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

TL;DR

This work establishes uniform bounds on prime levels p for which the cyclotomic division field F(E[p]) equals the cyclotomic field F(ζ_p) over any number field F, and, under GRH with no rational CM, uniform bounds for when F(E[p])/F is abelian. The authors develop a framework combining Serre’s inertia description, the Dickson classification of subgroups of GL_2(p), and isogeny-character ideas to constrain mod-p Galois images and force p to lie in finite sets dependent only on F. They generalize results previously known over Q to arbitrary number fields, providing explicit uniformity statements and sharpening our understanding of how Galois representations of elliptic curves control cyclotomic and abelian division fields. The approach yields both a general bound for cyclotomic division fields and a GRH-conditional bound for abelian division fields, with proofs decomposed into an abelian-division-field case and a small-division-field case.

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 $F(E[p])=F(ζ_p)$. 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 González-Jiménez and Lozano-Robledo to general number fields.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Classification of subgroups of $\textrm{GL}_2(p)$
• Theorem 5
• Proposition 6
• proof
• Remark 7
• Theorem 8
• Remark 9