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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Division by $ 2$ of rational points on elliptic curves

Authors: B. M. Bekker and Yu. G. Zarhin
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 29 (2017), nomer 4.
Journal: St. Petersburg Math. J. 29 (2018), 683-713
MSC (2010): Primary 14H52
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.

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Additional Information

B. M. Bekker
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetsky prospekt 28, Peterhof, St. Petersburg 198504, Russia

Yu. G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Keywords: Torsion subgroup, 2-descent, Mordell--Weil theorem
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.
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