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Introduction: Motivations from Geometry 1
Introduction 1
Analogies Between Arithmetic and Geometry 2
Zeta Function for Curves 3
The Riemann-Roch Theorem 5
The Castelnuovo-Severi Inequality 7
Zeta Functions for Number Fields 10
Weil's Explicit Sum Formula 14
Gamma and Beta Measures 19
Quotients Z[subscript p]/Z*[subscript p] and P[superscript 1] (Q[subscript p])/Z*[subscript p] x Z[subscript p] 20
Z[subscript p]/Z*[subscript p] 20
P[superscript 1] (Q[subscript p])/Z*[subscript p] x Z[subscript p] 21
[gamma]-Measure on Q[subscript p] 24
p-[gamma]-Integral 24
[eta]-[gamma]-Integral 25
[gamma]-Measure on Q[subscript p] 25
[beta]-Measure on P[superscript 1](Q[subscript p]) 25
The Projective Space P[superscript 1](Q[subscript p]) 25
[beta]-Integral 27
[beta]-Measure on P[superscript 1](Q[subscript p]) 28
Remarks on the [gamma] and [beta]-Measure 29
[beta]-Measure Gives [gamma]-Measure 29
[gamma]-Measure Gives [beta]-Measure 30
Special Case [alpha] = [beta] = 1 31
Markov Chains 33
Markov Chain on Trees 34
Probability Measures on [partial differential]X 34
Hilbert Spaces 35
Symmetric p-Adic [beta]-Chain 36
Non-Symmetric p-Adic [beta]-Chain 37
p-Adic [gamma]-Chain 40
Markov Chain on Non-Trees 41
Non-Tree 41
Harmonic Functions 42
Martin Kernel 44
Real Beta Chain and q-Interpolation 47
Real [beta]-Chain 47
Probability Measure 48
Green Kernel and Martin Kernel 49
Boundary 50
Harmonic Measure 51
q-Interpolation 52
Complex [beta]-Chain 52
q-Zeta Functions 53
q-[beta]-Chain 55
q-Binomial Theorem 56
Probability Measure 57
Green Kernel and Martin Kernel 58
Boundary 59
Harmonic Measure 60
Ladder Structure 63
Ladder for Trees 67
Ladder for the q-[beta]-Chain 70
Finite Layer: The q-Hahn Basis 70
Boundary: The q-Jacobi Basis 74
Ladder for q-[gamma]-Chain 77
Finite Layer: The Finite q-Laguerre Basis 77
Boundary: The q-Laguerre Basis 78
Ladder for [eta]-[beta]-Chain 81
Finite Layer: The [eta]-Hahn Basis 81
Boundary: The [eta]-Jacobi Basis 82
The [eta]-Laguerre Basis 87
Real Units 89
q-Interpolation of Local Tate Thesis 95
Mellin Transforms 98
Classical Cases 98
q-Interpolations 103
Fourier-Bessel Transforms 106
Fourier Transform on H[superscript beta subscript p] 106
q-Fourier Transform 107
Convolutions 109
The Basic Basis 111
Pure Basis and Semi-Group 117
The Pure Basis 118
The Semi-Group G[superscript beta] 121
Global Tate-Iwasawa Theory 125
Higher Dimensional Theory 131
Higher Dimensional Cases 132
q-[beta]-Chain 132
The p-Adic Limit of the q-[beta]-Chain 136
The Real Limit of the q-[beta]-Chain 136
Representations of GL[subscript d](Z[subscript p]), p [greater than or equal] [eta], on Rank-1 Symmetric Spaces 137
Real Grassmann Manifold 143
Measures on the Higher Rank Spaces 143
Grassmann Manifolds 143
Measures on O[subscript m], X[superscript d subscript m] and V[superscript d subscript m] 145
Measures on [Omega subscript m] 148
Explicit Calculations 148
Measures 148
Metrics 151
Higher Rank Orthogonal Polynomials 153
Real Case 153
General Case 155
p-Adic Grassmann Manifold 157
Representation of GL[subscript d](Z[subscript p]) 157
Measures on GL[subscript d](Z[subscript p]), V[superscript d subscript m] and X[superscript d subscript m] 157
Unitary Representations of GL[subscript d](Z[subscript p]) and G[subscript N superscript d] 160
Harmonic Measure 164
Notations 164
Harmonic Measure on [Omega superscript d subscript m] 165
Basis for the Hecke Algebra 169
q-Grassmann Manifold 173
q-Selberg Measures 173
The p-Adic Limit of the q-Selberg Measures 174
The Real Limit of the q-Selberg Measures 175
Higher Rank q-Jacobi Basis 176
Quantum Groups 178
Higher Rank Quantum Groups 178
The Universal Enveloping Algebra 181
Quantum Grassmann Manifolds 182
Quantum Group U[subscript q](su(1,1)) and the q-Hahn Basis 185
The Quantum Universal Enveloping Algebra U[subscript q](su(1,1)) 185
Deformation of U(sl(2, C)) 185
The [beta]-Highest Weight Representation 187
Limits of the Subalgebras U[superscript plus or minus subscript q] 189
The Hopf Algebra Structure 190
Tensor Product Representation 193
The Universal R-Matrix 196
Problems and Questions 199
Orthogonal Polynomials 203
Bibliography 209
Index 215
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Add Arithmetical Investigations, In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to , Arithmetical Investigations to the inventory that you are selling on WonderClubX
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Add Arithmetical Investigations, In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to , Arithmetical Investigations to your collection on WonderClub |