Delta Characters and Crystalline Cohomology

TL;DR

This paper develops a theory of shifted Witt vectors with delta structures, applies it to arithmetic jet spaces and Picard-Fuchs operators, and studies associated filtered isocrystals for abelian schemes and elliptic curves.

Contribution

It introduces $m$-shifted $ extpi$-typical Witt vectors with delta structures and explores their applications to arithmetic jet spaces and crystalline cohomology.

Findings

Shifted Witt vectors admit a delta structure satisfying canonical identities.

Generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level.

The filtered isocrystal $ extbf{H}_ extdelta(A)$ is of finite rank for abelian schemes and matches crystalline cohomology in certain cases.

Abstract

The first part of the paper develops the theory of $m$-shifted $\pi$-typical Witt vectors which can be viewed as subobjects of the usual $\pi$-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy a canonical identity with the delta structure of the usual $\pi$-typical Witt vectors. Using this theory, we prove that the generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level. This also allows us to interpret the arithmetic Picard-Fuchs operator geometrically. For a $\pi$-formal group scheme $G$, by a previous construction, one attaches a canonical filtered isocrystal $\mathbf{H}_\delta(G)$ associated to the arithmetic jet spaces of $G$. In the second half of our paper, we show that $\mathbf{H}_\delta(A)$ is of finite rank if $A$ is an abelian scheme. We also prove a strengthened version of a result of Buium on delta characters on abelian schemes. As an application, for an elliptic curve $A$ defined over $\mathbb{Z}_p$, we show that our canonical filtered isocrystal $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is weakly admissible. In particular, if $A$ does not admit a lift of Frobenius, we show that $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is isomorphic to the first crystalline cohomology $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$ in the category of filtered isocrystals. On the other hand, if $A$ admits a lift of Frobenius, then $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is isomorphic to the sub-isocrystal $H^0(A,\Omega_A) \otimes \mathbb{Q}_p$ of $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$.