ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

• 来源期刊: 计算数学(英文版)
• 关键词: Courbe elliptique Isogénie.
• 年，卷（期）: 2002,（4）
• 页码范围: 337-348
• 页数: 12页
• 语种: 英文

摘要

An elliptic curve is a pair (E, O), where E is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE: y2 + a1xy + a3y = x3 + a2x2 + a4x + a6.Let Q be the set of rationals. E is said to be difined over Q if the coefficients ai, i =1, 2, 3, 4, 6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E defined over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsWe say that an elliptic curve E′/Q is isogenous to the elliptic curve E if there is an isogeny,i.e. a morphism φ: E → E′ such that φ(O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E(Q)tors is in the form Z/mZ where m = 9, 10, 12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem.Morever, for every family of such elliptic curves, we give an explicit model of all their isogenous curves with cyclic kernels consisting of rational points.