Modern Analysis of Automorphic Forms By Example

Volume 1 of a two-volume introduction to the analytical aspects of automorphic forms, featuring proofs of critical results with examples.

Paul Garrett (Author)

9781107154001, Cambridge University Press

Hardback, published 20 September 2018

406 pages
23.6 x 15.8 x 2.6 cm, 0.67 kg

Review of Multi-volume Set: 'It is marvelous to see how Garrett goes about presenting such deep and broad material in what is certainly a superbly holistic manner. It's really a wonderful example of what I think is the right pedagogy for this part of number theory. The examples he uses are lynchpins for an increasingly elaborate development of the subject, and the reader has a number of accessible places to hang his hat as the story unfolds.' Michael Berg, MAA Reviews

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.

Subject Areas: Number theory [PBH]