Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- 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
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
18 - Effective convergence bounds
from Part V - Frobenius Structures
Published online by Cambridge University Press: 06 August 2022
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- Part III p-adic Differential Equations on Discs and Annuli
- Part IV Difference Algebra and Frobenius Modules
- 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
- Part VII Global theory
- Appendix A Picard–Fuchs modules
- Appendix B Rigid cohomology
- Appendix C p-adic Hodge theory
- References
- Index of notation
- Subject index
Summary
In this chapter, we discuss some effective bounds on the solutions of p-adic differential equations with nilpotent singularities. These come in two forms. We start by discussing bounds that make no reference to a Frobenius structure, due to Christol, Dwork, and Robba. These could have been presented earlier; we chose to postpone them until this point so that we can better contrast them against the bounds available in the presence of a Frobenius structure. The latter are original, though they are strongly inspired by some recent results of Chiarellotto and Tsuzuki. These results carry both theoretical and practical interest. Besides their application in the study of p-adic exponents mentioned above, another theoretical point of interest is their use in the study of logarithmic growth of horizontal sections at a boundary. We discuss some recent advances in this study due to André, Chiarellotto–Tsuzuki, and Ohkubo. A point of practical interest is that effective convergence bounds are useful for carrying out rigorous numerical calculations, e.g., in the machine computation of zeta functions of varieties over finite fields. See the notes for Appendix B for further discussion.
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- Chapter
- Information
- p-adic Differential Equations , pp. 323 - 337Publisher: Cambridge University PressPrint publication year: 2022