Uniform bounds on the level of cyclotomic division fields of elliptic curves

Abstract

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on $p$ for which $F(E[p])/F$ is abelian, provided that $F$ has no rational complex multiplication. These are generalizations of results of Gonz\'{a}lez-Jim\'{e}nez and Lozano-Robledo to general number fields.