Order and Pascal depth of Pascal finite automorphisms of the plane

Abstract

Let $K$ be a field of characteristic $p>0$. For a Pascal finite automorphism $F$ of the affine plane we show that its order is determined by its Pascal depth, $|F|=p^{\lceil {\log }_p{\tau }_K(F)\rceil }$, and that, combined with Dolgachev's theorem on the plane Cremona group, this pins the order spectrum of Pascal finite plane automorphisms to $\{1,p,p^2\}$ and bounds the Pascal depth by ${\tau }_K(F)\le p^2$. For the polynomial group ${\text{GA}}_2(K)$ we give a second, independent proof of the order-$p^2$ ceiling, a purely group-theoretic argument from the Jung--van der Kulk amalgam and Serre's tree theorem, using no birational geometry. We prove that the bound is sharp in two independent senses. Order~$p^2$ is attained by the length-two Witt vectors, and Pascal depth $p^2$ is attained by an explicit tame automorphism $G_p$, for which we give a characteristic-free proof that ${\tau }_K(G_p)=p^2$. We contrast the plane with higher dimensions, where both order and depth are unbounded.