Incarnations of Berthelot's conjecture

TL;DR

This paper surveys various forms of Berthelot's conjecture, explores their interrelations, and proves new comparison results that confirm some cases of the conjecture, especially regarding overconvergent isocrystals and arithmetic D-modules.

Contribution

It provides a comprehensive survey of Berthelot's conjecture variants and establishes new comparison results linking overconvergent isocrystals and arithmetic D-modules, confirming specific cases.

Findings

Ogus' convergent pushforward of an overconvergent F-isocrystal is overconvergent.

Comparison results between pushforwards of overconvergent isocrystals and arithmetic D-modules.

Some cases of Berthelot's conjecture are confirmed through these results.

Abstract

In this article we give a survey of the various forms of Berthelot's conjecture and some of the implications between them. By proving some comparison results between pushforwards of overconvergent isocrystals and those of arithmetic $\mathcal{D}$-modules, we manage to deduce some cases of the conjecture from Caro's results on the stability of overcoherence under pushforward via a smooth and proper morphism of varieties. In particular, we show that Ogus' convergent pushforward of an overconvergent $F$-isocrystal under a smooth and projective morphism is overconvergent.