This is Volume 1 of a two-volume book that provides a self-contained introduction to the theory and application of automorphic forms, using examples to illustrate several critical analytical concepts surrounding and supporting the theory of automorphic forms. The two-volume book treats three instances, starting with some small unimodular examples, followed by adelic GL2, and finally GLn. Volume 1 features critical results, which are proven carefully and in detail, including discrete decomposition of cuspforms, meromorphic continuation of Eisenstein series, spectral decomposition of pseudo-Eisenstein series, and automorphic Plancherel theorem. Volume 2 features automorphic Green's functions, metrics and topologies on natural function spaces, unbounded operators, vector-valued integrals, vector-valued holomorphic functions, and asymptotics. With numerous proofs and extensive examples, this classroom-tested introductory text is meant for a second-year or advanced graduate course in automorphic forms, and also as a resource for researchers working in automorphic forms, analytic number theory, and related fields.
1. Four small examples; 2. The quotient Z+GL2(k)/GL2(A); 3. SL3(Z), SL5(Z); 4. Invariant differential operators; 5. Integration on quotients; 6. Action of G on function spaces on G; 7. Discrete decomposition of cuspforms; 8. Moderate growth functions, theory of the constant term; 9. Unbounded operators on Hilbert spaces; 10. Discrete decomposition of pseudo-cuspforms; 11. Meromorphic continuation of Eisenstein series; 12. Global automorphic Sobolev spaces, Green's functions; 13. Examples – topologies on natural function spaces; 14. Vector-valued integrals; 15. Differentiable vector-valued functions; 16. Asymptotic expansions.
Paul Garrett is Professor of Mathematics at the University of Minnesota. His research focuses on analytical issues in the theory of automorphic forms. He has published numerous journal articles as well as five books.
Collana: Cambridge Studies in Advanced Mathematics
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