Polarized K3 surfaces with an automorphism of order 3 and low Picard number

In this paper, for each $d>0$, we study the minimum integer $h_{3,2d}\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ of degree $H^2=2d$ and Picard number $\rho (X):=\text{rank Pic }X=h_{3,2d}$ admitting an automorphism of order $3$. We show that $h_{3,2}\in \{4,6\}$ and $h_{3,2d}=2$ for $d>1$. Analogously, we study the minimum integer $h_{3,2d}^{\ast }\in \mathbb{N}$ for which there exists a complex polarized K3 surface $(X,H)$ as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of $X$. We show that $h_{3,2d}^{\ast }$ is equal to $2$ if $d>1$ and equal to $6$ if $d=1$. We provide explicit examples of K3 surfaces defined over $\mathbb{Q}$ realizing these bounds.