Canonical vector heights on $K3$ surfaces – A nonexistence result

Abstract

A. Baragar introduced a canonical vector height on a $K3$ surface $X$ defined over a number field, and showed its existence if $X$ has Picard rank two with infinite automorphism group. In another paper, A. Baragar and R. van Lujik performed numerical computation on certain $K3$ surfaces with Picard rank three, which strongly suggests that, in general, a canonical vector height does not exist. In this note, we prove this last assertion. We compare the set of periodic points of one automorphism with another on certain $K3$ surfaces.