Automorphic Forms and Representations

This book covers both the classical and representation theoretic views of automorphic forms in a style that is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin-Selberg method and the triple L-function, and examines this subject matter from many different and complementary viewpoints. Researchers as well as students in algebra and number theory will find this a valuable guide to a notoriously difficult subject.

Preface

ix

(2)

Advice to the Reader

xi

(3)

Prerequisities

xiv

(1)

Notations

xv

1 Modular Forms

1

(126)

1.1 Dirichlet L-Functions

3

(14)

1.2 The Modular Group

17

(9)

1.3 Modular Forms for SL(2, Z)

26

(15)

1.4 Hecke Operators

41

(13)

1.5 Twisting

54

(11)

1.6 The Rankin-Selberg Method

65

(11)

1.7 Hecke Characters and Hilbert Modular Forms

76

(14)

1.8 Artin L-Functions and Langlands Functoriality

90

(13)

1.9 Maass Forms

103

(16)

1.10 Base Change

119

(8)

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

127

(125)

2.1 Maass Forms and the Spectral Problem

128

(17)

2.2 Basic Lie Theory

145

(20)

2.3 Discreteness of the Spectrum

165

(21)

2.4 Basic Representation Theory

186

(17)

2.5 Irreducible (g, K)-Modules for GL(2, R)

203

(19)

2.6 Unitaricity and Intertwining Integrals

222

(19)

2.7 Representations and the Spectral Problem

241

(2)

2.8 Whittaker Models

243

(3)

2.9 A Theorem of Harish-Chandra

246

(6)

3 Automorphic Representations

252

(145)

3.1 Tate's Thesis

254

(24)

3.2 Classical Automorphic Forms and Representations

278

(13)

3.3 Automorphic Representations of GL(n)

291

(16)

3.4 The Tensor Product Theorem

307

(14)

3.5 Whittaker Models and Automorphic Forms

321

(20)

3.6 Adelization of Classical Automorphic Forms

341

(5)

3.7 Eisenstein Series and Intertwining Integrals

346

(22)

3.8 The Rankin-Selberg Method

368

(7)

3.9 The Global Langlands Conjectures

375

(10)

3.10 The Triple Convolution

385

(12)

4 Representations of GL(2) Over a p-adic Field

397

(161)

4.1 GL(2) Over a Finite Field

398

(26)

4.2 Smooth and Admissible Representations

424

(12)

4.3 Distributions and Sheaves

436

(16)

4.4 Whittaker Models and the Jacquet Functor

452

(17)

4.5 The Principal Series Representations

469

(21)

4.6 Spherical Representations

490

(21)

4.7 Local Functional Equations

511

(12)

4.8 Supercuspidals and the Weil Representation

523

(26)

4.9 The Local Langlands Correspondence

549

(9)

Bibliography

558

(10)

Index

568

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Automorphic Forms and Representations

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Table of Contents

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