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

by Dorian Goldfeld

• ISBN13: 9780521837712
• ISBN10: 0521837715
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-08-21
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Introduction	xi
1 Discrete group actions	1	(37)
1.1 Action of a group on a topological space	3	(5)
1.2 Iwasawa decomposition	8	(7)
1.3 Siegel sets	15	(4)
1.4 Haar measure	19	(4)
1.5 Invariant measure on coset spaces	23	(4)
1.6 Volume of SL(n,Z)\SL(n,R)/SO(n,R)	27	(11)
2 Invariant differential operators	38	(16)
2.1 Lie algebras	39	(3)
2.2 Universal enveloping algebra of g(n,R)	42	(4)
2.3 The center of the universal enveloping algebra of g(n,R)	46	(4)
2.4 Eigenfunctions of invariant differential operators	50	(4)
3 Automorphic forms and L—functions for SL(2,Z)	54	(45)
3.1 Eisenstein series	55	(4)
3.2 Hyperbolic Fourier expansion of Eisenstein series	59	(3)
3.3 Maass forms	62	(1)
3.4 Whittaker expansions and multiplicity one for GL(2, R)	63	(4)
3.5 Fourier—Whittaker expansions on GL(2,R)	67	(1)
3.6 Ramanujan—Petersson conjecture	68	(2)
3.7 Selberg eigenvalue conjecture	70	(1)
3.8 Finite dimensionality of the eigenspaces	71	(2)
3.9 Even and odd Maass forms	73	(1)
3.10 Hecke operators	74	(3)
3.11 Hermite and Smith normal forms	77	(3)
3.12 Hecke operators for L²(SL(2,Z))\h²	80	(4)
3.13 L-functions associated to Maass forms	84	(5)
3.14 L-functions associated to Eisenstein series	89	(2)
3.15 Converse theorems for SL(2,Z)	91	(3)
3.16 The Selberg spectral decomposition	94	(5)
4 Existence of Maass forms	99	(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