Automorphic representations and L-functions for the general linear group. Volume I /

This modern, graduate-level textbook does not assume prior knowledge of representation theory. Includes numerous concrete examples and over 250 exercises.

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Goldfeld, D.
Otros Autores: Hundley, Joseph
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge ; New York : Cambridge University Press, 2011.
Colección:Cambridge studies in advanced mathematics ; 129.
Temas:
Automorphic forms.
L-functions.
Representations of groups.
Formes automorphes.
Fonctions L.
Représentations de groupes.
MATHEMATICS > Complex Analysis.
Automorphic forms
L-functions
Representations of groups
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Half-title
  • Series-title
  • Title
  • Copyright
  • Dedication
  • Contents for Volume I
  • Contents for Volume II
  • Introduction
  • Preface to the Exercises
  • 1 Adeles over Q
  • 1.1 Absolute values
  • 1.2 The field Qp of p-adic numbers
  • 1.3 Adeles and ideles over Q
  • 1.4 Action of Q on the adeles and ideles
  • 1.5 p-adic integration
  • 1.6 p-adic Fourier transform
  • 1.7 Adelic Fourier transform
  • 1.8 Fourier expansion of periodic adelic functions
  • 1.9 Adelic Poisson summation formula
  • Exercises for Chapter 1
  • 2 Automorphic representations and L-functions for GL(1, AQ)
  • 2.1 Automorphic forms for GL (1, AQ)
  • 2.2 The L-function of an automorphic form
  • 2.3 The local L-functions and their functional equations
  • 2.4 Classical L-functions and root numbers
  • 2.5 Automorphic representations for GL(1, AQ)
  • 2.6 Hecke operators for GL(1, AQ)
  • 2.7 The Rankin-Selberg method
  • 2.8 The p-adic Mellin transform
  • Exercises for Chapter 2
  • 3 The classical theory of automorphic forms for GL(2)
  • 3.1 Automorphic forms in general
  • 3.2 Congruence subgroups of the modular group
  • 3.3 Automorphic functions of integral weight k
  • 3.4 Fourier expansion at ... of holomorphic modular forms
  • 3.5 Maass forms
  • 3.6 Whittaker functions
  • 3.7 Fourier-Whittaker expansions of Maass forms
  • 3.8 Eisenstein series
  • 3.9 Maass raising and lowering operators
  • 3.10 The bottom of the spectrum
  • 3.11 Hecke operators, oldforms, and newforms
  • 3.12 Finite dimensionality of the eigenspaces
  • Exercises for Chapter 3
  • 4 Automorphic forms for GL(2, AQ)
  • 4.1 Iwasawa and Cartan decompositions for GL(2, R)
  • 4.2 Iwasawa and Cartan decompositions for GL(2, Qp)
  • 4.3 The adele group GL(2, AQ)
  • 4.4 The action of GL (2, Q) on GL(2, AQ)
  • 4.5 The universal enveloping algebra of gl(2,C).
  • 4.6 The center of the universal enveloping algebra of gl(2, C)
  • 4.7 Automorphic forms for GL(2, AQ)
  • 4.8 Adelic lifts of weight zero, level one, Maass forms
  • 4.9 The Fourier expansion of adelic automorphic forms
  • 4.10 Global Whittaker functions for GL(2, AQ)
  • 4.11 Strong approximation for congruence subgroups
  • 4.12 Adelic lifts with arbitrary weight, level, and character
  • 4.13 Global Whittaker functions for adelic lifts with arbitrary weight, level, and character
  • Exercises for Chapter 4
  • 5 Automorphic representations for GL(2, AQ)
  • 5.1 Adelic automorphic representations for GL(2, AQ)
  • 5.2 Explicit realization of actions defining a (g, K ...)-module
  • 5.3 Explicit realization of the action of GL(2, Afinite)
  • 5.4 Examples of cuspidal automorphic representations
  • 5.5 Admissible (g, K ...) × GL(2, Afinite)-modules
  • Exercises for Chapter 5
  • 6 Theory of admissible representations of GL(2, Qp)
  • 6.0 Short roadmap to chapter 6
  • 6.1 Admissible representations of GL(2, Qp)
  • 6.2 Ramified versus unramified
  • 6.3 Local representation coming from a level 1 Maass form
  • 6.4 Jacquet's local Whittaker function
  • 6.5 Principal series representations
  • 6.6 Jacquet's map: Principal series larrow Whittaker functions
  • 6.7 The Kirillov model
