Kernel of Arithmetic Jet Spaces

TL;DR

This paper develops a theory of arithmetic jet spaces and Witt vectors over Dedekind domains, establishing structural results and applications to formal group schemes and point subgroups, with implications for algebraic geometry and number theory.

Contribution

It introduces $m$-shifted $ ext{π}$-typical Witt vectors and a lift of Frobenius, providing new structural insights into arithmetic jet spaces and formal group schemes.

Findings

$N^{mn} ext{G}$ is isomorphic to $J^{n-1}(N^{m1} ext{G})$ for $ ext{π}$-formal groups.

$J^n ext{G}$ is a canonical extension of $ ext{G}$ by Witt vector schemes.

Characterization of subgroup points $G( ext{π}^{n+1}R)$ in terms of reduction modulo $ ext{π}^{n+1}$.

Abstract

Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only.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.