K3 surfaces with an automorphism of order 66, the maximum possible

Abstract: In each characteristic $p\neq 2, 3$, it was shown in a previous work that the order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it is shown that in each characteristic $p\neq 2, 3$ a K3 surface with a cyclic action of order 66 is unique up to isomorphism. The equation of the unique surface is given explicitly in the tame case ($p\nmid 66$) and in the wild case ($p=11$).