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El. knyga: p-adic Differential Equations

(University of California, San Diego)
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Now in its second edition, this volume provides a uniquely detailed study of $P$-adic differential equations. Assuming only a graduate-level background in number theory, the text builds the theory from first principles all the way to the frontiers of current research, highlighting analogies and links with the classical theory of ordinary differential equations. The author includes many original results which play a key role in the study of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge theory, perfectoid spaces, and algorithms for L-functions of arithmetic varieties. This updated edition contains five new chapters, which revisit the theory of convergence of solutions of $P$-adic differential equations from a more global viewpoint, introducing the Berkovich analytification of the projective line, defining convergence polygons as functions on the projective line, and deriving a global index theorem in terms of the Laplacian of the convergence polygon.

Recenzijos

' the book under review is unique in the sense that it can serve as a comprehensive introduction to the subject (the monograph assumes just a graduate-level background in algebraic number theory) and as a roadmap for researchers in the area.' Alexander B. Levin, MathSciNet

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A detailed and unified treatment of $P$-adic differential equations, from the basic principles to the current frontiers of research.
Preface xiv
Acknowledgments xx
0 Introductory remarks
1(12)
0.1 Why p-adic differential equations?
1(2)
0.2 Zeta functions of varieties
3(2)
0.3 Zeta functions and p-adic differential equations
5(2)
0.4 A word of caution
7(6)
Notes
8(1)
Exercises
9(4)
Part I Tools of p-adic Analysis
1 Norms on algebraic structures
13(23)
1.1 Norms on abelian groups
13(3)
1.2 Valuations and nonarchimedean norms
16(1)
1.3 Norms on modules
17(9)
1.4 Examples of nonarchimedean norms
26(2)
1.5 Spherical completeness
28(8)
Notes
32(2)
Exercises
34(2)
2 Newton polygons
36(10)
2.1 Newton polygons
36(3)
2.2 Slope factorizations and a master factorization theorem
39(3)
2.3 Applications to nonarchimedean field theory
42(4)
Notes
44(1)
Exercises
45(1)
3 Ramification theory
46(10)
3.1 Defect
47(1)
3.2 Unramified extensions
48(2)
3.3 Tamely ramified extensions
50(3)
3.4 The case of local fields
53(3)
Notes
54(1)
Exercises
55(1)
4 Matrix analysis
56(23)
4.1 Singular values and eigenvalues (archimedean case)
57(4)
4.2 Perturbations (archimedean case)
61(2)
4.3 Singular values and eigenvalues (nonarchimedean case)
63(5)
4.4 Perturbations (nonarchimedean case)
68(4)
4.5 Horn's inequalities
72(7)
Notes
73(2)
Exercises
75(4)
Part II Differential Algebra
5 Formalism of differential algebra
79(18)
5.1 Differential rings and differential modules
79(3)
5.2 Differential modules and differential systems
82(1)
5.3 Operations on differential modules
83(4)
5.4 Cyclic vectors
87(1)
5.5 Differential polynomials
88(2)
5.6 Differential equations
90(1)
5.7 Cyclic vectors: a mixed blessing
91(2)
5.8 Taylor series
93(4)
Notes
94(1)
Exercises
95(2)
6 Metric properties of differential modules
97(26)
6.1 Spectral radii of bounded endomorphisms
97(2)
6.2 Spectral radii of differential operators
99(7)
6.3 A coordinate-free approach
106(2)
6.4 Newton polygons for twisted polynomials
108(1)
6.5 Twisted polynomials and spectral radii
109(2)
6.6 The visible decomposition theorem
111(2)
6.7 Matrices and the visible spectrum
113(3)
6.8 A refined visible decomposition theorem
116(3)
6.9 Changing the constant field
119(4)
Notes
120(1)
Exercises
121(2)
7 Regular and irregular singularities
123(20)
7.1 Irregularity
124(1)
7.2 Exponents in the complex analytic setting
125(2)
7.3 Formal solutions of regular differential equations
127(4)
7.4 Index and irregularity
131(2)
7.5 The Turrittin-Levelt-Hukuhara decomposition theorem
133(3)
7.6 Asymptotic behavior
136(7)
Notes
137(2)
Exercises
139(4)
Part III P-adic Differential Equations on Discs and Annuli
8 Rings of functions on discs and annuli
143(18)
8.1 Power series on closed discs and annuli
144(2)
8.2 Gauss norms and Newton polygons
146(2)
8.3 Factorization results
148(3)
8.4 Open discs and annuli
151(1)
8.5 Analytic elements
152(4)
8.6 More approximation arguments
156(5)
Notes
158(1)
Exercises
159(2)
9 Radius and generic radius of convergence
161(18)
9.1 Differential modules have no torsion
162(1)
9.2 Antidifferentiation
163(1)
9.3 Radius of convergence on a disc
164(1)
9.4 Generic radius of convergence
165(3)
9.5 Some examples in rank 1
168(1)
9.6 Transfer theorems
168(2)
9.7 Geometric interpretation
170(2)
9.8 Subsidiary radii
172(1)
9.9 Another example in rank 1
173(1)
9.10 Comparison with the coordinate-free definition
174(1)
9.11 An explicit convergence estimate
175(4)
Notes
176(1)
Exercises
177(2)
10 Frobenius pullback and pushforward
179(17)
