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Automorphic Representations and L-Functions for the General Linear Group: Volume 1 [Hardback]

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(Columbia University, New York), (Southern Illinois University, Carbondale)
  • Formāts: Hardback, 572 pages, height x width x depth: 229x152x37 mm, weight: 1010 g, Worked examples or Exercises
  • Sērija : Cambridge Studies in Advanced Mathematics
  • Izdošanas datums: 21-Apr-2011
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 052147423X
  • ISBN-13: 9780521474238
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Automorphic Representations and L-Functions for the General Linear Group: Volume 1Palielināt
  • Formāts: Hardback, 572 pages, height x width x depth: 229x152x37 mm, weight: 1010 g, Worked examples or Exercises
  • Sērija : Cambridge Studies in Advanced Mathematics
  • Izdošanas datums: 21-Apr-2011
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 052147423X
  • ISBN-13: 9780521474238
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521474238.html
Keywords:
This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

Recenzijas

'In this book, the authors give a thorough yet elementary introduction to the theory of automorphic forms and L-functions for the general linear group of rank two over rational adeles The exposition is accompanied by exercises after every chapter. Definitions are repeated when needed, and previous results are always cited, so the book is very accessible.' Marcela Hanzer, Zentralblatt MATH

Papildus informācija

This modern, graduate-level textbook does not assume prior knowledge of representation theory. Includes numerous concrete examples and over 250 exercises.
Introduction xv
Preface to the Exercises xix
1 Adeles over Q 1(38)
1.1 Absolute values
1(1)
1.2 The field Qp of p-adic numbers
2(5)
1.3 Adeles and ideles over Q
7(1)
1.4 Action of on the adeles and ideles
8(4)
1.5 p-adic integration
12(3)
1.6 p-adic Fourier transform
15(3)
1.7 Adelic Fourier transform
18(5)
1.8 Fourier expansion of periodic adelic functions
23(7)
1.9 Adelic Poisson summation formula
30(1)
Exercises for
Chapter 1
31(8)
2 Automorphic representations and L-functions for GL (1, AQ) 39(37)
2.1 Automorphic forms for GL (1, AQ)
39(6)
2.2 The L-function of an automorphic form
45(10)
2.3 The local L-functions and their functional equations
55(5)
2.4 Classical L-functions and root numbers
60(5)
2.5 Automorphic representations for GL (1, AQ)
65(3)
2.6 Hecke operators for GL (1, AQ)
68(1)
2.7 The Rankin-Selberg method
69(1)
2.8 The p-adic Mellin transform
70(2)
Exercises for
Chapter 2
72(4)
3 The classical theory of automorphic forms for GL (2) 76(27)
3.1 Automorphic forms in general
76(1)
3.2 Congruence subgroups of the modular group
77(1)
3.3 Automorphic functions of integral weight k
78(2)
3.4 Fourier expansion at oo of holomorphic modular forms
80(1)
3.5 Maass forms
81(3)
3.6 Whittaker functions
84(3)
3.7 Fourier-Whittaker expansions of Maass forms
87(2)
3.8 Eisenstein series
89(1)
3.9 Maass raising and lowering operators
90(2)
3.10 The bottom of the spectrum
92(2)
3.11 Hecke operators, oldforms, and newforms
94(3)
3.12 Finite dimensionality of the eigenspaces
97(1)
Exercises for
Chapter 3
98(5)
4 Automorphic forms for GL (2, AQ) 103(49)
4.1 Iwasawa and Cartan decompositions for GL (2, R)
103(2)
4.2 Iwasawa and Cartan decompositions for GL(2, Qp)
105(2)
4.3 The adele group GL (2, AQ)
107(1)
4.4 The action of GL(2, Q) on GL(2, AQ)
108(4)
4.5 The universal enveloping algebra of gl(2, C)
112(5)
4.6 The center of the universal enveloping algebra of g[ (2, C)
117(1)
4.7 Automorphic forms for GL(2, AQ)
117(2)
4.8 Adelic lifts of weight zero, level one, Maass forms
119(7)
4.9 The Fourier expansion of adelic automorphic forms
126(2)
4.10 Global Whittaker functions for GL(2, AQ)
128(6)
4.11 Strong approximation for congruence subgroups
134(2)
4.12 Adelic lifts with arbitrary weight, level, and character
136(5)
4.13 Global Whittaker functions for adelic lifts with arbitrary weight, level, and character
141(6)
Exercises for
Chapter 4
147(5)
5 Automorphic representations for GL (2, AQ) 152(31)
5.1 Adelic automorphic representations for GL (2, AQ)
152(9)
5.2 Explicit realization of actions defining a (g, Kinfinity)-module
161(7)
5.3 Explicit realization of the action of GL (2, Affinite)
168(4)
5.4 Examples of cuspidal automorphic representations
172(1)
5.5 Admissible (g, Kinfinity) x GL(2, Afinite)-modules
173(5)
Exercises for
Chapter 5
178(5)
6 Theory of admissible representations of GL (2, Qp) 183(76)
6.0 Short roadmap to chapter 6
183(1)
6.1 Admissible representations of GL (2, Qp)
183(9)
6.2 Ramified versus unramified
