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# 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

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n4.a7

#### Authors

#### Abstract

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