This book surveys the theory of $P$-adic differential equations, from the foundations of $P$-adic numbers to the current frontiers of research. It assumes only a graduate-level background in number theory, and includes detailed chapter notes as well as numerous exercises. This second edition features new material on global theory.
This book surveys the theory of $P$-adic differential equations, from the foundations of $P$-adic numbers to the current frontiers of research. It assumes only a graduate-level background in number theory, and includes detailed chapter notes as well as numerous exercises. This second edition features new material on global theory.
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.
Inhaltsangabe
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.
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.
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