  • 6.8 The Kirillov model of the principal series representation
  • 6.9 Haar measure on GL(2, Qp)
  • 6.10 The special representations
  • 6.11 Jacquet modules
  • 6.12 Induced representations and parabolic induction
  • 6.13 The supercuspidal representations of GL(2, Qp)
  • 6.14 The uniqueness of the Kirillov model
  • 6.15 The Kirillov model of a supercuspidal representation
  • 6.16 The classification of the irreducible and admissible representations of GL(2, Qp)
  • Exercises for Chapter 6
  • 7 Theory of admissible (g, K8) modules for GL(2, R)
  • 7.1 Admissible (g, K8)-modules.
  • 7.2 Ramified versus unramified
  • 7.3 Jacquet's local Whittaker function
  • 7.4 Principal series representations
  • 7.5 Classification of irreducible admissible (g, K8)-modules
  • Exercises for Chapter 7
  • 8 The contragredient representation for GL(2)
  • 8.1 The contragredient representation for GL(2, Qp)
  • 8.2 The contragredient representation of a principal series representation of GL(2, Qp)
  • 8.3 Contragredient of a special representation of GL(2, Qp)
  • 8.4 Contragredient of a supercuspidal representation
  • 8.5 The contragredient representation for GL(2, R)
  • 8.6 The contragredient representation of a principal series representation of GL(2, R)
  • 8.7 Global contragredients for GL(2, AQ)
  • 8.8 Integration on GL(2, AQ)
  • 8.9 The contragredient representation of a cuspidal automorphic representation of GL(2, AQ)
  • 8.10 Growth of matrix coefficients
  • 8.11 Asymptotics of matrix coefficients of (g, K8)-modules
  • 8.12 Matrix coefficients of GL(2, Qp) via the Jacquet module
  • Exercises for Chapter 8
  • 9 Unitary representations of GL (2)
  • 9.1 Unitary representations of GL(2, Qp)
  • 9.2 Unitary principal series representations of GL(2, Qp)
  • 9.3 Unitary and irreducible special or supercuspidal representations of GL(2, Qp)
  • 9.4 Unitary (g, K8)-modules
  • 9.5 Unitary (g, K8) × GL(2,Afinite)-modules
  • Exercises for Chapter 9
  • 10 Tensor products of local representations
  • 10.1 Euler products
  • 10.2 Tensor product of (g, K8)-modules and representations
  • 10.3 Infinite tensor products of local representations
  • 10.4 The factorization of unramified irreducible admissible cuspidal automorphic representations
  • 10.5 Decomposition of representations of locally compact groups into finite tensor products
  • 10.6 The spherical Hecke algebra for GL(2, Qp)
  • 10.7 Initial decomposition of admissible (g, K8) × GL(2,Afinite)-modules.
  • 10.8 The tensor product theorem
  • 10.9 The Ramanujan and Selberg conjectures for GL(2, AQ)
  • Exercises for Chapter 10
  • 11 The Godement-Jacquet L-function for GL(2, AQ)
  • 11.1 Historical remarks
  • 11.2 The Poisson summation formula for GL(2, AQ)
  • 11.3 Haar measure
  • 11.4 The global zeta integral for GL(2, AQ)
  • 11.5 Factorization of the global zeta integral
  • 11.6 The local functional equation
  • 11.7 The local L-function for GL(2, Qp) (unramified case)
  • 11.8 The local L-function for irreducible supercuspidal representations of GL(2, Qp)
  • 11.9 The local L-function for irreducible principal series representations of GL(2, Qp)
  • 11.10 Local L-function for unitary special representations of GL(2, Qp)
  • 11.11 Proof of the local functional equation for principal series representations of GL(2, Qp)
  • 11.12 The local functional equation for the unitary special representations of GL(2, Qp)
  • 11.13 Proof of the local functional equation for the supercuspidal representations of GL(2, Qp)
  • 11.14 The local L-function for irreducible principal series representations of GL(2, R)
  • 11.15 Proof of the local functional equation for principal series representations of GL(2, R)
  • 11.16 The local L-function for irreducible discrete series representations of GL(2, R)
  • Exercises for Chapter 11
  • Solutions to Selected Exercises
  • References
  • Symbols Index
  • Index.

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