p-adic Differential Equations

A detailed and unified treatment of $P$-adic differential equations, from the basic principles to the current frontiers of research.

Kiran S. Kedlaya (Author)

9781009123341, Cambridge University Press

Hardback, published 9 June 2022

420 pages
23.5 x 15.7 x 3.3 cm, 0.91 kg

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.

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.

Subject Areas: Differential calculus & equations [PBKJ], Number theory [PBH]