Chapter
2 Automorphic representations and L-functions for GL (1,A subscript Q)
2.1 Automorphic forms for GL (1,A subscript Q)
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,A subscript Q)
2.6 Hecke operators for GL (1,A subscript Q)
2.7 The Rankin-Selberg method
2.8 The p-adic Mellin transform
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 infinite of holomorphic modular forms
3.7 Fourier-Whittaker expansions of Maass forms
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
4 Automorphic forms for GL (2,A subscript Q)
4.1 Iwasawa and Cartan decompositions for GL (2,R)
4.2 Iwasawa and Cartan decompositions for GL (2,Q subscript p)
4.3 The adele group GL (2,A subscript Q)
4.4 The action of GL (2,Q) on GL (2,A subscript Q)
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,A subscript Q)
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,A subscript Q)
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
5 Automorphic representations for GL (2,A subscript Q)
5.1 Adelic automorphic representations for GL (2,A subscript Q)
5.2 Explicit realization of actions defining a (g, K subscript infinite)-module
5.3 Explicit realization of the action of GL (2, A subscript finite )
5.4 Examples of cuspidal automorphic representations
5.5 Admissible (g, K subscript infinite) × GL(2, A subscript finite)-modules
6 Theory of admissible representations of GL (2,Q subscript p)
6.0 Short roadmap to chapter 6
6.1 Admissible representations of GL (2,Q subscript p)
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→Whittaker functions
6.8 The Kirillov model of the principal series representation
6.9 Haar measure on GL (2,Q subscript p)
6.10 The special representations
6.12 Induced representations and parabolic induction
6.13 The supercuspidal representations of GL (2,Q subscript p)
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,Q subscript p)
7 Theory of admissible (g, K subscript infinite) modules for GL (2,R)
7.1 Admissible (g, K subscript infinite)-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, K subscript infinite)-modules
8 The contragredient representation for GL (2)
8.1 The contragredient representation for GL (2,Q subscript p)
8.2 The contragredient representation of a principal series representation of GL (2,Q subscript p)
8.3 Contragredient of a special representation of GL (2,Q subscript p)
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,A subscript Q)
8.8 Integration on GL (2,A subscript Q)
8.9 The contragredient representation of a cuspidal automorphic representation of GL (2,A subscript Q)
8.10 Growth of matrix coefficients
8.11 Asymptotics of matrix coefficients of (g, K subscript infinite)-modules
8.12 Matrix coefficients of GL (2,Q subscript p) via the Jacquet module
9 Unitary representations of GL (2)
9.1 Unitary representations of GL (2,Q subscript p)
9.2 Unitary principal series representations of GL (2,Q subscript p)
9.3 Unitary and irreducible special or supercuspidal representations of GL (2,Q subscript p)
9.4 Unitary (g, K subscript infinite)-modules
9.5 Unitary (g, K subscript infinite) × GL(2,A subscript finite)-modules
10 Tensor products of local representations
10.2 Tensor product of (g, K subscript infinite)-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,Q subscript p)
10.7 Initial decomposition of admissible (g, K subscript infinite) × GL(2,A subscript finite)-modules
10.8 The tensor product theorem
10.9 The Ramanujan and Selberg conjectures for GL (2,A subscript Q)
11 The Godement-Jacquet L-function for GL (2,A subscript Q)
11.2 The Poisson summation formula for GL (2,A subscript Q)
11.4 The global zeta integral for GL (2,A subscript Q)
11.5 Factorization of the global zeta integral
11.6 The local functional equation
11.7 The local L-function for GL (2,Q subscript p) (unramified case)
11.8 The local L-function for irreducible supercuspidal representations of GL (2,Q subscript p)
11.9 The local L-function for irreducible principal series representations of GL (2,Q subscript p)
11.10 Local L-function for unitary special representations of GL (2,Q subscript p)
11.11 Proof of the local functional equation for principal series representations of GL (2,Q subscript p)
11.12 The local functional equation for the unitary special representations of GL (2,Q subscript p)
11.13 Proof of the local functional equation for the supercuspidal representations of GL (2,Q subscript p)
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)
Solutions to Selected Exercises
Automorphic Representations and L-Functions for the General Linear Group: Volume 1
( Cambridge Studies in Advanced Mathematics )
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