Slopes of $F$-isocrystals over abelian varieties

TL;DR

This work proves that any $F$-isocrystal on an abelian variety over a perfect field of characteristic $p$ has constant slopes, extending Tsuzuki’s finite-field result. The authors leverage monodromy groups of convergent isocrystals and establish that the Tannaka group of $ ext{Isoc}(A)$ is commutative via a Künneth-based Eckmann–Hilton argument, enabling global constancy of slopes. A Künneth formula for convergent isocrystals is developed, yielding exact sequences for fundamental groups of products, which underpins the commutativity results. The paper also derives Albanese-variety comparisons and finite-étale-constancy phenomena in finite-field and purity contexts, and connects these to $p$-adic representation theory and the structure of Isoc on Albanese varieties, with broader implications for diagonalizable and unipotent monodromy components.

Abstract

We prove that an $F$-isocrystal over an abelian variety defined over a perfect field of positive characteristic has constant slopes. This recovers and extends a theorem of Tsuzuki for abelian varieties over finite fields. Our proof exploits the theory of monodromy groups of convergent isocrystals.

Theorems & Definitions (22)

• Theorem 1.1: Theorem \ref{['main:t']}
• Proposition 1.2: Proposition \ref{['comm-fund-group:p']}
• Theorem 1.3: Theorem \ref{['fini:t']}
• Theorem 1.4: Theorem \ref{['alb:t']}
• Theorem 2.1: LP17
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof