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Automorphic Representations and L-Functions for the General Linear Group

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright Page
  4. Dedication
  5. Contents for Volume I
  6. Contents for Volume II
  7. Introduction
  8. Preface to the Exercises
  9. 1 - Adeles Over Q
    1. 1.1 Absolute Values
    2. 1.2 The Field Qp of p-adic Numbers
    3. 1.3 Adeles and Ideles Over Q
    4. 1.4 Action of Q on the Adeles and Ideles
    5. 1.5 p-adic Integration
    6. 1.6 p-adic Fourier Transform
    7. 1.7 Adelic Fourier Transform
    8. 1.8 Fourier Expansion of Periodic Adelic Functions
    9. 1.9 Adelic Poisson Summation Formula
    10. Exercises for Chapter 1
  10. 2 - Automorphic Representations and L-functions for GL (1, AQ)
    1. 2.1 Automorphic Forms for GL (1, AQ)
    2. 2.2 The L-function of an Automorphic Form
    3. 2.3 The Local L-functions and their Functional Equations
    4. 2.4 Classical L-functions and Root Numbers
    5. 2.5 Automorphic Representations for GL (1, AQ)
    6. 2.6 Hecke Operators for GL (1, AQ)
    7. 2.7 The Rankin-Selberg Method
    8. 2.8 The p-adic Mellin Transform
    9. Exercises for Chapter 2
  11. 3 - The Classical Theory of Automorphic Forms for GL (2)
    1. 3.1 Automorphic Forms in General
    2. 3.2 Congruence Subgroups of the Modular Group
    3. 3.3 Automorphic Functions of Integral Weight k
    4. 3.4 Fourier Expansion at ∞ of Holomorphic Modular Forms
    5. 3.5 Maass Forms
    6. 3.6 Whittaker Functions
    7. 3.7 Fourier-Whittaker Expansions of Maass Forms
    8. 3.8 Eisenstein Series
    9. 3.9 Maass Raising and Lowering Operators
    10. 3.10 The Bottom of the Spectrum
    11. 3.11 Hecke Operators, Oldforms, and Newforms
    12. 3.12 Finite Dimensionality of the Eigenspaces
    13. Exercises for Chapter 3
  12. 4 - Automorphic Forms for GL (2, AQ)
    1. 4.1 Iwasawa and Cartan Decompositions for GL (2, R)
    2. 4.2 Iwasawa and Cartan Decompositions for GL (2, Qp)
    3. 4.3 The Adele Group GL (2, AQ)
    4. 4.4 The Action of GL (2, Q) on GL (2, AQ)
    5. 4.5 The Universal Enveloping Algebra of gl(2, C)
    6. 4.6 The Center of the Universal Enveloping Algebra of gl(2, C)
    7. 4.7 Automorphic Forms for GL (2, AQ)
    8. 4.8 Adelic Lifts of Weight Zero, Level One, Maass Forms
    9. 4.9 The Fourier Expansion of Adelic Automorphic Forms
    10. 4.10 Global Whittaker Functions for GL (2, AQ)
    11. 4.11 Strong Approximation for Congruence Subgroups
    12. 4.12 Adelic Lifts with Arbitrary Weight, Level, and Character
    13. 4.13 Global Whittaker Functions for Adelic Lifts with Arbitrary Weight, Level, and Character
    14. Exercises for Chapter 4
  13. 5 - Automorphic Representations for GL (2, AQ)
    1. 5.1 Adelic Automorphic Representations for GL (2, AQ)
    2. 5.2 Explicit Realization of Actions Defining a (gl, K∞)-module
    3. 5.3 Explicit Realization of the Action of GL (2, Afinite)
    4. 5.4 Examples of Cuspidal Automorphic Representations
    5. 5.5 Admissible (gl, K∞) × GL (2, Afinite)-modules
    6. Exercises for Chapter 5
  14. 6 - Theory of Admissible Representations of GL (2, Qp)
    1. 6.0 Short Roadmap to Chapter 6
    2. 6.1 Admissible Representations of GL (2, Qp)
    3. 6.2 Ramified Versus Unramified
    4. 6.3 Local Representation Coming from a Level 1 Maass Form
    5. 6.4 Jacquet’s Local Whittaker Function
    6. 6.5 Principal Series Representations
    7. 6.6 Jacquet’s map: Principal Series → Whittaker Functions
    8. 6.7 The Kirillov Model
    9. 6.8 The Kirillov Model of the Principal Series Representation
    10. 6.9 Haar Measure on GL (2, Qp)
    11. 6.10 The Special Representations
    12. 6.11 Jacquet Modules
    13. 6.12 Induced Representations and Parabolic Induction
    14. 6.13 The Supercuspidal Representations of GL (2, Qp)
    15. 6.14 The Uniqueness of the Kirillov Model
