p-adic Differential Equations

Preface; 0. Introductory remarks; Part I. Tools of $P$-adic Analysis: 1. Norms on algebraic structures; 2. Newton polygons; 3. Ramification theory; 4. Matrix analysis; Part II. Differential Algebra: 5. Formalism of differential algebra; 6. Metric properties of differential modules; 7. Regular and irregular singularities; Part III. $P$-adic Differential Equations on Discs and Annuli: 8. Rings of functions on discs and annuli; 9. Radius and generic radius of convergence; 10. Frobenius pullback and pushforward; 11. Variation of generic and subsidiary radii; 12. Decomposition by subsidiary radii; 13. $P$-adic exponents; Part IV. Difference Algebra and Frobenius Modules: 14. Formalism of difference algebra; 15. Frobenius modules; 16. Frobenius modules over the Robba ring; Part V. Frobenius Structures: 17. Frobenius structures on differential modules; 18. Effective convergence bounds; 19. Galois representations and differential modules; Part VI. The $P$-adic Local Monodromy Theorem: 20. The $P$-adic local monodromy theorem; 21. The $P$-adic local monodromy theorem: proof; 22. $P$-adic monodromy without Frobenius structures; Part VII. Global Theory: 23. Banach rings and their spectra; 24. The Berkovich projective line; 25. Convergence polygons; 26. Index theorems; 27. Local constancy at type-4 points; Appendix A: Picard-Fuchs modules; Appendix B: Rigid cohomology Appendix C: $P$-adic Hodge theory; References; Index of notations; Index.