Delta characters and crystalline cohomology of abelian schemes

TL;DR

This paper explicitly describes the smallest filtered sub-isocrystal generated by the Hodge filtered piece of crystalline cohomology for abelian schemes over p-adic rings, using arithmetic jet spaces and delta characters.

Contribution

It proves the delta isocrystal by Borger and Saha is isomorphic to the fundamental smallest sub-isocrystal in the category of filtered F-isocrystals.

Findings

Established a comparison isomorphism between delta isocrystal and crystalline cohomology.

Connected delta characters to the structure of the smallest sub-isocrystal.

Validated the construction of delta isocrystals via explicit isomorphism.

Abstract

We provide an explicit description of the smallest filtered sub-isocrystal generated by the Hodge filtered piece of the crystalline cohomology for an abelian scheme over a $p$-adic ring. Our method is based on the theory of arithmetic jet spaces and delta characters associated to the abelian scheme, introduced by Buium and later studied by Borger and Saha using a functor of points approach. In particular, we prove that the delta isocrystal constructed by Borger and Saha is indeed isomorphic to the fundamental smallest sub-isocrystal of the crystalline cohomology in the category of filtered $F$-isocrystals. As an application, we establish a comparison isomorphism between the delta isocrystal and the crystalline cohomology of abelian schemes, which is governed by the group of order $1$ delta characters of the abelian scheme.