A cut-by-curves criterion for overconvergent of $F$-isocrystals

TL;DR

This paper proposes a new criterion for overconvergence of $F$-isocrystals on smooth schemes over finite fields, based on their behavior on curves and conditions on local monodromy, advancing understanding in $p$-adic cohomology.

Contribution

It establishes a weaker overconvergence criterion for $F$-isocrystals using the theory of companions and monodromy conditions, extending previous conjectures.

Findings

Proves a criterion linking overconvergence to restrictions on curves.

Uses étale and crystalline companion theories to support the criterion.

Shows trivialization of wild local monodromy via a single dominant morphism.

Abstract

Let $X$ be a smooth scheme over a finite field. It is conjectured that a convergent $F$-isocrystal on $X$ is overconvergent if its restriction to every curve contained in $X$ is overconvergent. Using the theory of \'etale and crystalline companions, we establish a weaker version of this criterion in which we also assume that the wild local monodromy of the restrictions to curves is trivialized by pullback along a single dominant morphism to $X$.