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

TL;DR

This paper establishes uniform bounds on prime levels of elliptic curves over any number field, extending previous results and providing new bounds under the Generalized Riemann Hypothesis for specific cases.

Contribution

It generalizes existing bounds on cyclotomic division fields of elliptic curves to all number fields and under GRH, offers new uniform bounds for abelian extensions.

Findings

Existence of uniform bounds on prime levels for elliptic curves over any number field.

Under GRH, bounds on primes where the extension is abelian for non-CM elliptic curves.

Extension of previous results to broader classes of number fields.

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.