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Logarithmic Dieudonné theory and overconvergent extensions

Marco D'Addezio

TL;DR

This work addresses Crew's parabolicity framework by examining the slopes of $\dag$-hulls of $F$-isocrystals and establishing an overconvergence phenomenon for certain quotients under a slope condition $[s,s+1]$. Central to the approach is logarithmic Dieudonné theory, which links $p$-divisible groups to overconvergent $F$-isocrystals, and a new overconvergent Dieudonné equivalence that on curves identifies potentially semi-stable $p$-divisible groups with overconvergent $F$-isocrystals whose slopes lie in $[0,1]$. The proof combines Tate’s local extension property for étale subgroups, the overconvergent Dieudonné framework, and Tsuzuki’s unit-root characterization to deduce finiteness results for associated Galois representations after base change. These results shed light on Crew’s parabolicity conjecture and have implications for arithmetic questions tied to Shimura varieties and abelian varieties, including specific finiteness statements for $A(\ ext{Ω}^{\mathrm{sep}})[p^\infty]$ under endomorphism constraints.

Abstract

In the proof of Crew's parabolicity conjecture, we established a key property concerning the slopes of $\dagger$-hulls of $F$-isocrystals, extending a result of Tsuzuki. This article presents an alternative proof of this theorem for a specific class of $F$-isocrystals. The central ingredient is a local extension property for étale $p$-divisible subgroups. To relate $p$-divisible groups and overconvergent $F$-isocrystals, we employ logarithmic Dieudonné theory, as introduced by Kato and further developed by Inoue. Over curves, this leads to an equivalence between the category of potentially semi-stable $p$-divisible groups and overconvergent $F$-isocrystals with slopes in the interval $[0,1]$.

Logarithmic Dieudonné theory and overconvergent extensions

TL;DR

This work addresses Crew's parabolicity framework by examining the slopes of -hulls of -isocrystals and establishing an overconvergence phenomenon for certain quotients under a slope condition . Central to the approach is logarithmic Dieudonné theory, which links -divisible groups to overconvergent -isocrystals, and a new overconvergent Dieudonné equivalence that on curves identifies potentially semi-stable -divisible groups with overconvergent -isocrystals whose slopes lie in . The proof combines Tate’s local extension property for étale subgroups, the overconvergent Dieudonné framework, and Tsuzuki’s unit-root characterization to deduce finiteness results for associated Galois representations after base change. These results shed light on Crew’s parabolicity conjecture and have implications for arithmetic questions tied to Shimura varieties and abelian varieties, including specific finiteness statements for under endomorphism constraints.

Abstract

In the proof of Crew's parabolicity conjecture, we established a key property concerning the slopes of -hulls of -isocrystals, extending a result of Tsuzuki. This article presents an alternative proof of this theorem for a specific class of -isocrystals. The central ingredient is a local extension property for étale -divisible subgroups. To relate -divisible groups and overconvergent -isocrystals, we employ logarithmic Dieudonné theory, as introduced by Kato and further developed by Inoue. Over curves, this leads to an equivalence between the category of potentially semi-stable -divisible groups and overconvergent -isocrystals with slopes in the interval .
Paper Structure (5 sections, 24 theorems, 32 equations)

This paper contains 5 sections, 24 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Logarithmic flatness
  3. Logarithmic Dieudonné theory
  4. The overconvergent Dieudonné equivalence
  5. Main theorem

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a $\dag$-extendable $F$-isocrystal over $X$ whose $p$-adic slopes lie in an interval $[s,s+1]$ for some $s\in\mathbb{Q}$. Then every quotient of $(\mathcal{E},\Phi_\mathcal{E})$ in $\mathbf{F\textrm{-}Isoc}(X)$ which is isoclinic of slope $s$ is again $\dag$-extendable.

Theorems & Definitions (48)

  • Theorem 1.1: Theorem \ref{['a-UR01:t']}
  • Proposition 1.2: Tat67
  • Theorem 1.3: Theorem \ref{['DM:t']}
  • Theorem 1.4: Theorem \ref{['et-sub-psemi:t']}
  • Theorem 1.5: Tsu19, DA23
  • Theorem 1.6: DA23
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • ...and 38 more