The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called ?liftings.? This in-depth book concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of Sp(2, C) in SL(4, C). It develops the technique of comparing twisted and stabilized trace formulae. It gives a detailed classification of the automorphic and admissible representation of the rank two symplectic PGSp(2) by means of a definition of packets and quasi-packets, using character relations and trace formulae identities. It also shows multiplicity one and rigidity theorems for the discrete spectrum. Applications include the study of the d

Lifting Automorphic Forms of PGSp(2) to PGL(4): Basic Facts; Trace Formulae; Lifting from SO(4) to PGL(4); Lifting from PGSp(2) to PGL(4); Fundamental Lemma; Zeta Functions of Shimura Varieties of PGSp(2): Automorphic Representations; Local Terms; Real Representations; Galois Representations; Background: On Automorphic Forms; On Artin's Conjecture.