  Since the results here have been superseded by another paper cowritten by the
author, this article is available for reference purposes only.
  Fix a Dedekind domain $\mathcal{O}$ and a non-zero prime $\mathfrak{p}$ in it
along with a uniformizer $\pi$. In the first part of the paper, we construct
$m$-shifted $\pi$-typical Witt vectors $W_{mn}(B)$ for any $\mathcal{O}$
algebra $B$ of length $m+n+1$. They are a generalization of the usual
$\pi$-typical Witt vectors. Along with it we construct a lift of Frobenius,
called the lateral Frobenius $\tilde{F}: W_{mn}(B) \rightarrow W_{m(n-1)}(B)$
and show that it satisfies a natural identity with the usual Frobenius map. Now
given a group scheme $G$ defined over $\mathrm{Spec}~ R$, where $R$ is an
$\mathcal{O}$-algebra with a fixed $\pi$-derivation $\delta$ on it, one
naturally considers the $n$-th arithmetic jet space $J^nG$ whose points are the
Witt ring valued points of $G$. This leads to a natural projection map of group
schemes $u: J^{m+n}G \rightarrow J^mG$. Let $N^{mn}G$ denote the kernel of $u$.
  One of our main results imply that for any $\pi$-formal group scheme
$\hat{G}$ over $\mathrm{Spf}~ R$, $N^{mn}\hat{G}$ is isomorphic to
$J^{n-1}(N^{m1}G)$. As an application, if $\hat{G}$ is a smooth commutative
$\pi$-formal group scheme of dimension $d$ and $R$ is of characteristic 0 whose
ramification is bounded above by $p-2$, then our result implies that $J^nG$ is
a canonical extension of $\hat{G}$ by $(\mathbb{W}_{n-1})^d$ where
$\mathbb{W}_{n-1}$ is the $\pi$-formal group scheme $\hat{\mathbb{A}}^n$
endowed with the group law of addition of Witt vectors. Our results also give a
geometric characterization of $G(\pi^{n+1}R)$ which is the subgroup of points
of $G(R)$ that reduces to identity under the modulo $\pi^{n+1}$ map.
