Arithmetical Investigations Book

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