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 decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations.

To put these results in a general context, the book concludes with a technical introduction to Langlands’ program in the area of automorphic representations. It includes a proof of known cases of Artin’s conjecture.

Contents:

• 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

“The insider will find a lot of conjectures and valuable concrete material to think and mediate about. The interested layman will get a view onto rapidly moving important topics, starting with the study in Part 3 where an exposition of developments in number theory up to Artin's conjecture is given with the intention of helping the reader understand Parts 1 and 2.”