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Kernels of Arithmetic Jet Spaces and Frobenius Morphism

Rajat Kumar Mishra, Arnab Saha

TL;DR

The paper develops a unified framework connecting arithmetic jet spaces to shifted Witt vectors by showing that Frobenius on Witt vectors induces morphisms on generalized kernels $N^{[m]n}G$ via natural ring maps $E_{[m]}$. It proves that for smooth $ obreak π$-formal groups with ramification bound $v_ obreakπ(p) obreak\\le p-2$, the kernels are isomorphic to $( obreak ext{W}_{n-1})^g$ and fit into a short exact sequence with the jet spaces, with the induced map $oldsymbol{Φ}_{[m]}$ corresponding to multiplication by $ obreak π$ in the Witt-vector setting. A key contribution is the construction of shifted Witt vectors $W_{[m]n}$ and the Frobenius lift $F_{[m]}$, along with the map $E_{[m]}$, which together encode the Frobenius- and $ ext{π}$-multiplication structure in a way that mirrors the classical Witt vector calculus (Frobenius, Verschiebung, and multiplication by $ obreak π$). The results establish a robust, functorial approach to kernels of jet spaces and their Frobenius actions, enabling broader delta-geometry techniques and potential Diophantine applications.

Abstract

For any $π$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $π$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $π$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $π$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $π$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $π$.

Kernels of Arithmetic Jet Spaces and Frobenius Morphism

TL;DR

The paper develops a unified framework connecting arithmetic jet spaces to shifted Witt vectors by showing that Frobenius on Witt vectors induces morphisms on generalized kernels via natural ring maps . It proves that for smooth -formal groups with ramification bound , the kernels are isomorphic to and fit into a short exact sequence with the jet spaces, with the induced map corresponding to multiplication by in the Witt-vector setting. A key contribution is the construction of shifted Witt vectors and the Frobenius lift , along with the map , which together encode the Frobenius- and -multiplication structure in a way that mirrors the classical Witt vector calculus (Frobenius, Verschiebung, and multiplication by ). The results establish a robust, functorial approach to kernels of jet spaces and their Frobenius actions, enabling broader delta-geometry techniques and potential Diophantine applications.

Abstract

For any -formal group scheme , the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted -typical Witt vectors. In the special case when , the arithmetic jet space, as well as the generalized kernels are affine -formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any -formal group scheme along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by .
Paper Structure (5 sections, 10 theorems, 63 equations)

This paper contains 5 sections, 10 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Arithmetic Jet Spaces
  4. Shifted Witt Vectors
  5. Kernel of Jet Spaces

Key Result

Theorem 1.1

For all $m \geq 1$, the ring map $E_{[m]}$ induces a natural morphism of prolongation sequences $\Phi_{[m]}: N^{[m]*}G \rightarrow N^{[m-1]*}G$ such that $\mathfrak{f}_m$ and $\Phi_{[m]}$ commutes with each other. In particular, the following diagram commutes for every $m,n \geq 1$\xymatrix{ N^{

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 4.1
  • proof
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • proof
  • ...and 12 more