Order and Pascal depth of Pascal finite automorphisms of the plane

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.