10.1 Why Frobenius?
180(1)
10.2 P-th powers and roots
180(2)
10.3 Moving along Frobenius
182(2)
10.4 Frobenius antecedents
184(2)
10.5 Frobenius descendants and subsidiary radii
186(2)
10.6 Decomposition by spectral radius
188(4)
10.7 Integrality of the generic radius
192(1)
10.8 Off-center Frobenius antecedents and descendants
193(3)
Notes
194(1)
Exercises
195(1)
11 Variation of generic and subsidiary radii
196(18)
11.1 Harmonicity of the valuation function
197(1)
11.2 Variation of Newton polygons
198(3)
11.3 Variation of subsidiary radii: statements
201(2)
11.4 Convexity for the generic radius
203(1)
11.5 Measuring small radii
204(1)
11.6 Larger radii
205(3)
11.7 Monotonicity
208(1)
11.8 Radius versus generic radius
209(1)
11.9 Subsidiary radii as radii of optimal convergence
210(4)
Notes
212(1)
Exercises
212(2)
12 Decomposition by subsidiary radii
214(19)
12.1 Metrical detection of units
215(1)
12.2 Decomposition over a closed disc
216(4)
12.3 Decomposition over a closed annulus
220(2)
12.4 Partial decomposition over a closed disc or annulus
222(2)
12.5 Decomposition over an open disc or annulus
224(1)
12.6 Modules solvable at a boundary
225(1)
12.7 Solvable modules of rank 1
226(1)
12.8 Clean modules
227(6)
Notes
231(1)
Exercises
232(1)
13 P-adic exponents
233(28)
13.1 P-adic Liouville numbers
233(3)
13.2 P-adic regular singularities
236(1)
13.3 The Robba condition
237(1)
13.4 Abstract p-adic exponents
238(2)
13.5 Exponents for annuli
240(6)
13.6 The p-adic Fuchs theorem for annuli
246(4)
13.7 Transfer to a regular singularity
250(3)
13.8 Liouville partitions
253(8)
Notes
256(1)
Exercises
257(4)
Part IV Difference Algebra and Frobenius Modules
14 Formalism of difference algebra
261(20)
14.1 Difference algebra
261(3)
14.2 Twisted polynomials
264(1)
14.3 Difference-closed fields
265(2)
14.4 Difference algebra over a complete field
267(5)
14.5 Hodge and Newton polygons
272(2)
14.6 The Dieudonne-Manin classification theorem
274(7)
Notes
277(2)
Exercises
279(2)
15 Frobenius modules
281(13)
15.1 A multitude of rings
281(3)
15.2 Substitutions and Frobenius lifts
284(2)
15.3 Generic versus special Frobenius
286(3)
15.4 A reverse filtration
289(3)
15.5 Substitution maps in the Robba ring
292(2)
Notes
292(1)
Exercises
293(1)
16 Frobenius modules over the Robba ring
294(17)
16.1 Frobenius modules on open discs
294(2)
16.2 More on the Robba ring
296(2)
16.3 Pure difference modules
298(2)
16.4 The slope filtration theorem
300(2)
16.5 Harder-Narasimhan filtrations
302(1)
16.6 Extended Robba rings
303(1)
16.7 Proof of the slope filtration theorem
304(7)
Notes
306(2)
Exercises
308(3)
Part V Frobenius Structures
17 Frobenius structures on differential modules
311(12)
17.1 Frobenius structures
311(3)
17.2 Frobenius structures and the generic radius of convergence
314(2)
17.3 Independence from the Frobenius lift
316(2)
17.4 Slope filtrations and differential structures
318(1)
17.5 Extension of Frobenius structures
319(1)
17.6 Frobenius intertwiners
320(3)
Notes
321(1)
Exercises
322(1)
18 Effective convergence bounds
323(15)
18.1 A first bound
324(1)
18.2 Effective bounds for solvable modules
324(4)
18.3 Better bounds using Frobenius structures
328(3)
18.4 Logarithmic growth
331(3)
18.5 Nonzero exponents
334(4)
Notes
335(1)
Exercises
336(2)
19 Galois representations and differential modules
338(15)
19.1 Representations and differential modules
339(2)
19.2 Finite representations and overconvergent differential modules
341(2)
19.3 The unit-root p-adic local monodromy theorem
343(3)
19.4 Ramification and differential slopes
346(7)
Notes
348(2)
Exercises
350(3)
Part VI The p-adic local monodromy theorem
20 The p-adic local monodromy theorem
