On elements of prime order in the plane Cremona group over a perfect field

TL;DR

This paper characterizes elements of prime order in the plane Cremona group over a perfect field, linking their existence to algebraic tori and roots of unity, and classifies elements of order 7.

Contribution

It establishes a precise criterion for the existence of prime order elements in the Cremona group based on algebraic tori and roots of unity.

Findings

Elements of prime order  in  are characterized by associated algebraic tori.

No elements of prime order  > 7 exist over  without primitive roots of unity.

All elements of order 7 are conjugate when  is algebraically closed and contains no primitive 7th roots of unity.

Abstract

We show that the plane Cremona group over a perfect field $k$ of characteristic $p \ge 0$ contains an element of prime order $\ell\ge 7$ not equal to $p$ if and only if there exists a 2-dimensional algebraic torus $T$ over $k$ such that $T(k)$ contains an element of order $\ell$. If $p = 0$ and $k$ does not contain a primitive $\ell$-th root of unity, we show that there are no elements of prime order $\ell > 7$ in $\Cr_2(k)$ and all elements of order 7 are conjugate.