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Eisenstein Series and Automorphic Representations
With Applications in String Theory

$117.00 (C)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: July 2018
  • availability: Available
  • format: Hardback
  • isbn: 9781107189928

$ 117.00 (C)

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About the Authors
  • This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects and extends many results scattered throughout the literature, in particular the Langlands constant term formula for Eisenstein series on G(A) as well as the Casselman–Shalika formula for the p-adic spherical Whittaker function. This book also covers more advanced topics such as spherical Hecke algebras and automorphic L-functions. Many of these mathematical results have natural interpretations in string theory, and so some basic concepts of string theory are introduced with an emphasis on connections with automorphic forms. Throughout the book special attention is paid to small automorphic representations, which are of particular importance in string theory but are also of independent mathematical interest. Numerous open questions and conjectures, partially motivated by physics, are included to prompt the reader's own research.

    • This book will be useful and interesting to both number theorists and string theorists, who will each gain insight into the other's work
    • Engages the reader with lots of examples, perfect for learning the techniques outlined in the book
    • Gives a good overview of recent research topics to help guide the reader's future research
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    Reviews & endorsements

    'This book provides a bridge between two very active and important parts of mathematics and physics, namely the theory of automorphic forms on reductive groups and string theory. The authors have masterfully presented both aspects and their connections, and have provided examples and details at all levels to make the book available to a large readership, including non-experts in both fields. This is a valuable contribution and a welcoming text for graduate students as well.’ Freydoon Shahidi, Purdue University, Indiana

    'The book is a valuable addition to the literature, and it may inspire more exchange between mathematics and physics at an advanced level.' Anton Deitmar, MathSciNet

    ‘The prerequisites for a profitable reading this book are enormous. Readers without a solid background in algebraic and analytic number theory, classfield theory, modular forms and representation theory will only be able to read a couple of sections. Researchers in these fields will be grateful to the authors and the publisher for providing access to some rather advanced mathematics. The material is presented in a very clear and lucid way; there is an extensive index and a list of references containing 634 items.’ Franz Lemmermeyer, zbMATH

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    Product details

    • Date Published: July 2018
    • format: Hardback
    • isbn: 9781107189928
    • length: 584 pages
    • dimensions: 235 x 157 x 38 mm
    • weight: 0.92kg
    • contains: 20 b/w illus.
    • availability: Available
  • Table of Contents

    1. Motivation and background
    Part I. Automorphic Representations:
    2. Preliminaries on p-adic and adelic technology
    3. Basic notions from Lie algebras and Lie groups
    4. Automorphic forms
    5. Automorphic representations and Eisenstein series
    6. Whittaker functions and Fourier coefficients
    7. Fourier coefficients of Eisenstein series on SL(2, A)
    8. Langlands constant term formula
    9. Whittaker coefficients of Eisenstein series
    10. Analysing Eisenstein series and small representations
    11. Hecke theory and automorphic L-functions
    12. Theta correspondences
    Part II. Applications in String Theory:
    13. Elements of string theory
    14. Automorphic scattering amplitudes
    15. Further occurrences of automorphic forms in string theory
    Part III. Advanced Topics:
    16. Connections to the Langlands program
    17. Whittaker functions, crystals and multiple Dirichlet series
    18. Automorphic forms on non-split real forms
    19. Extension to Kac–Moody groups
    Appendix A. SL(2, R) Eisenstein series and Poisson resummation
    Appendix B. Laplace operators on G/K and automorphic forms
    Appendix C. Structure theory of su(2, 1)
    Appendix D. Poincaré series and Kloosterman sums

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    Eisenstein Series and Automorphic Representations

    Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson

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  • Authors

    Philipp Fleig, Max-Planck-Institut für Dynamik und Selbstorganisation, Germany
    Philipp Fleig is a Postdoctoral Researcher at the Max-Planck-Institut für Dynamik und Selbstorganisation, Germany.

    Henrik P. A. Gustafsson, Stanford University, California
    Henrik P. A. Gustafsson is a Postdoctoral Researcher in the Department of Mathematics at Stanford University, California.

    Axel Kleinschmidt, Max-Planck-Institut für Gravitationsphysik, Germany
    Axel Kleinschmidt is a Senior Scientist at the Max-Planck-Institut für Gravitationsphysik, Germany (Albert Einstein Institute) and at the International Solvay Institutes, Brussels.

    Daniel Persson, Chalmers University of Technology, Gothenberg
    Daniel Persson is an Associate Professor in the Department of Mathematical Sciences at Chalmers University of Technology, Gothenburg.

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