On the minimal degree of definition of p-primary torsion subgroups of elliptic curves

In this article, we study the minimal degree [K(T) : K] of a p-subgroup T ⊆ E(K̄)tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation ρE,p∞ : Gal(K̄/K) GL(2, Zp). However, we are able to show that there are certain uniform bounds for the minimal degree of definition of T. When the results are applied to K = Q and p = 2, we obtain a divisibility condition on the minimal degree of definition of any subgroup of E[2n] that is best possible