Eisenstein Series and Automorphic Representations
Eisenstein Series and Automorphic Representations
Henrik P. A. Gustafsson, Stanford University, California
Axel Kleinschmidt, Max-Planck-Institut für Gravitationsphysik, Germany
Daniel Persson, Chalmers University of Technology, Gothenberg
$117.00
USDThis 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
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
Product details
July 2018Hardback
9781107189928
584 pages
235 × 157 × 38 mm
0.92kg
20 b/w illus.
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
- References
- Index.
