p-adic Differential Equations

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
0.1 Why p-adic differential equations?......Page 21
0.2 Zeta functions of varieties......Page 23
0.3 Zeta functions and p-adic differential equations......Page 25
0.4 A word of caution......Page 27
Notes......Page 28
Exercises......Page 29
Part I Tools of p-adic Analysis......Page 31
1.1 Norms on abelian groups......Page 33
1.2 Valuations and nonarchimedean norms......Page 36
1.3 Norms on modules......Page 37
1.4 Examples of nonarchimedean norms......Page 45
1.5 Spherical completeness......Page 48
Notes......Page 51
Exercises......Page 53
2.1 Introduction to Newton polygons......Page 55
2.2 Slope factorizations and a master factorization theorem......Page 58
2.3 Applications to nonarchimedean field theory......Page 61
Notes......Page 62
Exercises......Page 63
3 Ramification theory......Page 65
3.1 Defect......Page 66
3.2 Unramified extensions......Page 67
3.3 Tamely ramified extensions......Page 69
3.4 The case of local fields......Page 72
Notes......Page 73
Exercises......Page 74
4 Matrix analysis......Page 75
4.1 Singular values and eigenvalues (archimedean case)......Page 76
4.2 Perturbations (archimedean case)......Page 80
4.3 Singular values and eigenvalues (nonarchimedean case)......Page 82
4.4 Perturbations (nonarchimedean case)......Page 88
4.5 Horn's inequalities......Page 91
Notes......Page 92
Exercises......Page 94
Part II Differential Algebra......Page 95
5.1 Differential rings and differential modules......Page 97
5.2 Differential modules and differential systems......Page 100
5.3 Operations on differential modules......Page 101
5.4 Cyclic vectors......Page 104
5.5 Differential polynomials......Page 105
5.7 Cyclic vectors: a mixed blessing......Page 107
5.8 Taylor series......Page 110
Exercises......Page 111
6.1 Spectral radii of bounded endomorphisms......Page 113
6.2 Spectral radii of differential operators......Page 115
6.3 A coordinate-free approach......Page 122
6.4 Newton polygons for twisted polynomials......Page 124
6.5 Twisted polynomials and spectral radii......Page 125
6.6 The visible decomposition theorem......Page 127
6.7 Matrices and the visible spectrum......Page 129
6.8 A refined visible decomposition theorem......Page 132
6.9 Changing the constant field......Page 134
Notes......Page 136
Exercises......Page 137
7 Regular singularities......Page 138
7.1 Irregularity......Page 139
7.2 Exponents in the complex analytic setting......Page 140
7.3 Formal solutions of regular differential equations......Page 143
7.4 Index and irregularity......Page 146
7.5 The Turrittin–Levelt–Hukuhara decomposition theorem......Page 147
Notes......Page 149
Exercises......Page 150
Part III p-adic Differential Equations on Discs and Annuli......Page 153
8 Rings of functions on discs and annuli......Page 155
8.1 Power series on closed discs and annuli......Page 156
8.2 Gauss norms and Newton polygons......Page 158
8.3 Factorization results......Page 160
8.4 Open discs and annuli......Page 163
8.5 Analytic elements......Page 164
8.6 More approximation arguments......Page 167
Notes......Page 169
Exercises......Page 170
9 Radius and generic radius of convergence......Page 171
9.1 Differential modules have no torsion......Page 172
9.2 Antidifferentiation......Page 173
9.3 Radius of convergence on a disc......Page 174
9.4 Generic radius of convergence......Page 175
9.5 Some examples in rank 1......Page 177
9.6 Transfer theorems......Page 178
9.7 Geometric interpretation......Page 180
9.9 Another example in rank 1......Page 182
9.10 Comparison with the coordinate-free definition......Page 184
Notes......Page 185
Exercises......Page 186
10.1 Why Frobenius descent?......Page 188
10.2 pth powers and roots......Page 189
10.3 Frobenius pullback and pushforward operations......Page 190
10.4 Frobenius antecedents......Page 192
10.5 Frobenius descendants and subsidiary radii......Page 194
10.6 Decomposition by spectral radius......Page 196
10.7 Integrality of the generic radius......Page 200
10.8 Off-center Frobenius antecedents and descendants......Page 201
Notes......Page 202
Exercises......Page 203
11 Variation of generic and subsidiary radii......Page 204
11.1 Harmonicity of the valuation function......Page 205
