Chapter
2 Invariant differential operators
2.2 Universal enveloping algebra of gl(n, R)
2.3 The center of the universal enveloping algebra of gl(n, R)
2.4 Eigenfunctions of invariant differential operators
3 Automorphic forms and L–functions for SL(2, Z)
3.2 Hyperbolic Fourier expansion of Eisenstein series
3.4 Whittaker expansions and multiplicity one for GL(2, R)
3.5 Fourier–Whittaker expansions on GL(2, R)
3.6 Ramanujan–Petersson conjecture
3.7 Selberg eigenvalue conjecture
3.8 Finite dimensionality of the eigenspaces
3.9 Even and odd Maass forms
3.11 Hermite and Smith normal forms
3.12 Hecke operators for…
3.13 L–functions associated to Maass forms
3.14 L-functions associated to Eisenstein series
3.15 Converse theorems for SL(2, Z)
3.16 The Selberg spectral decomposition
4 Existence of Maass forms
4.1 The infinitude of odd Maass forms for SL(2, Z)
4.4 How to interpret: an explicit operator with purely cuspidal image
4.5 There exist infinitely many even cusp forms for SL(2, Z)
4.7 Interpretation via wave equation and the role of finite propagation speed
4.8 Interpretation via wave equation: higher rank case
5 Maass forms and Whittaker functions for SL(n, Z)
5.2 Whittaker functions associated to Maass forms
5.3 Fourier expansions on SL(n, Z)\h
5.4 Whittaker functions for SL(n, R)
5.5 Jacquet’s Whittaker function
5.6 The exterior power of a vector space
5.7 Construction of the I function using wedge products
5.8 Convergence of Jacquet’s Whittaker function
5.9 Functional equations of Jacquet’s Whittaker function
5.10 Degenerate Whittaker functions
6 Automorphic forms and L-functions for SL(3, Z)
6.1 Whittaker functions and multiplicity one for SL(3, Z)
6.2 Maass forms for SL(3, Z)
6.3 The dual and symmetric Maass forms
6.4 Hecke operators for SL(3, Z)
6.5 The Godement–Jacquet L-function
6.6 Bump’s double Dirichlet series
7 The Gelbart–Jacquet lift
7.1 Converse theorem for SL(3, Z)
7.2 Rankin–Selberg convolution for GL(2)
7.3 Statement and proof of the Gelbart–Jacquet lift
7.4 Rankin–Selberg convolution for GL(3)
8 Bounds for L-functions and Siegel zeros
8.2 Convexity bounds for the Selberg class
8.3 Approximate functional equations
8.4 Siegel zeros in the Selberg class
8.6 The Siegel zero lemma
8.7 Non-existence of Siegel zeros for Gelbart–Jacquet lifts
8.8 Non-existence of Siegel zeros on GL(n)
9 The Godement–Jacquet L-function
9.1 Maass forms for SL(n, Z)
9.2 The dual and symmetric Maass forms
9.3 Hecke operators for SL(n, Z)
9.4 The Godement–Jacquet L-function
10 Langlands Eisenstein series
10.2 Langlands decomposition of parabolic subgroups
10.3 Bruhat decomposition
10.4 Minimal, maximal, and general parabolic Eisenstein series
10.5 Eisenstein series twisted by Maass forms
10.6 Fourier expansion of minimal parabolic Eisenstein series
10.7 Meromorphic continuation and functional equation of maximal parabolic Eisenstein series
10.8 The L-function associated to a minimal parabolic Eisenstein series
10.9 Fourier coefficients of Eisenstein series twisted by Maass forms
10.11 The constant term of SL(3, Z) Eisenstein series twisted by SL(2, Z)-Maass forms
10.12 An application of the theory of Eisenstein series to the non-vanishing of L-functions on the line…
10.13 Langlands spectral decomposition for SL(3, Z)\h
11 Poincaré series and Kloosterman sums
11.1 Poincaré series for SL(n, Z)
11.3 Plücker coordinates and the explicit evaluation of Kloosterman sums
11.4 Properties of Kloosterman sums
11.5 Fourier expansion of Poincaré series
11.6 Kuznetsov’s trace formula for SL(n, Z)
12 Rankin–Selberg convolutions
12.1 The GL(n) × GL(n) convolution
12.2 The GL(n) × GL(n + 1) convolution
12.3 The GL(n) × GL(n´) convolution with n < n´
12.4 Generalized Ramanujan conjecture
12.5 The Luo–Rudnick–Sarnak bound for the generalized Ramanujan conjecture
12.6 Strong multiplicity one theorem
13.2 Langlands functoriality
Appendix The GL(n)pack Manual
A.1.3 Assistance for users new to computers or Mathematica
A.1.4 Mathematica functions
A.1.5 The data type CRE (Canonical Rational Expression)
A.1.6 The algorithms in this package
A.2 Functions for GL(n)pack
A.3 Function descriptions and examples
Automorphic Forms and L-Functions for the Group GL(n,R)
( Cambridge Studies in Advanced Mathematics )
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