TY - JOUR

T1 - Division by 2 of rational points on elliptic curves

AU - Bekker, B. M.

AU - Zarhin, Yu G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 & sect;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 F-q with a prescribed (small) group E(F-q). The last two sections are devoted to the cases of 3- and 5-torsion.

AB - 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 & sect;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 F-q with a prescribed (small) group E(F-q). The last two sections are devoted to the cases of 3- and 5-torsion.

KW - 2-descent

KW - Mordell-Weil theorem

KW - Torsion subgroup

KW - QUADRATIC FIELDS

KW - TORSION POINTS

UR - http://www.scopus.com/inward/record.url?scp=85048044360&partnerID=8YFLogxK

UR - https://www.elibrary.ru/item.asp?id=35745842

U2 - 10.1090/spmj/1512

DO - 10.1090/spmj/1512

M3 - Article

AN - SCOPUS:85048044360

VL - 29

SP - 683

EP - 713

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 4

ER -