Disputation Alirzea Shavali Kohshor On the Image of Automorphic Galois Representations

Under the Langlands philosophy, there should be a correspondence between certain automorphic representations of GL(n), certain n-dimensional Galois representations, and motives over number fields. There is a folklore heuristic that the image of the Galois representation should be as big as possible unless there is an automorphic reason for it not to be. In this thesis, we will formulate a precise conjecture in this direction, assuming some standard conjectures in the literature. In the n=2 case, this follows from the work of Ribet, Momose, and Nekovar. We are able to prove this conjecture unconditionally in the n=3 case.