On exponentiality of automorphisms of ${\bf A}^n$ of order $p$ in characteristic $p>0$

TL;DR

This paper explores when automorphisms of affine space of order p in characteristic p are exponential, revealing cases where they are and constructing examples where they are not, thus clarifying their structure.

Contribution

It characterizes exponential automorphisms of order p in affine space, providing conditions under which they are exponential and constructing counterexamples.

Findings

Triangular automorphisms of order p are exponential in low dimensions.

Existence of non-exponential automorphisms of order p for any dimension n ≥ 2.

Analysis of G_a-actions inducing elementary automorphisms.

Abstract

Let $X$ be an integral affine scheme of characteristic $p>0$, and $\sigma $ a non-identity automorphism of $X$. If $\sigma $ is $\textit{exponential}$, i.e., induced from a ${\bf G}_a$-action on $X$, then $\sigma $ is obviously of order $p$. It is easy to see that the converse is not true in general. In fact, there exists $X$ which admits an automorphism of order $p$, but admits no non-trivial ${\bf G}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${\bf A}_R^n$, because ${\bf A}_R^n$ admits various ${\bf G}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${\bf A}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${\bf G}_a$-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ${\bf A}_R^n$ of order $p$ are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${\bf A}_R^n$ of order $p$ for each $n\ge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${\bf G}_a$-actions inducing an elementary automorphism of ${\bf A}_R^n$.