Torsion subgroups of elliptic curves with non-cyclic torsion over Q in elementary abelian 2-extensions of Q

Abstract

Introduction.Let E be an elliptic curve over Q and F the maximal elementary abelian 2-extension of Q, that is, F := Q({ √ m; m ∈ Z}). It is known that the torsion subgroup E(F ) tors of E(F ) is finite (Ribet [8]). More precisely, Laska and Lorenz showed that there exist at most thirty-one possibilities for E(F ) tors (see [3, Theorem] or Theorem 2.1). However, it is not known whether all the groups listed in Theorem 2.1 can happen as E(F ) tors . Now assume that E has non-cyclic torsion over Q; then by Mazur's theorem ([4]), the group E(Q) tors is isomorphic to Z/2Z ⊕ Z/mZ, where m = 2, 4, 6 or 8. Such an elliptic curve has a Weierstrass model E : y 2 = x(x + M )(x + N ), where M and N are non-zero integers with M > N . Further we may assume that the greatest common divisor (M, N ) of M and N is a square-free integer or 1, since for any positive integer d, E is isomorphic over Q to an elliptic curve E d 2 given by y 2 = x(x+d 2 M )(x+d 2 N ) by replacing x with x/d 2 and y with y/d 3 , respectively. Then using the result of Ono ([6, Main Theorem 1], see also Theorem 2.2), Kwon classified the torsion subgroup of E over all quadratic fields ([2, Theorem 1]); Qiu and Zhang classified the torsion subgroup of E for a certain elliptic curve E with E(Q) tors Z/2Z ⊕ Z/2Z over all elementary abelian 2-extensions of Q, i.e., over all number fields of type (2, . . . , 2) ([7, Theorems 3 and 4]); Ohizumi classified the torsion subgroup of E for an elliptic curve E with E(Q) tors Z/2Z ⊕ Z/8Z or Z/2Z ⊕ Z/6Z over all bicyclic biquadratic fields, i.e., over all number fields of type (2, 2) ([5, Main Theorems 4.1 and 4.2]).In this paper, first we completely determine the structure of the torsion subgroup E(F ) tors when E(Q) tors is non-cyclic:

A complete diophantine characterization of the rational torsion of an elliptic curve

“…Instead, while trying to cope with the odd order case we eventually arrived to a global strategy for finding equations which led to the previous theorem. Both Ono's and Qiu-Zhang's results have been used to study the behaviour of torsion structures when a bigger ground field is considered (typically a quadratic extension of Q), for instance in [5,2,3]. We hope that our characterization will shed some light also in this matter, where much is yet to be known.…”

“…Assume (x 1 , w 1 ) ∈ E(Q) has order five. Then the first coordinate of [4](x 1 , w 1 ) must be again x 1 , and if we write [2](x 1 , w 1 ) = (x 2 , w 2 ), then it must hold x 1 = x 2 . We will have, from duplication formula,…”

“…Assume (x 1 , w 1 ) ∈ E(Q) is a point of order seven, and denote [2](x 1 , w 1 ) = (x 2 , w 2 ) and [3](x 1 , w 1 ) = (x 3 , w 3 ), being x 1 , x 2 and x 3 different integers. From the duplication formula, as above, we have…”

“…and, since (x, y) ∈ E(Q), there exist z 1 , z 2 ∈ Z such that x = 3z 2 1 and y = ±(9z 3 1 + z 2 ) with z 2 = 18z 3 1 q/u. Therefore,…”

“…is a point of order nine, the first coordinate of [3](x, y) must be 3z 2 1 , as [3](x, y) has order three.…”