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

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 $\mathbb{Q}$.