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

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.)

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.

Received 10 May 2015

Accepted 23 November 2015

Published 9 November 2017