192(1)
6.3 Local representation coming from a level 1 Maass form
193(2)
6.4 Jacquet's local Whittaker function
195(5)
6.5 Principal series representations
200(5)
6.6 Jacquet's map: Principal series -> Whittaker functions
205(9)
6.7 The Kirillov model
214(7)
6.8 The Kirillov model of the principal series representation
221(7)
6.9 Haar measure on GL (2, Qp)
228(4)
6.10 The special representations
232(4)
6.11 Jacquet modules
236(2)
6.12 Induced representations and parabolic induction
238(2)
6.13 The supercuspidal representations of GL(2, Qp)
240(3)
6.14 The uniqueness of the Kirillov model
243(9)
6.15 The Kirillov model of a supercuspidal representation
252(1)
6.16 The classification of the irreducible and admissible representations of GL (2, Qp)
252(1)
Exercises for
Chapter 6
253(6)
7 Theory of admissible (g, K infinity) modules for GL (2, R) 259(18)
7.1 Admissible (g, Kinfinity)-modules
259(1)
7.2 Ramified versus unramified
260(1)
7.3 Jacquet's local Whittaker function
260(3)
7.4 Principal series representations
263(6)
7.5 Classification of irreducible admissible (g, Kinfinity)-modules
269(6)
Exercises for
Chapter 7
275(2)
8 The contragredient representation for GL (2) 277(81)
8.1 The contragredient representation for GL (2, Qp)
277(4)
8.2 The contragredient representation of a principal series representation of GL (2, Qp)
281(2)
8.3 Contragredient of a special representation of GL (2, Qp)
283(2)
8.4 Contragredient of a supercuspidal representation
285(4)
8.5 The contragredient representation for GL (2, R)
289(5)
8.6 The contragredient representation of a principal series representation of GL (2, R)
294(9)
8.7 Global contragredients for GL (2, AQ)
303(3)
8.8 Integration on GL (2, AQ)
306(5)
8.9 The contragredient representation of a cuspidal automorphic representation of GL (2, AQ)
311(5)
8.10 Growth of matrix coefficients
316(14)
8.11 Asymptotics of matrix coefficients of (g, Kinfinity)-modules
330(13)
8.12 Matrix coefficients of GL (2, Qp) via the Jacquet module
343(10)
Exercises for
Chapter 8
353(5)
9 Unitary representations of GL (2) 358(20)
9.1 Unitary representations of GL (2, Qp)
358(2)
9.2 Unitary principal series representations of GL(2, Qp)
360(4)
9.3 Unitary and irreducible special or supercuspidal representations of GL (2, Q)
364(1)
9.4 Unitary (g, Kinfinity)-modules
365(3)
9.5 Unitary (g, Kinfinity) x GL(2, Afinite)-modules
368(6)
Exercises for
Chapter 9
374(4)
10 Tensor products of local representations 378(40)
10.1 Euler products
378(1)
10.2 Tensor product of (g, Kinfinity)-modules and representations
379(2)
10.3 Infinite tensor products of local representations
381(2)
10.4 The factorization of unramified irreducible admissible cuspidal automorphic representations
383(5)
10.5 Decomposition of representations of locally compact groups into finite tensor products
388(8)
10.6 The spherical Hecke algebra for GL (2, Qp)
396(7)
10.7 Initial decomposition of admissible (g, Kinfinity) x GL(2, Afinite)-modules
403(3)
10.8 The tensor product theorem
406(7)
10.9 The Ramanujan and Selberg conjectures for GL (2, AQ)
413(2)
Exercises for
Chapter 10
415(3)
11 The Godement-Jacquet L-function for GL (2, AQ) 418(60)
11.1 Historical remarks
418(1)
11.2 The Poisson summation formula for GL (2, AQ)
419(4)
11.3 Haar measure
423(2)
11.4 The global zeta integral for GL (2, AQ)
425(5)
11.5 Factorization of the global zeta integral
430(2)
11.6 The local functional equation
432(2)
11.7 The local L-function for GL (2, Qp) (unramified case)
434(6)
11.8 The local L-function for irreducible supercuspidal representations of GL (2, Qp)
440(1)
11.9 The local L-function for irreducible principal series representations of GL (2, Qp)
441(3)
11.10 Local L-function for unitary special representations of GL (2, Qp)
444(2)
11.11 Proof of the local functional equation for principal series representations of GL (2, Qp)
446(4)
11.12 The local functional equation for the unitary special representations of GL (2, Qp)
450(2)
11.13 Proof of the local functional equation for the supercuspidal representations of GL (2, Qp)
452(11)
11.14 The local L-function for irreducible principal series representations of GL (2, R)
463(4)
11.15 Proof of the local functional equation for principal series representations of GL (2, R)
467(4)
11.16 The local L-function for irreducible discrete series representations of GL (2, R)
471(3)
Exercises for
Chapter 11
474(4)
Solutions to Selected Exercises 478(53)
References 531(6)
Symbols Index 537(4)
Index 541
Dorian Goldfeld is a Professor in the Department of Mathematics at Columbia University, New York. Joseph Hundley is an Assistant Professor in the Department of Mathematics at Southern Illinois University, Carbondale.