Kedlaya K.S. p-adic Differential Equations

Cambridge: Cambridge University Press, 2010. — 400 p.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.Why p-adic differential equations?
Zeta functions of varieties
Zeta functions and p-adic differential equations
A word of caution
Notes
Exercises
Part I Tools of p-adic Analysis
Norms on abelian groups
Valuations and nonarchimedean norms
Norms on modules
Examples of nonarchimedean norms
Spherical completeness
Notes
Exercises
Introduction to Newton polygons
Slope factorizations and a master factorization theorem
Applications to nonarchimedean field theory
Notes
Exercises
Ramification theory
Defect
Unramified extensions
Tamely ramified extensions
The case of local fields
Notes
Exercises
Matrix analysis
Singular values and eigenvalues (archimedean case)
Perturbations (archimedean case)
Singular values and eigenvalues (nonarchimedean case)
Perturbations (nonarchimedean case)
Horn's inequalities
Notes
Exercises
Part II Differential Algebra
Differential rings and differential modules
Differential modules and differential systems
Operations on differential modules
Cyclic vectors
Differential polynomials
Cyclic vectors: a mixed blessing
Taylor series
Exercises
Spectral radii of bounded endomorphisms
Spectral radii of differential operators
A coordinate-free approach
Newton polygons for twisted polynomials
Twisted polynomials and spectral radii
The visible decomposition theorem
Matrices and the visible spectrum
A refined visible decomposition theorem
Changing the constant field
Notes
Exercises
Regular singularities
Irregularity
Exponents in the complex analytic setting
Formal solutions of regular differential equations
Index and irregularity
The Turrittin–Levelt–Hukuhara decomposition theorem
Notes
Exercises
Part III p-adic Differential Equations on Discs and Annuli
Rings of functions on discs and annuli
Power series on closed discs and annuli
Gauss norms and Newton polygons
Factorization results
Open discs and annuli
Analytic elements
More approximation arguments
Notes
Exercises
Radius and generic radius of convergence
Differential modules have no torsion
Antidifferentiation
Radius of convergence on a disc
Generic radius of convergence
Some examples in rank
Transfer theorems
Geometric interpretation
Another example in rank
Comparison with the coordinate-free definition
Notes
Exercises
Why Frobenius descent?
pth powers and roots
Frobenius pullback and pushforward operations
Frobenius antecedents
Frobenius descendants and subsidiary radii
Decomposition by spectral radius
Integrality of the generic radius
Off-center Frobenius antecedents and descendants
Notes
Exercises
Variation of generic and subsidiary radii
Harmonicity of the valuation function
Variation of Newton polygons
Variation of subsidiary radii: statements
Convexity for the generic radius
Measuring small radii
Larger radii
Monotonicity
Radius versus generic radius
Subsidiary radii as radii of optimal convergence
Notes
Exercises
Decomposition by subsidiary radii
Metrical detection of units
Decomposition over a closed disc
Decomposition over a closed annulus
Decomposition over an open disc or annulus
Partial decomposition over a closed disc or annulus
Modules solvable at a boundary
Solvable modules of rank
Clean modules
Exercises
p-adic Liouville numbers
p-adic regular singularities
The Robba condition
Abstract p-adic exponents
Exponents for annuli
The p-adic Fuchs theorem for annuli
Transfer to a regular singularity
Notes
Exercises
Part IV Difference Algebra and Frobenius Modules
Difference algebra
Twisted polynomials
Difference-closed fields
Difference algebra over a complete field
Hodge and Newton polygons
The Dieudonné–Manin classification theorem
Notes
Exercises
A multitude of rings
Frobenius lifts
Generic versus special Frobenius lifts
A reverse filtration
Notes
Exercises
Frobenius modules on open discs
More on the Robba ring
Pure difference modules
The slope filtration theorem
Proof of the slope filtration theorem
Notes
Exercises
Part V Frobenius Structures
Frobenius structures
Frobenius structures and the generic radius of convergence
Independence from the Frobenius lift
Extension of Frobenius structures
Notes
Exercises
A first bound
Effective bounds for solvable modules
Better bounds using Frobenius structures
Logarithmic growth
Notes
Exercises
Galois representations and differential modules
Representations and differential modules
Finite representations and overconvergent differential modules
The unit-root p-adic local monodromy theorem
Ramification and differential slopes
Notes
Exercises
Statement of the theorem
An example
Descent of sections
Local duality
When the residue field is imperfect
Notes
Exercises
Running hypotheses
Modules of differential slope
Modules of rank
Modules of rank prime to p
Notes
Exercises
Part VI Areas of Application
Origin of Picard–Fuchs modules
Frobenius structures on Picard–Fuchs modules
Relationship to zeta functions
Notes
Isocrystals on the affine line
Crystalline and rigid cohomology
Machine computations
Notes
A few rings
(phigamma)-modules
Galois cohomology
Differential equations from (phi gamma)-modules
Beyond Galois representations
NotesNotation