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El. knyga: Automorphic Representations and L-Functions for the General Linear Group: Volume 2

(Southern Illinois University, Carbondale), (Columbia University, New York)
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This modern approach to the theory of automorphic representations keeps definitions to a minimum, focusing instead on providing concrete examples and detailed proofs of the key theorems. This book is the perfect introduction for students at the advanced undergraduate level and beyond, and for researchers new to the field.

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, this book will motivate students and researchers to begin working in this fertile field of research.

Recenzijos

"Much of the material presented here is not easily available elsewhere. This brief volume will be of value to mathematicians seeking an introduction to the theory of automorphic forms, automorphic representationa, and L-functions." Solomon Friedberg for Mathematical Reviews

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This modern, graduate-level textbook does not assume prior knowledge of representation theory. Includes numerous concrete examples and exercises.
Contents for Volume I ix
Introduction xv
Preface to the Exercises xix
12 The classical theory of automorphic forms for G L(n, R)
1(15)
12.1 Iwasawa decomposition for G L(n, R)
1(1)
12.2 Congruence subgroups of S L(n, Z)
2(1)
12.3 Automorphic functions of arbitrary weight, level, and character
3(13)
Exercises for
Chapter 12
13(3)
13 Automorphic forms and representations for G L(n, AQ)
16(36)
13.1 Cartan, Bruhat decompositions for G L(n, R)
16(1)
13.2 Iwasawa, Cartan, Bruhat decompositions for G L(n, Qp)
17(5)
13.3 Strong approximation for G L(n)
22(2)
13.4 Adelic lifts and automorphic forms for G L(n, AQ)
24(7)
13.5 The Fourier expansion of adelic automorphic forms
31(5)
13.6 Adelic automorphic representations for G L(n, AQ)
36(5)
13.7 Tensor product theorem for G L(n)
41(2)
13.8 Newforms for G L(n)
43(9)
Exercises for
Chapter 13
47(5)
14 Theory of local representations for G L(n)
52(62)
14.1 Generalities on representations of G L(n, Qp)
52(4)
14.2 Generic representations of G L(n, Qp)
56(4)
14.3 Parabolic induction for G L(n, Qp)
60(6)
14.4 Supercuspidal representations of G L(n, Qp)
66(4)
14.5 The Bernstein-Zelevinsky classification for G L(n, Qp)
70(5)
14.6 Classification of smooth irreducible representations of G L(n, Qp) via the growth of matrix coefficients
75(3)
14.7 Unitary representations of G L(n, Qp)
78(2)
14.8 Generalities on (g, K∞)-modules of G L(n, R)
80(5)
14.9 Generic representations of G L(n, R)
85(3)
14.10 Parabolic induction for G L(n, R)
88(12)
14.11 Classification of the unitary and the generic unitary representations of G L(n, Qp)
100(2)
14.12 Unramified representations of G L(n, Qp) and G L(n, R)
102(3)
14.13 Unitary duals and other duals
105(1)
14.14 The Ramanujan conjecture for G L(n, AQ)
106(8)
Exercises for
Chapter 14
106(8)
15 The Godement-Jacquet L-function for G L(n, AQ)
114(39)
15.1 The Poisson summation formula for G L(n, AQ)
114(4)
15.2 The global zeta integral for G L(n, AQ)
118(6)
15.3 Factorization of the global zeta integral for G L(n, AQ)
124(1)
15.4 The local functional equation for G L(n, Qp)
125(3)
15.5 The L-function and local functional equation for the supereuspidal representations of G L(n, Qp)
128(1)
15.6 The local functional equation for tensor products
128(2)
15.7 The local zeta integral for a parabolically induced representation of G L(n, Qp)
130(8)
15.8 The local zeta integral for discrete series (square integrable) representations of G L(n, Qp)
138(5)
15.9 The local zeta integral for irreducible unitary generic representations of G L(n, R)
143(10)
Exercises for
Chapter 15
151(2)
Solutions to Selected Exercises 153(16)
References 169(6)
Symbols Index 175(4)
Index 179
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.
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