p-adic differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Pr.
2010
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
125 |
| Subjects: | |
| Online Access: | Table of contents only Publisher description Contributor biographical information Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 380 S. graph. Darst. |
| ISBN: | 9780521768795 |
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MARC
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| 100 | 1 | |a Kedlaya, Kiran Sridhara |d 1974- |e Verfasser |0 (DE-588)141747633 |4 aut | |
| 245 | 1 | 0 | |a p-adic differential equations |c Kiran S. Kedlaya |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Pr. |c 2010 | |
| 300 | |a XVII, 380 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 125 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a p-adic analysis | |
| 650 | 4 | |a Differential equations | |
| 650 | 0 | 7 | |a p-adische Differentialgleichung |0 (DE-588)4398266-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a p-adische Differentialgleichung |0 (DE-588)4398266-9 |D s |
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| 830 | 0 | |a Cambridge studies in advanced mathematics |v 125 |w (DE-604)BV000003678 |9 125 | |
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| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy1006/2010004489-d.html |3 Publisher description | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy1006/2010004489-b.html |3 Contributor biographical information | |
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Record in the Search Index
| _version_ | 1804143156587921408 |
|---|---|
| adam_text | Contents
Preface
page
xiii
0
Introductory remarks
1
0.1
Why
p-aáic
differential equations?
1
0.2
Zeta
functions of varieties
3
0.3
Zeta
functions and p-adic differential equations
5
0.4
A word of caution
7
Notes
8
Exercises
9
Part I Tools of p-adic Analysis
11
1
Norms on algebraic structures
13
1.1
Norms on abelian groups
13
1.2
Valuations and nonarchimedean norms
16
1.3
Norms on modules
17
1.4
Examples of nonarchimedean norms
25
1.5
Spherical completeness
28
Notes
31
Exercises
33
2
Newton polygons
35
2.1
Introduction to Newton polygons
35
2.2
Slope factorizations and a master factorization theorem
38
2.3
Applications to nonarchimedean field theory
41
Notes
42
Exercises
43
3
Ramification theory
45
3.1
Defect
46
3.2
Unramified extensions
47
Contents
3.3
Tamely ramified extensions
49
3.4
The case of local fields
52
Notes
53
Exercises
54
4
Matrix analysis
55
4.1
Singular values and eigenvalues (archimedean case)
56
4.2
Perturbations (archimedean case)
60
4.3
Singular values and eigenvalues (nonarchimedean case)
62
4.4
Perturbations (nonarchimedean case)
68
4.5
Horn s inequalities
71
Notes
72
Exercises
74
Part II Differential Algebra
75
77
77
80
81
5
Formalism of differential algebra
5.1
Differential rings and differential modules
5.2
Differential modules and differential systems
5.3
Operations on differential modules
5.4
Cyclic vectors
5.5
Differential polynomials
5.6
Differential equations
5.7
Cyclic vectors: a mixed blessing
5.8
Taylor series
Notes
Exercises
6
Metric properties of differential modules
6.1
Spectral radii of bounded endomorphisms
6.2
Spectral radii of differential operators
6.3
A coordinate-free approach
6.4
Newton polygons for twisted polynomials
6.5
Twisted polynomials and spectral radii
6.6
The visible decomposition theorem
6.7
Matrices and the visible spectrum
6.8
A refined visible decomposition theorem
6.9
Changing the constant field
Notes
Exercises
7
Regular singularities
7.1
Irregularity
85
87
87
90
91
91
93
93
95
102
104
105
107
109
112
114
116
117
118
119
Contents
vii
7.2
bxponents in the complex analytic setting
120
7.3
Formal solutions of regular differential equations
123
7.4
Index and irregularity
126
7.5
The Turrittin-Levelt-Hukuhara decomposition theorem
127
Notes
129
Exercises
130
Part III p-adic Differential Equations on Discs and
Annuli
133
8
Rings of functions on discs and
annuii
135
8.1
Power series on closed discs and
annuii
136
8.2
Gauss norms and Newton polygons
138
8.3
Factorization results
140
8.4
Open discs and
annuii
143
8.5
Analytic elements
144
8.6
More approximation arguments
HI
Notes
149
Exercises
150
9
Radius and generic radius of convergence
151
9.1
Differential modules have no torsion
152
9.2
Antidifferentiation
153
9.3
Radius of convergence on a disc
154
9.4
Generic radius of convergence
155
9.5
Some examples in rank
1
157
9.6
Transfer theorems
158
9.7
Geometric interpretation
160
9.8
Subsidiary radii
162
9.9
Another example in rank
1
162
9.10
Comparison with the coordinate-free definition
164
Notes
165
Exercises
166
10
Frobenius pullback and pushforward
168
10.1
Why Frobenius descent?
