Thesis (Ph.D., Mathematics) -- University of Idaho, 2017 | Let E/L be a real quadratic extension of number fields. This dissertation contains the construction of an explicit map from an irreducible cuspidal automorphic representation of GL(2,E) which contains a Hilbert modular form with Gamma_0 level to an irreducible automorphic representation of GSP(4,L) which contains a Siegel paramodular form. We discuss how to construct an orthogonal representation from a character and a representation of a quaternion algebra, in some generality. There is a well known global theta correspondence for the pair (GSO(4), GSP(4)). We discuss the local theta correspondence and discuss its invariance properties. Finally, we exhibit local data which produces a paramodular invariant vector for the local theta lift at every place, except when the local extension has wild ramification.
