Mathematical Research Letters

Volume 24 (2017)

Number 4

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

Pages: 1067 – 1096



Enrique González-Jiménez (Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain)

Álvaro Lozano-Robledo (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)


In this article, we study the minimal degree $[K(T) : K]$ of a $p$-subgroup $T \subseteq E{(\overline{K})}_{\mathrm{tors}}$ 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 $\rho_{E, p^{\infty}} : \mathrm{Gal}(\overline{K} / K) \to \mathrm{GL} (2, \mathbb{Z}_p)$. 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 $K = \mathbb{Q}$ 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.

2010 Mathematics Subject Classification

Primary 11G05. Secondary 14H52.

The first author was partially supported by the grant MTM2012–35849.

Received 10 May 2015

Accepted 23 November 2015

Published 9 November 2017