Arithmetic Geometry And Number Theory

Contents......Page 10
Foreword......Page 6
Preface......Page 8
1 Introduction......Page 11
2 Basic Properties of Local y-Factors......Page 15
2.2 Stability......Page 16
3.1 The case of GLn(F)......Page 17
3.2 A conjectural LCT......Page 19
3.3 The case of SO2n+1(F)......Page 22
4 Poles of Local y-Factors......Page 25
4.1 The case of G = SO2n+1......Page 27
4.2 Other classical groups......Page 30
1 WP Metrics and TZ Metrics......Page 39
2 Line Bundles over Moduli Spaces......Page 42
3 Fundamental Relations on MgN' Algebraic Story......Page 43
4 Fundamental Relation on MgN- Arithmetic Story......Page 44
5 Deligne Tuple in General......Page 48
6 Degeneration of TZ Metrics: Analytic Story......Page 49
References......Page 53
2 Complex Vector Bundles......Page 57
3 Fundamental Groups of p-Adic Curves......Page 58
4 Finite Vector Bundles......Page 59
5 A Bigger Category of Vector Bundles......Page 60
6 Parallel Transport on Bundles in Bxcp......Page 62
7 Working Outside a Divisor on Xcp......Page 63
8 Properties of Parallel Transport......Page 64
10 A Simpler Description of Bxcp D......Page 65
11 Strongly Semistable Reduction......Page 66
12 How Big are our Categories of Bundles?......Page 70
14 Mumford Curves......Page 71
References......Page 72
1 Introduction......Page 75
2.1 Complex Theory......Page 77
2.2 p-Adic Theory......Page 97
References......Page 109
1 Introduction......Page 113
2 Carlitz Theory......Page 115
3 Anderson-Thakur Theory......Page 120
4 t-Motives......Page 123
5 Algebraic Independence of the Special Zeta Values......Page 127
References......Page 131
Automorphic Forms & Eisenstein Series and Spectral Decompositions......Page 133
1.1 Langlands Decomposition......Page 137
1.2 Reduction Theory: Siegel Sets......Page 144
1.3 Moderate Growth and Rapidly Decreasing......Page 146
1.4 Automorphic Forms......Page 149
2.1 Moderate Growth and Rapid Decreasing......Page 151
2.3 3-Finiteness......Page 152
2.4 Philosophy of Cusp Forms......Page 155
2.5 L2-Automorphic Forms......Page 156
3.1 Equivalence Classes of Automorphic Representations......Page 157
3.2 Eisenstein Series and Intertwining Operators......Page 159
3.3 Convergence......Page 160
4 Constant Terms of Eisenstein Series......Page 162
5 Fundamental Properties of Eisenstein Series......Page 166
6.2 Fourier Transforms......Page 168
6.3 Paley-Wiener on p......Page 169
7 Pseudo-Eisenstein Series......Page 171
8.1 Inner Product Formula for P-ESes......Page 173
8.2 Decomposition of L2-Spaces According to Cuspidal Data......Page 175
8.3 Constant Terms of P-SEes......Page 176
9.1 Main Result......Page 178
9.2 Langlands Operators......Page 179
9.3 Key Bridge......Page 181
10.1 Pseudo-Eisenstein Series and Residual Process......Page 183
10.2 What do we have?......Page 185
11.1 Functional Analysis......Page 187
11.2 Main Theorem: Rough Version......Page 188
11.3 Main Theorem: Refined Version......Page 189
11.4 How to Prove?......Page 191
12.1 Relative Theory......Page 194
12.2 Discrete Spectrum......Page 195
13.1 Bridge......Page 196
13.2 Basic Facts......Page 197
14 Spectrum Decomposition: Levi Interpretation......Page 200
14.1 Spaces Ae......Page 0
15.1 In Terms of Levi......Page 201
15.2 Relation with Residual Approach: The Proof......Page 203
16.1 Positive Chamber and Positive Cone......Page 205
16.2 Arthur's Analytic Truncation......Page 207
17 Meromorphic Continuation of Eisenstein Series: Deduction......Page 208
17.1 Preparation......Page 209
17.2 Deduction to Relative Rank 1......Page 210
18.1 Working Site......Page 211
18.2 Constant Terms......Page 212
18.4 Functional Equation for Truncated Eisenstein Series......Page 213
18.5 Application of Resolvent Theory......Page 214
18.6 Injectivity......Page 216
18.7 Meromorphic Continuation......Page 218
18.8 Functional Equation......Page 219
References......Page 220
Geometric Arithmetic: A Program......Page 221
A.2. Non-Abelian CFT for Function Fields over C......Page 231
A.3. Towards Non-Abelian CFT for Global Fields......Page 253
B.l. Non-Abelian Zeta Functions for Curves......Page 261
Appendix to B.l: Weierstrass Groups......Page 272
B.2. New Non-Abelian Zeta Functions for Number Fields......Page 279
B.3. Non-Abelian L-Functions for Number Fields......Page 302
B.4. Geometric and Analytic Truncations: A Bridge......Page 312
B.5. Rankin-Selberg Method......Page 329
C.l. The Riemann Hypothesis for Curves......Page 343
C.2. Geo-Ari Intersection: A Mathematics Model......Page 346
C.3. Towards a Geo-Ari Cohomology in Lower Dimensions......Page 360
C.4. Riemann Hypothesis in Rank Two......Page 369
References......Page 392
2. An Example......Page 402
3. Reciprocity Map......Page 403
5. Proof......Page 404
6. An Application to Inverse Galois Problem......Page 406
References......Page 409