Improved Bounds on Entanglement of Torsion Point Fields

Abstract

Serre's Open Image Theorem states that the image of the adelic Galois representation of a non-CM elliptic curve $E$ over a number field $k$ is open in $\GL_2(\hat{\Z})$, and hence has finite index. Thus, the adelic image is completely determined by the image of the mod $N$ Galois representation for a positive integer $N$, the smallest of which is called the level of the adelic image. Notably, while the theorem holds for a fixed non-CM elliptic curve, there is no uniform bound on the level when considering the collection of all non-CM elliptic curves over $k$, as it can be arbitrarily large. Nevertheless, it has been shown that, for a fixed positive integer $m$, there is a uniform and explicit bound on the level of the $m$-adic Galois representation that applies for all non-CM elliptic curves over $k$. In this thesis, we will show how introducing good reduction assumptions allows us to improve this uniform bound on the level of the $m$-adic representation for certain families of non-CM elliptic curves over $\Q$ and, therefore, obtain more precise information on the entanglement of the torsion point fields.