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=x2+a2x2+...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=x2+a2x2+a4x+a6.Let Q be the set of rationals. E is said to be dinned 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.展开更多
基金This research was supported by the TMR programme of the European Community under contract ERBFMBICT960848.
文摘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=x2+a2x2+a4x+a6.Let Q be the set of rationals. E is said to be dinned 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.
