p adic Differential Equations 2nd Edition by Kiran Kedlaya ISBN 9781009123341 1009123343

0 Introductory remarks

0.1 Why p-adic differential equations?

0.2 Zeta functions of varieties

0.3 Zeta functions and p-adic differential equations

0.4 A word of caution

Notes

Exercises

Part I Tools of p-adic Analysis

1 Norms on algebraic structures

1.1 Norms on abelian groups

1.2 Valuations and nonarchimedean norms

1.3 Norms on modules

1.4 Examples of nonarchimedean norms

1.5 Spherical completeness

Notes

Exercises

2 Newton polygons

2.1 Newton polygons

2.2 Slope factorizations and a master factorization theorem

2.3 Applications to nonarchimedean field theory

Notes

Exercises

3 Ramification theory

3.1 Defect

3.2 Unramified extensions

3.3 Tamely ramified extensions

3.4 The case of local fields

Notes

Exercises

4 Matrix analysis

4.1 Singular values and eigenvalues (archimedean case)

4.2 Perturbations (archimedean case)

4.3 Singular values and eigenvalues (nonarchimedean case)

4.4 Perturbations (nonarchimedean case)

4.5 Horn’s inequalities

Notes

Exercises

Part II Differential Algebra

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 and irregular singularities

7.1 Irregularity

7.2 Exponents in the complex analytic setting

7.3 Formal solutions of regular differential equations

7.4 Index and irregularity

7.5 The Turrittin–Levelt–Hukuhara decomposition theorem

7.6 Asymptotic behavior

Notes

Exercises

Part III p-adic Differential Equations on Discs and Annuli

8 Rings of functions on discs and annuli

8.1 Power series on closed discs and annuli

8.2 Gauss norms and Newton polygons

8.3 Factorization results

8.4 Open discs and annuli

8.5 Analytic elements

8.6 More approximation arguments

Notes

Exercises

9 Radius and generic radius of convergence

9.1 Differential modules have no torsion

9.2 Antidifferentiation

9.3 Radius of convergence on a disc

9.4 Generic radius of convergence

9.5 Some examples in rank 1

9.6 Transfer theorems

9.7 Geometric interpretation

9.8 Subsidiary radii

9.9 Another example in rank 1

9.10 Comparison with the coordinate-free definition

9.11 An explicit convergence estimate

Notes

Exercises

10 Frobenius pullback and pushforward

10.1 Why Frobenius?

10.2 p-th powers and roots

10.3 Moving along Frobenius

10.4 Frobenius antecedents

10.5 Frobenius descendants and subsidiary radii

10.6 Decomposition by spectral radius

10.7 Integrality of the generic radius

10.8 Off-center Frobenius antecedents and descendants

Notes

Exercises

11 Variation of generic and subsidiary radii

11.1 Harmonicity of the valuation function

11.2 Variation of Newton polygons

11.3 Variation of subsidiary radii: statements

11.4 Convexity for the generic radius

11.5 Measuring small radii

11.6 Larger radii

11.7 Monotonicity

11.8 Radius versus generic radius

11.9 Subsidiary radii as radii of optimal convergence

Notes

Exercises

12 Decomposition by subsidiary radii

12.1 Metrical detection of units

12.2 Decomposition over a closed disc

12.3 Decomposition over a closed annulus

12.4 Partial decomposition over a closed disc or annulus

12.5 Decomposition over an open disc or annulus

12.6 Modules solvable at a boundary

12.7 Solvable modules of rank 1

12.8 Clean modules

Notes

Exercises

13 p-adic exponents

13.1 p-adic Liouville numbers

13.2 p-adic regular singularities

13.3 The Robba condition

13.4 Abstract p-adic exponents

13.5 Exponents for annuli

13.6 The p-adic Fuchs theorem for annuli

13.7 Transfer to a regular singularity

13.8 Liouville partitions

Notes

Exercises

Part IV Difference Algebra and Frobenius Modules

14 Formalism of difference algebra

14.1 Difference algebra

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 Substitutions and Frobenius lifts

