Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences. This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n, E) and GL(n, F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms Contents Introduction Chapter 1. Local Results 1. The norm map and the geometry of σ-conjugacy 2. Harmonic analysis on the non-connected group 3. Transfer of orbital integrals of smooth functions 4. Orbital integrals of Hecke functions 5. Orbital integrals: non-inert primes 6. Base change lifting of local representations 7. Archimedean case Chapter 2. The Global Comparison 1. Preliminary remarks 2. Normalization factors and the trace formula 3. The distributions IM(γ) and 4. Convolution and the differentia! equation 5. Statement of Theorem A 6. Comparison of and IM(γ, f ) 7. Comparison of germs 8. The distributions IM(π, X) and 9. Statement of Theorem B 10. Comparison of (π, X, f) and IM(π, X, f) 11. More on normalizing factors 12. A formula for (f) 13. The map εM 14. Cancellation of singularities 15. Separation by infinitesimal character 16. Elimination of restrictions on f 17. Completion of the proofs of Theorems A and B Chapter 3. Base Change 1. Weak and strong base change: definitions 2. Some results of Jacquet and Shalika 3. Fibers of global base change 4. Weak lifting 5. Strong lifting 6. Base change lift of automorphic forms in cyclic extensions 7. The strong Artin conjecture for nilpotent groups Bibliography