353(14)
20.1 Statement of the theorem
353(2)
20.2 An example
355(1)
20.3 Descent of horizontal sections
356(3)
20.4 Local duality
359(1)
20.5 When the residue field is imperfect
360(2)
20.6 Minimal slope quotients
362(5)
Notes
363(3)
Exercises
366(1)
21 The p-adic local monodromy theorem: proof
367(7)
21.1 Running hypotheses
367(1)
21.2 Modules of differential slope 0
368(2)
21.3 Modules of rank 1
370(1)
21.4 Modules of rank prime to p
371(1)
21.5 The general case
372(2)
Notes
372(1)
Exercises
373(1)
22 P-adic monodromy without Frobenius structures
374(19)
22.1 The Robba ring revisited
374(1)
22.2 Modules of cyclic type
375(3)
22.3 A Tannakian construction
378(4)
22.4 Interlude on finite linear groups
382(2)
22.5 Back to the Tannakian construction
384(2)
22.6 Proof of the theorem
386(1)
22.7 Relation to Frobenius structures
387(6)
Notes
389(1)
Exercises
390(3)
Part VII Global theory
23 Banach rings and their spectra
393(6)
23.1 Banach rings
393(1)
23.2 The spectrum of a Banach ring
394(1)
23.3 Topological properties
394(2)
23.4 Complete residue fields
396(3)
Notes
397(1)
Exercises
397(2)
24 The Berkovich projective line
399(12)
24.1 Points
399(2)
24.2 Classification of points
401(1)
24.3 The domination relation
402(2)
24.4 The tree structure
404(1)
24.5 Skeleta
405(3)
24.6 Harmonic and subharmonic functions
408(3)
Notes
408(1)
Exercises
409(2)
25 Convergence polygons
411(12)
25.1 The normalized radius of convergence
411(1)
25.2 Normalized subsidiary radii and the convergence polygon
412(1)
25.3 A constancy criterion for convergence polygons
413(2)
25.4 Finiteness of the convergence polygon
415(2)
25.5 Effect of singularities
417(1)
25.6 Affinoid subspaces
418(1)
25.7 Meromorphic differential equations
419(1)
25.8 Open discs and annuli
420(3)
Notes
421(2)
26 Index theorems
423(19)
26.1 The index of a differential module
423(1)
26.2 More on affinoid subspaces of P#
424(1)
26.3 The Laplacian of the convergence polygon
425(2)
26.4 An index formula for algebraic differential equations
427(2)
26.5 Local analysis on a disc
429(2)
26.6 Local analysis on an annulus
431(2)
26.7 Some nonarchimedean functional analysis
433(3)
26.8 Plus and minus indices
436(2)
26.9 Global analysis on a disc
438(1)
26.10 A global index formula
439(3)
Notes
440(1)
Exercises
441(1)
27 Local constancy at type-4 points
442(7)
27.1 Geometry around a point of type 4
442(1)
27.2 Local constancy in the visible range
443(1)
27.3 Local monodromy at a point of type 4
444(2)
27.4 End of the proof
446(3)
Notes
446(3)
Appendix A Picard-Fuchs modules
449(5)
A.1 Picard-Fuchs modules
449(1)
A.2 Frobenius structures on Picard-Fuchs modules
450(1)
A.3 Relationship with zeta functions
451(3)
Notes
452(2)
Appendix B Rigid cohomology
454(6)
B.1 Isocrystals on the affine line
454(2)
B.2 Crystalline and rigid cohomology
456(1)
B.3 Machine computations
457(3)
Notes
458(2)
Appendix C P-adic Hodge theory
460(9)
C.1 A few rings
460(2)
C.2 (Φ, Γ)-modules
462(2)
C.3 Galois cohomology
464(1)
C.4 Differential equations from (Φ, Γ)-modules
465(2)
C.5 Beyond Galois representations
467(2)
Notes
467(2)
References 469(20)
Index of notation 489(2)
Subject Index 491
Kiran S. Kedlaya is the Stefan E. Warschawski Professor of Mathematics at University of California, San Diego. He has published over 100 research articles in number theory, algebraic geometry, and theoretical computer science, as well as several books, including two on the Putnam competition. He has received a Presidential Early Career Award, a Sloan Fellowship, and a Guggenheim Fellowship, and been named an ICM invited speaker and a fellow of the American Mathematical Society.
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