    16. 6.15 The Kirillov Model of a Supercuspidal Representation
    17. 6.16 The Classification of the Irreducible and Admissible Representations of GL (2, Qp)
    18. Exercises for Chapter 6
  15. 7 - Theory of Admissible (gl K∞) Modules for GL (2, R)
    1. 7.1 Admissible (gl, K∞)-modules
    2. 7.2 Ramified Versus Unramified
    3. 7.3 Jacquet’s Local Whittaker Function
    4. 7.4 Principal Series Representations
    5. 7.5 Classification of Irreducible Admissible (gl, K∞)-modules
    6. Exercises for Chapter 7
  16. 8 - The Contragredient Representation for GL (2)
    1. 8.1 The Contragredient Representation for GL (2, Qp)
    2. 8.2 The Contragredient Representation of a Principal Series Representation of GL (2, Qp)
    3. 8.3 Contragredient of a Special Representation of GL (2, Qp)
    4. 8.4 Contragredient of a Supercuspidal Representation
    5. 8.5 The Contragredient Representation for GL (2, R)
    6. 8.6 The Contragredient Representation of a Principal Series Representation of GL (2, R)
    7. 8.7 Global Contragredients for GL (2, AQ)
    8. 8.8 Integration on GL (2, AQ)
    9. 8.9 The Contragredient Representation of a Cuspidal Automorphic Representation of GL (2, AQ)
    10. 8.10 Growth of Matrix Coefficients
    11. 8.11 Asymptotics of Matrix Coefficients of (gl, K∞)-modules
    12. 8.12 Matrix Coefficients of GL (2, Qp) via the Jacquet module
    13. Exercises for Chapter 8
  17. 9 - Unitary Representations of GL (2)
    1. 9.1 Unitary Representations of GL (2, Qp)
    2. 9.2 Unitary Principal Series Representations of GL (2, Qp)
    3. 9.3 Unitary and Irreducible Special or Supercuspidal Representations of GL (2, Qp)
    4. 9.4 Unitary (gl, K∞)-modules
    5. 9.5 Unitary (gl, K∞) × GL (2, Afinite)-modules
    6. Exercises for Chapter 9
  18. 10 - Tensor Products of Local Representations
    1. 10.1 Euler Products
    2. 10.2 Tensor Product of (gl, K∞)-modules and representations
    3. 10.3 Infinite Tensor Products of local Representations
    4. 10.4 The Factorization of Unramified Irreducible Admissible Cuspidal Automorphic Representations
    5. 10.5 Decomposition of Representations of Locally Compact Groups into Finite Tensor Products
    6. 10.6 The Spherical Hecke Algebra for GL (2, Qp)
    7. 10.7 Initial Decomposition of Admissible (g, K∞) × GL (2, Afinite)-modules
    8. 10.8 The Tensor Product Theorem
    9. 10.9 The Ramanujan and Selberg Conjectures for GL (2, AQ)
    10. Exercises for Chapter 10
  19. 11 - The Godement-Jacquet L-function for GL (2, AQ)
    1. 11.1 Historical Remarks
    2. 11.2 The Poisson Summation Formula for GL (2, AQ)
    3. 11.3 Haar Measure
    4. 11.4 The Global Zeta Integral for GL (2, AQ)
    5. 11.5 Factorization of the Global Zeta Integral
    6. 11.6 The Local Functional Equation
    7. 11.7 The Local L-function for GL (2, Qp) (Unramified Case)
    8. 11.8 The Local L-function for Irreducible Supercuspidal Representations of GL (2, Qp)
    9. 11.9 The Local L-function for Irreducible Principal Series Representations of GL (2, Qp)
    10. 11.10 Local L-function for Unitary Special Representations of GL (2, Qp)
    11. 11.11 Proof of the Local Functional Equation for Principal Series Representations of GL (2, Qp)
    12. 11.12 The Local Functional Equation for the Unitary Special Representations of GL (2, Qp)
    13. 11.13 Proof of the Local Functional Equation for the Supercuspidal Representations of GL (2, Qp)
    14. 11.14 The Local L-function for Irreducible Principal Series Representations of GL (2, R)
    15. 11.15 Proof of the Local Functional Equation for Principal Series Representations of GL (2, R)
    16. 11.16 The Local L-function for Irreducible Discrete Series Representations of GL (2, R)
    17. Exercises for Chapter 11
  20. Solutions to Selected Exercises
  21. References
  22. Symbols Index
  23. Index