Modular Elliptic Curves over Quartic CM Fields

In this thesis I establish the modularity of a number of elliptic curves defined over quartic CM fields, by showing that the Galois representation attached to such curves (arising from the natural Galois action on the l-adic Tate module) is isomorphic to a representation attached to a cuspidal automorphic form for GL(2) over the CM field in question. This is achieved through the study of the Hecke action on the cohomology of certain symmetric spaces, which are known to be isomorphic to spaces of cuspidal automorphic forms by a generalization of the Eichler-Shimura isomorphism.