11.2 Variation of Newton polygons......Page 206
11.3 Variation of subsidiary radii: statements......Page 209
11.4 Convexity for the generic radius......Page 210
11.5 Measuring small radii......Page 211
11.6 Larger radii......Page 213
11.7 Monotonicity......Page 215
11.8 Radius versus generic radius......Page 217
11.9 Subsidiary radii as radii of optimal convergence......Page 218
Notes......Page 219
Exercises......Page 220
12 Decomposition by subsidiary radii......Page 221
12.1 Metrical detection of units......Page 222
12.2 Decomposition over a closed disc......Page 223
12.3 Decomposition over a closed annulus......Page 227
12.4 Decomposition over an open disc or annulus......Page 229
12.5 Partial decomposition over a closed disc or annulus......Page 230
12.6 Modules solvable at a boundary......Page 231
12.7 Solvable modules of rank 1......Page 232
12.8 Clean modules......Page 234
Exercises......Page 236
13.1 p-adic Liouville numbers......Page 238
13.2 p-adic regular singularities......Page 241
13.3 The Robba condition......Page 242
13.4 Abstract p-adic exponents......Page 243
13.5 Exponents for annuli......Page 245
13.6 The p-adic Fuchs theorem for annuli......Page 251
13.7 Transfer to a regular singularity......Page 254
Notes......Page 257
Exercises......Page 258
Part IV Difference Algebra and Frobenius Modules......Page 261
14.1 Difference algebra......Page 263
14.2 Twisted polynomials......Page 266
14.3 Difference-closed fields......Page 267
14.4 Difference algebra over a complete field......Page 268
14.5 Hodge and Newton polygons......Page 274
14.6 The Dieudonné–Manin classification theorem......Page 276
Notes......Page 278
Exercises......Page 280
15.1 A multitude of rings......Page 282
15.2 Frobenius lifts......Page 284
15.3 Generic versus special Frobenius lifts......Page 286
15.4 A reverse filtration......Page 289
Notes......Page 291
Exercises......Page 292
16.1 Frobenius modules on open discs......Page 293
16.2 More on the Robba ring......Page 295
16.3 Pure difference modules......Page 297
16.4 The slope filtration theorem......Page 299
16.5 Proof of the slope filtration theorem......Page 301
Notes......Page 304
Exercises......Page 306
Part V Frobenius Structures......Page 309
17.1 Frobenius structures......Page 311
17.2 Frobenius structures and the generic radius of convergence......Page 314
17.3 Independence from the Frobenius lift......Page 316
17.5 Extension of Frobenius structures......Page 318
Notes......Page 319
Exercises......Page 320
18.1 A first bound......Page 321
18.2 Effective bounds for solvable modules......Page 322
18.3 Better bounds using Frobenius structures......Page 326
18.4 Logarithmic growth......Page 328
Notes......Page 330
Exercises......Page 331
19 Galois representations and differential modules......Page 333
19.1 Representations and differential modules......Page 334
19.2 Finite representations and overconvergent differential modules......Page 336
19.3 The unit-root p-adic local monodromy theorem......Page 338
19.4 Ramification and differential slopes......Page 341
Notes......Page 343
Exercises......Page 345
20.1 Statement of the theorem......Page 346
20.2 An example......Page 348
20.3 Descent of sections......Page 349
20.4 Local duality......Page 352
20.5 When the residue field is imperfect......Page 353
Notes......Page 355
Exercises......Page 357
21.1 Running hypotheses......Page 358
21.2 Modules of differential slope 0......Page 359
21.3 Modules of rank 1......Page 361
21.4 Modules of rank prime to p......Page 362
Notes......Page 363
Exercises......Page 364
Part VI Areas of Application......Page 365
22.1 Origin of Picard–Fuchs modules......Page 367
22.2 Frobenius structures on Picard–Fuchs modules......Page 368
22.3 Relationship to zeta functions......Page 369
Notes......Page 370
23.1 Isocrystals on the affine line......Page 372
23.2 Crystalline and rigid cohomology......Page 373
23.3 Machine computations......Page 374
Notes......Page 375
24.1 A few rings......Page 377
24.2 (phi,gamma)-modules......Page 379
24.3 Galois cohomology......Page 381
24.4 Differential equations from (phi, gamma)-modules......Page 382
24.5 Beyond Galois representations......Page 383
Notes......Page 384
References......Page 385
Notation......Page 394
Index......Page 396