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Parabolicity conjecture of $F$-isocrystals

Marco D'Addezio

Abstract

In this article we prove Crew's parabolicity conjecture of $F$-isocrystals. For this purpose, we introduce and study the notion of $\dagger$-hull of a sub-$F$-isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent $F$-isocrystals.

Parabolicity conjecture of $F$-isocrystals

Abstract

In this article we prove Crew's parabolicity conjecture of -isocrystals. For this purpose, we introduce and study the notion of -hull of a sub--isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent -isocrystals.

Paper Structure

This paper contains 27 sections, 57 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. -hull of -isocrystals
  4. Lefschetz theorem
  5. Punctual -structures
  6. The structure of the article
  7. Acknowledgements
  8. Notation
  9. Tannakian categories
  10. Isocrystals
  11. Isocrystals with punctual -structure
  12. First definitions
  13. Basic properties
  14. Comparisons
  15. Main theorems
  16. ...and 12 more sections

Key Result

Theorem 1.1.1

The subgroup $G(\mathcal{M},\eta)\subseteq G({\mathcal{M}^{{\hbox{\larger[-4]$\dag$}}}},\eta)$ is the subgroup of $G({\mathcal{M}^{{\hbox{\larger[-4]$\dag$}}}},\eta)$ stabilising the slope filtration of $\mathcal{M}_\eta$. Moreover, when ${\mathcal{M}^{{\hbox{\larger[-4]$\dag$}}}}$ is semi-simple, $

Theorems & Definitions (122)

  • Theorem 1.1.1: Theorem \ref{['para-c:t']}
  • Theorem 1.1.2: Theorem \ref{['semi-simp:t']}
  • Theorem 1.1.3: Theorem \ref{['abelian:t']}
  • Corollary 1.1.4: Corollary \ref{['k-c:c']}
  • Theorem 1.1.5: Theorem \ref{['mult-one:t']}
  • Definition 1.2.1
  • Theorem 1.2.2: Theorem \ref{['main:t']}
  • Theorem 1.3.1: Theorem \ref{['tame-lef:t']}
  • Definition 3.1.2
  • Lemma 3.1.3
  • ...and 112 more