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

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.

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