Division by $2$ of rational points on elliptic curves
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B. M. Bekker and Yu. G. Zarhin
Translated by: B. M. Bekker - St. Petersburg Math. J. 29 (2018), 683-713
- DOI: https://doi.org/10.1090/spmj/1512
- Published electronically: June 1, 2018
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Abstract:
The well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion is reproved in a simple way. Next, the explicit formulas for division by $2^n$ obtained in §2 are used to construct versal families of elliptic curves that contain points of orders 4, 5, 6, and 8. These families are further employed to describe explicitly elliptic curves over certain finite fields $\mathbb F_q$ with a prescribed (small) group $E(\mathbb F_q)$. The last two sections are devoted to the cases of 3- and 5-torsion.References
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Bibliographic Information
- B. M. Bekker
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetsky prospekt 28, Peterhof, St. Petersburg 198504, Russia
- MR Author ID: 323935
- ORCID: 0000-0001-5481-8324
- Email: bekker.boris@gmail.com
- Yu. G. Zarhin
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
- MR Author ID: 200326
- Email: zarhin@math.psu.edu
- Received by editor(s): May 15, 2016
- Published electronically: June 1, 2018
- Additional Notes: The first author was partially supported by RFBR (grant no. 14-01-00393)
The second author was partially supported by a grant from the Simons Foundation (#246625 to Yuri Zarkhin). This work was started in May–June 2016 when he was a visitor at the Max-Planck-Institut für Mathematik (Bonn, Germany), whose hospitality and support are gratefully acknowledged. - © Copyright 2018 American Mathematical Society
- Journal: St. Petersburg Math. J. 29 (2018), 683-713
- MSC (2010): Primary 14H52
- DOI: https://doi.org/10.1090/spmj/1512
- MathSciNet review: 3708868
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