Galois Representations of Abelian Varieties

The main topic of this thesis is Galois representations of abelian varieties. Following the introduction, there are four largely independent chapters that each make progress around the main topic. Chapter 2 gives an explicit bound (conditional on the generalized Riemann hypothesis and in terms of the conductor) on the largest prime number ell for which the mod ell Galois representation of an elliptic curve over Q without complex multiplication is non-surjective (joint with Tian Wang). Chapter 3 applies Galois representations of elliptic curves to study rigidity in two well-known local-global principles. Chapter 4 considers principally polarized abelian varieties of arbitrary dimension whose adelic Galois image is open in the appropriate profinite group, giving a bound (in terms of standard invariants of the abelian variety) on the level at which the adelic Galois image is defined. Chapter 5 is a brief computational note on determining whether a given elliptic curve over Q is a Serre curve.