Delta characters and crystalline cohomology

The first part of the paper develops the theory of m-shifted π- typicalWitt vectors which can be viewed as subobjects of the usual π-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 π-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 π-formal group scheme G, by a previous construction, associated to the arithmetic jet spaces of G one attaches a canonical filtered module H<inf>δ</inf>(G) with a semilinear operator on it. In the second half of our paper, we show that H<inf>δ</inf>(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 Z<inf>p</inf>, we show that our canonical filtered isocrystal H<inf>δ</inf>(A) (Formula Presented) Q<inf>p</inf> is weakly admissible. In particular, if A does not admit a lift of Frobenius, we show that H<inf>δ</inf>(A) (Formula Presented) Q<inf>p</inf> is isomorphic to the first crystalline cohomology (Formula Presented) Q<inf>p</inf> in the category of filtered isocrystals. On the other hand, if A admits a lift of Frobenius, then H<inf>δ</inf>(A)(Formula Presented)Q<inf>p</inf> is isomorphic to the sub-isocrystal H⁰(A,ΩA) (Formula Presented) Q<inf>p</inf> of (Formula Presented). The above result can be viewed as a character theoretic interpretation of the crystalline cohomology. The difference between the integral structures of H<inf>δ</inf>(A) and (Formula Presented) is measured by a delta modular form f¹ constructed by Buium.