Slopes of $F$-isocrystals over abelian varieties

TL;DR

This paper proves that $F$-isocrystals over abelian varieties in positive characteristic have constant slopes, extending Tsuzuki's theorem and utilizing monodromy group theory of convergent isocrystals.

Contribution

It establishes the constancy of slopes for $F$-isocrystals over abelian varieties, generalizing previous results for finite fields.

Findings

$F$-isocrystals over abelian varieties have constant slopes

Extension of Tsuzuki's theorem to broader class of varieties

Application of monodromy group theory in proof

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.