# A crystalline incarnation of Berthelot's conjecture and K\"unneth formula for isocrystals

## TL;DR

This paper proves Berthelot's conjecture for crystals up to isogeny and establishes a K"unneth formula for the crystalline fundamental group, advancing understanding of overconvergent isocrystals in algebraic geometry.

## Contribution

It demonstrates the validity of Berthelot's conjecture for crystals up to isogeny and derives a K"unneth formula for the crystalline fundamental group, providing new tools in p-adic cohomology.

## Key findings

- Berthelot's conjecture holds for crystals up to isogeny.
- A K"unneth formula for the crystalline fundamental group is established.
- Advances the theory of overconvergent isocrystals in characteristic p.

## Abstract

Berthelot's conjecture predicts that under a proper and smooth morphism of schemes in characteristic $p$, the higher direct images of an overconvergent $F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that this is true for crystals up to isogeny. As an application we prove a K\"unneth formula for the crystalline fundamental group.