168
10.2
púi
powers and roots
169
10.3
Frobenius pullback and pushforward operations
170
10.4
Frobenius antecedents
172
10.5
Frobenius descendants and subsidiary radii
174
10.6
Decomposition by spectral radius
176
10.7
Integrality of the generic radius
180
10.8
Off-center Frobenius antecedents and descendants
181
viii Contents
Notes
182
Exercises
183
11
Variation
of generic and subsidiary radii
184
11.1
Harmonicity
of the valuation function
185
11.2
Variation of Newton polygons
186
11.3
Variation of subsidiary radii: statements
189
11.4
Convexity for the generic radius
190
11.5
Measuring small radii
191
11.6
Larger radii
193
11.7
Monotonicity
195
11.8
Radius versus generic radius
197
11.9
Subsidiary radii as radii of optimal convergence
198
Notes
199
Exercises
200
12
Decomposition by subsidiary radii
201
12.1
Metrical detection of units
202
12.2
Decomposition over a closed disc
203
12.3
Decomposition over a closed annulus
207
12.4
Decomposition over an open disc or annulus
209
12.5
Partial decomposition over a closed disc or annulus
210
12.6
Modules solvable at a boundary
211
12.7
Solvable modules of rank
1
212
12.8
Clean modules
214
Notes
216
Exercises
216
13
p-adie exponents
218
13.1
p-adic Liouville numbers
218
13.2
p-adic regular singularities
221
13.3
The Robba condition
222
13.4
Abstract p-adic exponents
223
13.5
Exponents for
annuii
225
13.6
The p-adic
Fuchs
theorem for
annuii
231
13.7
Transfer to a regular singularity
234
Notes
237
Exercises
238
Part IV Difference Algebra and Frobenius Modules
241
14
Formalism of difference algebra
243
14.1
Difference algebra
243
Contents ix
246
247
248
254
256
258
260
262
262
264
266
269
271
272
273
273
275
277
279
281
284
286
Part V
Frobenius
Structures
289
17
Frobenius
structures on differential modules
291
17.1
Frobenius structures
291
17.2
Frobenius structures and the generic radius of
convergence
294
17.3
Independence from the Frobenius lift
296
17.4
Slope filtrations and differential structures
298
17.5
Extension of Frobenius structures
298
Notes
299
Exercises
300
18
Effective convergence bounds
301
18.1
A first bound
301
18.2
Effective bounds for solvable modules
302
18.3
Better bounds using Frobenius structures
306
18.4
Logarithmic growth
308
18.5
Nonzero exponents
310
14.2
Twisted polynomials
14.3
Difference-closed fields
14.4
Difference algebra over a complete field
14.5
Hodge and Newton polygons
14.6
The
Dieudonné-Manin
classification theorem
Notes
Exercises
15
Frobenius modules
15.1
A multitude of rings
15.2
Frobenius lifts
15.3
Generic versus special Frobenius lifts
15.4
A reverse filtration
Notes
Exercises
16
Frobenius modules over the Robba ring
16.1
Frobenius modules on open discs
16.2
More on the Robba ring
16.3
Pure difference modules
16.4
The slope filtration theorem
16.5
Proof of the slope filtration theorem
Notes
Exercises
Contents
Notes
310
Exercises
311
19
Galois representations and differential modules
313
19.1
Representations and differential modules
314
19.2
Finite representations and overconvergent differential
modules
316
19.3
The unit-root p-adic local monodromy theorem
318
19.4
Ramification and differential slopes
321
Notes
323
Exercises
325
20
The p-adic local monodromy theorem
326
20.1
Statement of the theorem
326
20.2
An example
328
20.3
Descent of sections
329
20.4
Local duality
332
20.5
When the residue field is imperfect
333
Notes
335
Exercises
337
21
The p-adic local monodromy theorem: proof
