Automorphic Forms and Representations

This book, published by Cambridge University Press, is now shipping. The final proofs were corrected on 9/16/96.

Here is the official catalog description from Cambridge.

Errata

Here is the list of known errata which I am maintaining for the book.

I: Modular Forms

1. Dirichlet L-functions
2. The Modular Group
3. Modular Forms for SL(2,Z)
4. Hecke Operators
5. Twisting
6. The Rankin-Selberg Method
7. Hilbert Modular Forms
8. Langlands' Functoriality
9. Maass Forms
10. Base Change

II: Automorphic Forms and Representations of GL(2,R)

1. Maass Forms and the Spectral Problem
2. Basic Lie Theory
3. Discreteness of the Spectrum
4. Basic Representation Theory
5. Irreducible (g,K)-Modules
6. Unitaricity and Intertwining Integrals
7. Representations and the Spectral Problem
8. Whittaker Models
9. A Theorem of Harish-Chandra

III: Automorphic Representations

1. Tate's Thesis
2. Automorphic Forms and Representations
3. Automorphic Representations of GL(n)
4. The Tensor Product Theorem
5. Whittaker Models and Automorphic Forms
6. Adelization of Automorphic Forms
7. Eisenstein Series and Intertwining Integrals
8. The Rankin-Selberg Method
9. The Global Langlands Conjectures
10. The Triple Convolution

IV: GL(2) over a p-adic field

1. GL(2) over a Finite Field
2. Smooth and Admissible Representations
3. Distributions and Sheaves
4. Whittaker Models and the Jacquet Functor
5. The Principal Series Representations
6. Spherical Representations
7. Local Functional Equations
8. Supercuspidals and the Weil Representation
9. The Local Langlands Correspondence

From the PREFACE

The Theory of Automorphic Forms, rightly or wrongly, has a reputation of being difficult for the student. I felt that there was a need for a book which would present the subject in a style which was accessible, yet based on complete proofs, revealing clearly the uniqueness principles which underlie the basic constructions. Since 1990 I have been lecturing on automorphic forms and representation theory at Stanford and the MSRI, and this book is the end result.

The level of this book is intermediate between an advanced textbook and a monograph. I hope that it will be found interesting by experts as well as graduate students. Its aim is to cover a substantial portion of the theory of automorphic forms on GL(2). Both the ``classical'' and ``representation theoretic'' viewpoints are covered.

There are significant omissions from our treatment, most seriously the Selberg Trace Formula. It has not been my aim to achieve complete coverage of the topics treated, or to write a reference book. I feel that the existing reference material is adequate, and it was not feasible to cover any single topic with the thoroughness I would have liked. Rather, it was my aim to treat my subject matter with some degree of depth. I hope that the reader will begin studying the reference material (such as the Corvallis volume and above all Jacquet and Langlands in the course of reading this book. If I have done my job well, the task of approaching Jacquet and Langlands should be made easier by the current volume.

I would like to thank William Banks, Antonia Bluher, Aleksandr Brener, David Cardon, Jim Cogdell, Anton Deitmar, David Feldman, Solomon Friedberg, Masaaki Furusawa, Steve Gelbart, Tom Goetze, David Goldberg, Jiandong Guo, Jeffrey Hoffstein, Ozlem Imamoglu, David Joyner, Chris Judge, Par Kurlberg, Annette Klute, David Manderscheid, Greg Martin, Andrei Paraschivescu, Ralph Phillips, Freydoon Shahidi, Tom Shemanske, Trask Stalnaker, Steve Rallis, Ken Ribet, Dinakar Ramakrishnan, Julie Roskies, San Cao Vo, James Woodson---and probably others I've forgotten---for helpful comments, corrections, discussions or other feedback. Thanks also to Lauren Cowles of Cambridge University Press for her interest in the manuscript and for her guidance, to Ellen Tirpak and the staff at TechBooks for their expert handling of the manuscript, to Reid Augustin for helping me set up my Linux machine, and to the MSRI for their help and support during 1994-5. And thank you, my wife Kathi, and my parents Kenneth and Ellen Bump, for your support, which was always there when I needed it most.

Parts of this book were written during my support with the AMS Centennial Research Fellowship, and grants from the National Science Foundation.

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