Coincidences of Division Fields of an elliptic curve defined over a number field

TL;DR

This paper investigates when different division fields of an elliptic curve over a number field coincide, analyzing conditions for such coincidences and providing constructions beyond rational cases.

Contribution

It establishes necessary conditions for division field coincidences and constructs examples over number fields, extending beyond the rational case.

Findings

Necessary conditions for division field coincidences.

Analysis of vertical and horizontal coincidences.

Construction of non-trivial examples over number fields.

Abstract

For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the surjectivity of this representation are either local (i.e. at a prime), or due to nonsurjectivity on the product of local Galois images. In this article, we study an extreme case: the coincidence i.e. the equality of $n$-division fields, generated by the $n$-torsion points, attached to different positive integers $n$. We give necessary conditions for coincidences, dealing separately with vertical coincidences, at a given prime, and horizontal coincidences, across multiple primes, in particular when the Galois group on the $n$-torsion contains the special linear group. We also give a non-trivial construction for coincidences not occurring over $\mathbb{Q}$.