338
21.1
Running hypotheses
338
21.2
Modules of differential slope
0
339
21.3
Modules of rank
1
341
21.4
Modules of rank prime to
ρ
342
21.5
The general case
343
Notes
343
Exercises
344
Part VI Areas of Application
345
22
Picard-Fuchs modules
347
22.1
Origin of Picard-Fuchs modules
347
22.2
Frobenius structures on Picard-Fuchs modules
348
22.3
Relationship to
zeta
functions
349
Notes
350
23
Rigid cohomology
352
23.1
Isocrystals on the
affine line
352
23.2
Crystalline and rigid cohomology
353
23.3
Machine computations
354
Notes
355
Contents xi
24
p-adic Hodge theory
357
24.1
A few rings
357
24.2
(φ,
D-modules
359
24.3
Galois cohomology
361
24.4
Differential equations from
(φ,
П
-modules
362
24.5
Beyond Galois representations
363
Notes
364
References
365
Notation
374
Index
376
|
| any_adam_object | 1 |
| author | Kedlaya, Kiran Sridhara 1974- |
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| dewey-ones | 512 - Algebra |
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| dewey-search | 512.7/4 |
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| discipline | Mathematik |
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| id | DE-604.BV036573388 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:43:11Z |
| institution | BVB |
| isbn | 9780521768795 |
| language | English |
| lccn | 2010004489 |
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| owner_facet | DE-384 DE-11 DE-355 DE-BY-UBR DE-703 DE-188 |
| physical | XVII, 380 S. graph. Darst. |
| publishDate | 2010 |
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| publisher | Cambridge Univ. Pr. |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Kedlaya, Kiran Sridhara 1974- Verfasser (DE-588)141747633 aut p-adic differential equations Kiran S. Kedlaya 1. publ. Cambridge Cambridge Univ. Pr. 2010 XVII, 380 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 125 Includes bibliographical references and index p-adic analysis Differential equations p-adische Differentialgleichung (DE-588)4398266-9 gnd rswk-swf p-adische Differentialgleichung (DE-588)4398266-9 s DE-604 Cambridge studies in advanced mathematics 125 (DE-604)BV000003678 125 http://www.loc.gov/catdir/enhancements/fy1006/2010004489-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy1006/2010004489-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy1006/2010004489-b.html Contributor biographical information Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020494451&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kedlaya, Kiran Sridhara 1974- p-adic differential equations Cambridge studies in advanced mathematics p-adic analysis Differential equations p-adische Differentialgleichung (DE-588)4398266-9 gnd |
| subject_GND | (DE-588)4398266-9 |
| title | p-adic differential equations |
| title_auth | p-adic differential equations |
| title_exact_search | p-adic differential equations |
| title_full | p-adic differential equations Kiran S. Kedlaya |
| title_fullStr | p-adic differential equations Kiran S. Kedlaya |
| title_full_unstemmed | p-adic differential equations Kiran S. Kedlaya |
| title_short | p-adic differential equations |
| title_sort | p adic differential equations |
| topic | p-adic analysis Differential equations p-adische Differentialgleichung (DE-588)4398266-9 gnd |
| topic_facet | p-adic analysis Differential equations p-adische Differentialgleichung |
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