The divisibility by 2 of rational points on elliptic curves

2017/02/08 by Boris M. Bekker, Bekker, Boris M., Yuri G. Zarhin +1
Computer Science · Mathematics · #11G05 #14H52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1702.02255

openalex publication_date 2017/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2ⁿ formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields \mathbbF_q with a prescribed (small) group E(\mathbbF_q). In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .

Citations

• Division by 2 on elliptic curves

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