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Modern Analysis of Automorphic Forms By Example

Volume 1


Part of Cambridge Studies in Advanced Mathematics

  • Date Published: September 2018
  • availability: In stock
  • format: Hardback
  • isbn: 9781107154001

$ 79.99

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About the Authors
  • 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.

    • Provides an extensive and self-contained treatment of analytical aspects of automorphic forms
    • Provides a deep exploration of three different examples with varying degrees of complexity and generality, allowing a gradual approach to multi-faceted and highly technical issues
    • Collects the most important analytical ideas and proofs in the theory of automorphic forms
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    Reviews & endorsements

    Review of Multi-volume Set: 'Any researcher working in the analytic theory of automorphic forms on higher rank groups will want to own this book. It is a treasure trove of examples and proofs that are well known to experts but very difficult to find in the open literature.' Dorian Goldfeld, Columbia University

    Review of Multi-volume Set: 'Written by a leading expert in the field, this volume provides a valuable account of the analytic theory of automorphic forms. The author chooses his examples to provide a middle road between the general theory and the most classical cases that do not exhibit all of the subject's more general phenomena. What makes this book special is this compromise and the subsequent aim, 'to discuss analytical issues at a technical level truly sufficient to convert appealing heuristics to persuasive, genuine proofs'.' John Friedlander, University of Toronto

    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

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    Product details

    • Date Published: September 2018
    • format: Hardback
    • isbn: 9781107154001
    • length: 406 pages
    • dimensions: 236 x 158 x 26 mm
    • weight: 0.67kg
    • availability: In stock
  • Table of Contents

    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.

  • Author

    Paul Garrett, University of Minnesota
    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.

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