Mordell–Weil groups and automorphism groups of elliptic $K3$ surfaces

Abstract

We present a method to calculate the action of the Mordell–Weil group of an elliptic $K3$ surface on the numerical Néron–Severi lattice of the $K3$ surface. As an application, we compute a finite generating set of the automorphism group of a $K3$ surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type.