Incarnations of Berthelot's conjecture

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.

17 pages. Final version, published in J. Number Theory

References in corpus (1)

• Relative log convergent cohomology and relative rigid cohomology I

Cited by in corpus (5)

• Constancy of Newton polygons of $F$-isocrystals on Abelian varieties and isotriviality of families of curves
• On higher direct images of convergent isocrystals
• Deformations of overconvergent isocrystals on the projective line
• A crystalline incarnation of Berthelot's conjecture and Künneth formula for isocrystals
• Milnor K-theory, F-isocrystals and Syntomic Regulators