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

Abstract

In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_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 ρ_{E,p^infty}:Gal_K-->GL(2,Z_p). 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[2^n] that is best possible.

Math. Res. Lett., to appear

References in corpus (3)

• On the possible images of the mod ell representations associated to elliptic curves over Q
• Torsion of rational elliptic curves over quadratic fields II
• Anatomy of torsion in the CM case

Cited by in corpus (5)

• Growth of torsion groups of elliptic curves upon base change
• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• On the torsion of rational elliptic curves over quartic fields
• An algorithm for determining torsion growth of elliptic curves
• Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree