Uniform bounds on prime levels of abelian division fields of elliptic curves over number fields without rationally defined CM

Abstract

Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels $n$ for which the $n$-division field $\mathbb{Q}\left(E\left[n\right]\right) / \mathbb{Q}$ of an elliptic curve $E/\mathbb{Q}$ defined over $\mathbb{Q}$ is abelian. Assuming GRH (Generalized Riemann Hypothesis), Allen and Genao partially generalized this result by restricting to prime levels while allowing arbitrary number fields without rationally defined CM. In this paper, we remove the GRH assumption and prove that the property established by Allen and Genao is in fact equivalent to the absence of rationally defined CM.