductors to 130000, Algorithmic number theory (Lecture Notes in Computer Science), Volume 4076, Springer, 2006, pp. 11-29 | DOI | MR | Zbl

[8] Igor V. Dolgachev Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | DOI | MR | Zbl

[9] William Fulton Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer, 1998, xiv+470 pages | DOI | MR | Zbl

[10] Jonathan Love Computing Relations in Groups of Zero-cycles and Isogeny Graphs, Ph. D. Thesis, Stanford University (2021) (in preparation; working title)

[11] Loïc Merel Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math., Volume 124 (1996) no. 1-3, pp. 437-449 | DOI | MR | Zbl

[12] David Mumford Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ., Volume 9 (1968), pp. 195-204 | DOI | MR

[13] Kartik Prasanna; Vasudevan Srinivas Zero Cycles on a Product of Elliptic Curves (2018) (private correspondence)

[14] Michael Rosen; Joseph H. Silverman On the rank of an elliptic surface, Invent. Math., Volume 133 (1998) no. 1, pp. 43-67 | DOI | MR | Zbl

[15] Matthias Schütt; Tetsuji Shioda Elliptic surfaces, Algebraic geometry in East Asia (Seoul 2008) (Advanced Studies in Pure Mathematics), Volume 60, Mathematical Society of Japan, 2010, pp. 51-160 | DOI | MR | Zbl

[16] Tetsuji Shioda On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli, Volume 39 (1990) no. 2, pp. 211-240 | MR | Zbl

[17] Tetsuji Shioda Correspondence of elliptic curves and Mordell–Weil lattices of certain elliptic K3’s, Algebraic cycles and motives. Vol. 2 (London Mathematical Society Lecture Note Series), Volume 344, Cambridge University Press, 2007, pp. 319-339 | DOI | MR | Zbl

[18] Joseph H. Silverman Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math., Volume 342 (1983), pp. 197-211 | DOI | MR | Zbl

[19] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994, xiv+525 pages | DOI | MR | Zbl

[20] Mark Watkins Rank distribution in a family of cubic twists, Ranks of elliptic curves and random matrix theory (London Mathematical Society Lecture Note Series), Volume 341, Cambridge University Press, 2007, pp. 237-246 | DOI | MR

[21] Don Zagier; Gerhard Kramarz Numerical investigations related to the L-series of certain elliptic curves, J. Indian Math. Soc., New Ser., Volume 52 (1987), pp. 51-69 | MR | Zbl

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