| Dorian Goldfeld is a Professor in the Department of Mathematics at Columbia University
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| Introduction 1 Discrete group actions 1.1 Action of a group on a topological space 1.2 Iwasawa decomposition 1.3 Siegel sets 1.4 Haar measure 1.5 Invariant measure on coset spaces 1.6 Volume of SL(n, Z)\SL(n, R)/SO(n, R) 2 Invariant differential operators 2.1 Lie algebras 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) 3.1 Eisenstein series 3.2 Hyperbolic Fourier expansion of Eisenstein series 3.3 Maass forms 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 3.10 Hecke operators 3.11 Hermite and Smith normal forms 3.12 Hecke operators for L2(SL(2, Z))\h2 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) 4.2 Integral operators 4.3 The endomorphism 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) 4.6 A weak Weyl law 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) 5.1 Maass forms 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 In 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 9 The Godement-Jacquet L-function 10 Langlands Eisenstein series 11 Poincare series and Kloosterman sums 12 Rankin-Selberg convolutions 13 Langlands conjectures List of symbols Appebdux The GL(n)Pack Manual by Kevin A. Broughan References Index |
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