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Automorphic Forms on SL2 (R)

$58.00

Part of Cambridge Tracts in Mathematics

  • Date Published: August 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521072120

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  • This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup ^D*G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on ^D*G\G and its relationship with the classical automorphic forms on X, Poincaré series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2(^D*G/G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras.

    • Very famous author
    • Difficult subject treated in introductory fashion
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    Reviews & endorsements

    "...is a very carefully written book, which is accessible to the graduate student. ... Its size of 184 pages makes it even more possible to complete the material in one semester. Thus, it can be used as preparation for learning the `Langlands program' since the case of SL2(R) is concrete enough not to be lost in the maze of generality and typical enough to exemplify the general case. The topic is now a dominant theme in both algebraic and analytic number theory as well as algebraic geometry, and so the monograph is timely." Mathematical Reviews Clippings

    "...I would recommend [Bu] to graduate students or anyone else who wants to get into the subject. There is a great deal of foundational material here. Any effort put into reading it and working exercises will be rewarded with a good understanding of the basics." Bulletin of the American Mathematical Society

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    Product details

    • Date Published: August 2008
    • format: Paperback
    • isbn: 9780521072120
    • length: 208 pages
    • dimensions: 229 x 152 x 12 mm
    • weight: 0.31kg
    • availability: Available
  • Table of Contents

    Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane:
    1. Prerequisites and notation
    2. Review of SL2(R), differential operators, convolution
    3. Action of G on X, discrete subgroups of G, fundamental domains
    4. The unit disc model
    Part II. Automorphic Forms and Cusp Forms:
    5. Growth conditions, automorphic forms
    6. Poincare series
    7. Constant term:the fundamental estimate
    8. Finite dimensionality of the space of automorphic forms of a given type
    9. Convolution operators on cuspidal functions
    Part III. Eisenstein Series:
    10. Definition and convergence of Eisenstein series
    11. Analytic continuation of the Eisenstein series
    12. Eisenstein series and automorphic forms orthogonal to cusp forms
    Part IV. Spectral Decomposition and Representations:
    13.Spectral decomposition of L2(G\G)m with respect to C
    14. Generalities on representations of G
    15. Representations of SL2(R)
    16. Spectral decomposition of L2(G\SL2(R)):the discrete spectrum
    17. Spectral decomposition of L2(G\SL2(R)): the continuous spectrum
    18. Concluding remarks.

  • Author

    Armand Borel, Institute for Advanced Study, Princeton, New Jersey

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