p -adic Differential Equations

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ISBN: 9780521768795 | 0521768799
Cover: Hardcover
Copyright: 7/26/2010

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Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.

Preface	p. xiii
Introductory remarks	p. 1
Why p-adic differential equations?	p. 1
Zeta functions of varieties	p. 3
Zeta functions and p-adic differential equations	p. 5
A word of caution	p. 7
Notes	p. 8
Exercises	p. 9
Tools of p-adic Analysis	p. 11
Norms on algebraic structures	p. 13
Norms on abelian groups	p. 13
Valuations and nonarchimedean norms	p. 16
Norms on modules	p. 17
Examples of nonarchimedean norms	p. 25
Spherical completeness	p. 28
Notes	p. 31
Exercises	p. 33
Newton polygons	p. 35
Introduction to Newton polygons	p. 35
Slope factorizations and a master factorization theorem	p. 38
Applications to nonarchimedean field theory	p. 41
Notes	p. 42
Exercises	p. 43
Ramification theory	p. 45
Defect	p. 46
Unramified extensions	p. 47
Tamely ramified extensions	p. 49
The case of local fields	p. 52
Notes	p. 53
Exercises	p. 54
Matrix analysis	p. 55
Singular values and eigenvalues (archimedean case)	p. 56
Perturbations (archimedean case)	p. 60
Singular values and eigenvalues (nonarchimedean case)	p. 62
Perturbations (nonarchimedean case)	p. 68
Horn's inequalities	p. 71
Notes	p. 72
Exercises	p. 74
Differential Algebra	p. 75
Formalism of differential algebra	p. 77
Differential rings and differential modules	p. 77
Differential modules and differential systems	p. 80
Operations on differential modules	p. 81
Cyclic vectors	p. 84
Differential polynomials	p. 85
Differential equations	p. 87
Cyclic vectors: a mixed blessing	p. 87
Taylor series	p. 90
Notes	p. 91
Exercises	p. 91
Metric properties of differential modules	p. 93
Spectral radii of bounded endomorphisms	p. 93
Spectral radii of differential operators	p. 95
A coordinate-free approach	p. 102
Newton polygons for twisted polynomials	p. 104
Twisied polynomials and spectral radii	p. 105
The visible decomposition theorem	p. 107
Matrices and the visible spectrum	p. 109
A refined visible decomposition theorem	p. 112
Changing the constant field	p. 114
Notes	p. 116
Exercises	p. 117
Regular singularities	p. 118
Irregularity	p. 119
Exponents in the complex analytic setting	p. 120
Formal solutions of regular differential equations	p. 123
Index and irregularity	p. 126
The Turrittin-Levelt-Hukuhara decomposition theorem	p. 127
Notes	p. 129
Exercises	p. 130
p-adic Differential Equations on Discs and Annuli	p. 133
Rings of functions on discs and annuli	p. 135
Power series on closed discs and annuli	p. 136
Gauss norms and Newton polygons	p. 138
Factorization results	p. 140
Open discs and annuli	p. 143
Analytic elements	p. 144
More approximation arguments	p. 147
Notes	p. 149
Exercises	p. 150
Radius and generic radius of convergence	p. 151
Differential modules have no torsion	p. 152
Antidifferentiation	p. 153
Radius of convergence on a disc	p. 154
Generic radius of convergence	p. 155
Some examples in rank 1	p. 157
Transfer theorems	p. 158
Geometric interpretation	p. 160
Subsidiary radii	p. 162
Another example in rank 1	p. 162
Comparison with the coordinate-free definition	p. 164
Note	p. 165
Exercises	p. 166
Frobenius pullback and pushforward	p. 168
Why Frobenius descent?	p. 168
pth powers and roots	p. 169
Frobenius pullback and pushforward operations	p. 170
Frobenius antecedents	p. 172
Frobenius descendants and subsidiary radii	p. 174
Decomposition by spectral radius	p. 176
Integrality of the generic radius	p. 180
Off-center Frobenius antecedents and descendants	p. 181
Notes	p. 182
Exercises	p. 183
Variation of generic and subsidiary radii	p. 184
Harmonicity of the valuation function	p. 185
Variation of Newton polygons	p. 186
