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

Let X$X$ be a K3 surface admitting a non‐symplectic automorphism σ$\sigma$ of order 3. Building on work by Garbagnati and Salgado, we classify the Jacobian elliptic fibrations on X$X$ with respect to the action of σ$\sigma$ on their fibers. If the fiber class of a Jacobian elliptic fibration on NS(X)$\operatorname{NS}(X)$ is fixed by σ$\sigma$, we determine the possible configurations of its singular fibers and present the equation for its generic fiber. When the Picard number of X$X$ is at least 12 and σ$\sigma$ acts trivially on NS(X)$\operatorname{NS}(X)$, we apply the Kneser–Nishiyama method to find its Jacobian elliptic fibrations up to J2$\mathcal {J}_2$‐equivalence. We use our method to classify them with respect to any non‐symplectic automorphism of order 3 in Aut(X)$\operatorname{Aut}(X)$.