Torsion subgroups of elliptic curves in elementary abelian 2-extensions of Q

Abstract

Let E be an elliptic curve over Q and let F := Q({ √ m ; m ∈ Z}). Laska and Lorenz showed that there exist at most 31 possibilities for the type of the torsion subgroup E(F ) tors of E over F. In this paper, we showed that there exist exactly 20 possibilities for E(F ) tors .

Ranks of Elliptic Curves with Prescribed Torsion over Number Fields

“…Examine the behavior of the torsion of E 1 (K) and E 2 (K) as K varies through all quadratic fields. The torsion of E 2 (K) will always be Z/7Z (see [7…”

Torsion of rational elliptic curves over quartic Galois number fields

“…The proof of this theorem is broken up based on the structure of Gal(K/Q) and so, in fact, we have the following more specialized theorems: Theorem 1. 3. Let E/Q be an elliptic curve, and let K be a quartic Galois extension with Gal(K/Q) ∼ = Z/4Z.…”

Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

“…Suppose now E(K) has a point of order 8. It follows that E(L) [8] ≃ Z/4Z ⊕ Z/8Z. Let G := Gal(L/M ) and take the short exact sequence (10) 0…”

“…It is impossible that E has 3 twists with 3-torsion, as this would imply that, by [8,Lemma 9], E d (F 2 ) would contain (Z/3Z) 3 for some twist E d of E and some biquadratic extension F 2 of Q.…”

“…This is a natural question to consider as, apart from being interesting in itself, it is often important to study rational elliptic curves over extensions of Q when solving Diophantine equations (see for example [2]). Somewhat similar problems were studied by Fujita [8], who studied the possible torsion groups of rational elliptic curves over the compositum of all quadratic fields, by Lozano-Robledo [25], who studied the minimal degree of the field of definition of points of order p on rational elliptic curves and by González-Jiménez and Tornero [14], who studied how can the torsion of a rational elliptic curve grow upon base changing to a quadratic field.…”