15.3 Generic versus special Frobenius

15.4 A reverse filtration

15.5 Substitution maps in the Robba ring

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 Harder–Narasimhan filtrations

16.6 Extended Robba rings

16.7 Proof of the slope filtration theorem

Notes

Exercises

Part V Frobenius Structures

17 Frobenius structures on differential modules

17.1 Frobenius structures

17.2 Frobenius structures and the generic radius of convergence

17.3 Independence from the Frobenius lift

17.4 Slope filtrations and differential structures

17.5 Extension of Frobenius structures

17.6 Frobenius intertwiners

Notes

Exercises

18 Effective convergence bounds

18.1 A first bound

18.2 Effective bounds for solvable modules

18.3 Better bounds using Frobenius structures

18.4 Logarithmic growth

18.5 Nonzero exponents

Notes

Exercises

19 Galois representations and differential modules

19.1 Representations and differential modules

19.2 Finite representations and overconvergent differential modules

19.3 The unit-root p-adic local monodromy theorem

19.4 Ramification and differential slopes

Notes

Exercises

Part VI The p-adic local monodromy theorem

20 The p-adic local monodromy theorem

20.1 Statement of the theorem

20.2 An example

20.3 Descent of horizontal sections

20.4 Local duality

20.5 When the residue field is imperfect

20.6 Minimal slope quotients

Notes

Exercises

21 The p-adic local monodromy theorem: proof

21.1 Running hypotheses

21.2 Modules of differential slope 0

21.3 Modules of rank 1

21.4 Modules of rank prime to ..

21.5 The general case

Notes

Exercises

22 p-adic monodromy without Frobenius structures

22.1 The Robba ring revisited

22.2 Modules of cyclic type

22.3 A Tannakian construction

22.4 Interlude on finite linear groups

22.5 Back to the Tannakian construction

22.6 Proof of the theorem

22.7 Relation to Frobenius structures

Notes

Exercises

Part VII Global theory

23 Banach rings and their spectra

23.1 Banach rings

23.2 The spectrum of a Banach ring

23.3 Topological properties

23.4 Complete residue fields

Notes

Exercises

24 The Berkovich projective line

24.1 Points

24.2 Classification of points

24.3 The domination relation

24.4 The tree structure

24.5 Skeleta

24.6 Harmonic and subharmonic functions

Notes

Exercises

25 Convergence polygons

25.1 The normalized radius of convergence

25.2 Normalized subsidiary radii and the convergence polygon

25.3 A constancy criterion for convergence polygons

25.4 Finiteness of the convergence polygon

25.5 Effect of singularities

25.6 Affinoid subspaces

25.7 Meromorphic differential equations

25.8 Open discs and annuli

Notes

26 Index theorems

26.1 The index of a differential module

26.2 More on affinoid subspaces of [mathbb(P)][sub(K)]

26.3 The Laplacian of the convergence polygon

26.4 An index formula for algebraic differential equations

26.5 Local analysis on a disc

26.6 Local analysis on an annulus

26.7 Some nonarchimedean functional analysis

26.8 Plus and minus indices

26.9 Global analysis on a disc

26.10 A global index formula

Notes

Exercises

27 Local constancy at type-4 points

27.1 Geometry around a point of type 4

27.2 Local constancy in the visible range

27.3 Local monodromy at a point of type 4

27.4 End of the proof

Notes

APPENDICES AND BACK MATTER

Appendix A Picard–Fuchs modules

A.1 Picard–Fuchs modules

A.2 Frobenius structures on Picard–Fuchs modules

A.3 Relationship with zeta functions

Notes

Appendix B Rigid cohomology

B.1 Isocrystals on the affine line

B.2 Crystalline and rigid cohomology

B.3 Machine computations

Notes

Appendix C p-adic Hodge theory

C.1 A few rings

C.2 (φ, Γ)-modules

C.3 Galois cohomology

C.4 Differential equations from (φ, Γ)-modules

C.5 Beyond Galois representations

Notes

References

Index of notation

Subject index