Book contents
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- 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
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
10 - Frobenius pullback and pushforward
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 0 Introductory remarks
- Part I Tools of p-adic Analysis
- Part II Differential Algebra
- 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
- Part V Frobenius Structures
- Part VI Areas of Application
- References
- Notation
- Index
Summary
In this chapter, we introduce Dwork's technique of Frobenius descent to analyze the generic radius of convergence and subsidiary radii of a differential module, in the range where Newton polygons do not apply. In one direction we introduce a somewhat refined form of the Frobenius antecedents introduced by Christol and Dwork. These fail to apply in an important boundary case; we remedy this by introducing the new notion of Frobenius descendants, which covers the boundary case.
Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over Fρ. For instance we get a full decomposition by spectral radius, extending the visible decomposition theorem (Theorem 6.6.1) and the refined visible decomposition theorem (Theorem 6.8.2). We will use these results again to study the variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of this part.
Notation 10.0.1. Throughout this chapter we retain Hypothesis 9.0.1. We also continue to use Fρ to denote the completion of K(t) for the ρ-Gauss norm viewed as a differential field with respect to d = d/dt, unless otherwise specified.
Notation 10.0.2. Throughout this chapter we also assume p > 0 unless otherwise specified.
Why Frobenius descent?
Remark 10.1.1. It may be helpful to review the current state of affairs, in order to clarify why we need to descend along a Frobenius morphism.
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- p-adic Differential Equations , pp. 168 - 183Publisher: Cambridge University PressPrint publication year: 2010