Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3

We classify elliptic fibrations on elliptic K3 surfaces with a non-symplectic automorphism σ of order 3 according to the action of σ on their fibres, building on work by Garbagnati and Salgado for non-symplectic involutions. Furthermore, we apply the Kneser--Nishiyama method to such K3 surfaces under the hypothesis that their Picard number is at least 12 and the automorphism σ acts trivially on their Ner\'on--Severi group.