  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. 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 that $\mathbf{H}_\delta(A) \otimes \mathbb{Q}_p$ is
canonically 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
$\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$.
  The above result can be viewed as a character theoretic interpretation of the
crystalline cohomology. The difference between the integral structures of
$\mathbf{H}_\delta(A)$ and $\mathbf{H}_{\mathrm{cris}}^1(A)$ is measured by a
delta modular form $f^1$ constructed by Buium.
