A splitting criterion for Galois representations associated to exceptional modular forms

There is a standard way to get a two dimensional global Galois representation ρ_f from a newform f of weight k (mod p). There is a conjecture by Serre, proven in most cases, which says that the ramification of ρ_f at p is completely determined by whether f has a companion form. Furthermore, the existence of a companion form is in most cases equivalent to the splitting of the restriction of ρ_f to a decomposition group at p. However, there is an exceptional case in which the splitting of this restriction, ρ_f,p, is a more subtle issue. The goal of Part I of this thesis is to develop a criterion which determines the splitting of ρ_f,p in the exceptional case. The goal of Part II is to show the computability of this splitting criterion by working out some explicit examples. Also, in the process of showing computability, we develop some methods for obtaining very good models for modular curves and the maps between them which may have many other applications.