Variation of subsidiary radii: statements	p. 189
Convexity for the generic radius	p. 190
Measuring small radii	p. 191
Larger radii	p. 193
Monotonicity	p. 195
Radius versus generic radius	p. 197
Subsidiary radii as radii of optimal convergence	p. 198
Notes	p. 199
Exercises	p. 200
Decomposition by subsidiary radii	p. 201
Metrical detection of units	p. 202
Decomposition over a closed disc	p. 203
Decomposition over a closed annulus	p. 207
Decomposition over an open disc or annulus	p. 209
Partial decomposition over a closed disc or annulus	p. 210
Modules solvable at a boundary	p. 211
Solvable modules of rank 1	p. 212
Clean modules	p. 214
Notes	p. 216
Exercises	p. 216
p-adic exponents	p. 218
p-adic Liouville numbers	p. 218
p-adic regular singularities	p. 221
The Robba condition	p. 222
Abstract p-adic exponents	p. 223
Exponents for annuli	p. 225
The p-adic Fuchs theorem for annuli	p. 231
Transfer to a regular singularity	p. 234
Notes	p. 237
Exercises	p. 238
Difference Algebra and Frobenius Modules	p. 241
Formalism of difference algebra	p. 243
Difference algebra	p. 243
Twisted polynomials	p. 246
Difference-closed fields	p. 247
Difference algebra over a complete field	p. 248
Hodge and Newton polygons	p. 254
The Dieudonné-Manin classification theorem	p. 256
Notes	p. 258
Exercises	p. 260
Frobenius modules	p. 262
A multitude of rings	p. 262
Frobenius lifts	p. 264
Generic versus special Frobenius lifts	p. 266
A reverse filtration	p. 269
Notes	p. 271
Exercises	p. 272
Frobenius modules over the Robba ring	p. 273
Frobenius modules on open discs	p. 273
More on the Robba ring	p. 275
Pure difference modules	p. 277
The slope filtration theorem	p. 279
Proof of the slope filtration theorem	p. 281
Notes	p. 284
Exercises	p. 286
Frobenius Structures	p. 289
Frobenius structures on differential modules	p. 291
Frobenius structures	p. 291
Frobenius structures and the generic radius of convergence	p. 294
Independence from the Frobenius lift	p. 296
Slope filtrations and differential structures	p. 298
Extension of Frobenius structures	p. 298
Notes	p. 299
Exercises	p. 300
Effective convergence bounds	p. 301
A first bound	p. 301
Effective bounds for solvable modules	p. 302
Better bounds using Frobenius structures	p. 306
Logarithmic growth	p. 308
Nonzero exponents	p. 310
Notes	p. 310
Exercises	p. 311
Galois representations and differential modules	p. 313
Representation and differential modules	p. 314
Finite representations and overconvergent differential modules	p. 316
The unit-root p-adic local monodromy theorem	p. 318
Ramification and differential slopes	p. 321
Notes	p. 323
Exercises	p. 325
The p-adic local monodromy theorem	p. 326
Statement of the theorem	p. 326
An example	p. 328
Descent of sections	p. 329
Local duality	p. 332
When the residue field is imperfect	p. 333
Notes	p. 335
Exercises	p. 337
The p-adic local monodromy theorem: proof	p. 338
Running hypotheses	p. 338
Modules of differential slope 0	p. 339
Modules of rank 1	p. 341
Modules of rank prime to p	p. 342
The general case	p. 343
Notes	p. 343
Exercises	p. 344
Areas of Application	p. 345
Picard-Fuchs modules	p. 347
Origin of Picard-Fuchs modules	p. 347
Frobenius structures on Picard-Fuchs modules	p. 348
Relationship to zeta functions	p. 349
Notes	p. 350
Rigid cohomology	p. 352
Isocrystals on the affine line	p. 352
Crystalline and rigid cohomology	p. 353
Machine computations	p. 354
Notes	p. 355
p-adic Hodge theory	p. 357
A few rings	p. 357
(¿, ¿)-modules	p. 359
Galois cohomology	p. 361
Differential equations from (¿, ¿)-modules	p. 362
Beyond Galois representations	p. 363
Notes	p. 364
References	p. 365
Notation	p. 374
Index	p. 376
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p -adic Differential Equations

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