Automorphic Forms and L-Functions for the Group GL(n,R)

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ISBN: 9780521837712 | 0521837715
Cover: Hardcover
Copyright: 8/21/2006

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L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy to read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.

Introduction

1 Discrete group actions

(37)

1.1 Action of a group on a topological space

(5)

1.2 Iwasawa decomposition

(7)

1.3 Siegel sets

(4)

1.4 Haar measure

(4)

1.5 Invariant measure on coset spaces

(4)

1.6 Volume of SL(n,Z)\SL(n,R)/SO(n,R)

(11)

2 Invariant differential operators

(16)

2.1 Lie algebras

(3)

2.2 Universal enveloping algebra of g(n,R)

(4)

2.3 The center of the universal enveloping algebra of g(n,R)

(4)

2.4 Eigenfunctions of invariant differential operators

(4)

3 Automorphic forms and L—functions for SL(2,Z)

(45)

3.1 Eisenstein series

(4)

3.2 Hyperbolic Fourier expansion of Eisenstein series

(3)

3.3 Maass forms

(1)

3.4 Whittaker expansions and multiplicity one for GL(2, R)

(4)

3.5 Fourier—Whittaker expansions on GL(2,R)

(1)

3.6 Ramanujan—Petersson conjecture

(2)

3.7 Selberg eigenvalue conjecture

(1)

3.8 Finite dimensionality of the eigenspaces

(2)

3.9 Even and odd Maass forms

(1)

3.10 Hecke operators

(3)

3.11 Hermite and Smith normal forms

(3)

3.12 Hecke operators for L²(SL(2,Z))\h²

(4)

3.13 L-functions associated to Maass forms

(5)

3.14 L-functions associated to Eisenstein series

(2)

3.15 Converse theorems for SL(2,Z)

(3)

3.16 The Selberg spectral decomposition

(5)

4 Existence of Maass forms

(15)

4.1 The infinitude of odd Maass forms for SL(2,Z)

100

(1)

4.2 Integral operators

101

(4)

4.3 The endomorphism heart

105

(1)

4.4 How to interpret heart: an explicit operator with purely cuspidal image

106

(2)

4.5 There exist infinitely many even cusp forms for SL(2,Z)

108

(2)

4.6 A weak Weyl law

110

(1)

4.7 Interpretation via wave equation and the role of finite propagation speed

111

(1)

4.8 Interpretation via wave equation: higher rank case

111

(3)

5 Maass forms and Whittaker functions for SL(n,Z)

114

(39)

5.1 Maass forms

114

(2)

5.2 Whittaker functions associated to Maass forms

116

(2)

5.3 Fourier expansions on SL(n,Z)\hn

118

(10)

5.4 Whittaker functions for SL(n,R)

128

(1)

5.5 Jacquet's Whittaker function

129

(5)

5.6 The exterior power of a vector space

134

(4)

5.7 Construction of the Iv function using wedge products

138

(3)

5.8 Convergence of Jacquet's Whittaker function

141

(3)

5.9 Functional equations of Jacquet's Whittaker function

144

(6)

5.10 Degenerate Whittaker functions

150

(3)

6 Automorphic forms and L-functions for SL(3,Z)

153

(41)

6.1 Whittaker functions and multiplicity one for SL(3,Z)

153

(6)

6.2 Maass forms for SL(3,Z)

159

(2)

6.3 The dual and symmetric Maass forms

161

(2)

6.4 Hecke operators for SL(3,Z)

163

(9)

6.5 The Godement-Jacquet L-function

172

(14)

6.6 Bump's double Dirichlet series

186

(8)

7 The Gelbart-Jacquet lift

194

(41)

7.1 Converse theorem for SL(3,Z)

194

(16)

7.2 Rankin-Selberg convolution for GL(2)

210

(3)

7.3 Statement and proof of the Gelbart-Jacquet lift

213

(10)

7.4 Rankin-Selberg convolution for GL(3)

223

(12)

8 Bounds for L-functions and Siegel zeros

235

(24)

8.1 The Selberg class

235

(3)

8.2 Convexity bounds for the Selberg class

238

(3)

8.3 Approximate functional equations

241

(4)

8.4 Siegel zeros in the Selberg class

245

(4)

8.5 Siegel's theorem

249

(2)

8.6 The Siegel zero lemma

251

(1)

8.7 Non-existence of Siegel zeros for Gelbart–Jacquet lifts

252

(4)

8.8 Non-existence of Siegel zeros on GL(n)

256

(3)

9 The Godement–Jacquet L-function

259

(26)

9.1 Maass forms for SL(n,Z)

259

(2)

9.2 The dual and symmetric Maass forms

261

(5)

9.3 Hecke operators for SL(n,Z)

266

(11)

9.4 The Godement–Jacquet L-function

277

(8)

10 Langlands Eisenstein series

285

(52)

10.1 Parabolic subgroups

286

(2)

10.2 Langlands decomposition of parabolic subgroups

288

(4)

10.3 Bruhat decomposition

292

(3)

10.4 Minimal, maximal, and general parabolic Eisenstein series

295

(6)

10.5 Eisenstein series twisted by Maass forms

301

(2)

10.6 Fourier expansion of minimal parabolic Eisenstein series

303

(4)

10.7 Meromorphic continuation and functional equation of maximal parabolic Eisenstein series

307

(3)

10.8 The L-function associated to a minimal parabolic Eisenstein series

310

(5)

10.9 Fourier coefficients of Eisenstein series twisted by Maass forms

315

(4)

10.10 The constant term

319

(2)

10.11 The constant term of SL(3,Z) Eisenstein series twisted by SL(2,Z)-Maass forms

321

(1)

10.12 An application of the theory of Eisenstein series to the non-vanishing of L-functions on the line R(s) = 1

322

(2)

10.13 Langlands spectral decomposition for SL(3,Z)\h³

324

(13)

11 Poincaré series and Kloosterman sums

337

(28)

11.1 Poincaré series for SL(n,Z)

337

(2)

11.2 Kloosterman sums

339

(4)

11.3 Plücker coordinates and the evaluation of Kloosterman sums

343

(7)

11.4 Properties of Kloosterman sums

350

(2)

11.5 Fourier expansion of Poincaré series

352

(2)

11.6 Kuznetsov's trace formula for SL(n,Z)

354

(11)

12 Rankin–Selberg convolutions

365

(30)

12.1 The GL(n) x GL(n) convolution

366

(6)

12.2 The GL(n) x GL(n+1) convolution

372

(9)

12.3 The GL(n) x GL(n') convolution with n less than n' 376

12.4 Generalized Ramanujan conjecture

381

(3)

12.5 The Luo–Rudnick–Sarnak bound for the generalized Ramanujan conjecture

384

(9)

12.6 Strong multiplicity one theorem

393

(2)

13 Langlands conjectures

395

(12)

13.1 Artin L-functions

397

(5)

13.2 Langlands functoriality

402

(5)

List of symbols

407

(2)

Appendix The GL(n)pack Manual

409

(64)

Kevin A. Broughan

A.1 Introduction

409

(4)

A.2 Functions for GL(n)pack

413

(3)

A.3 Function descriptions and examples

416

(57)

References

473

(12)

Index

485

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Automorphic Forms and L-Functions for the Group GL(n,R)

Summary

Table of Contents

Supplemental Materials