Elliptic fibrations on quartic \(K3\) surfaces with large Picard numbers (Q1918395)

Let \(q_1\) and \(q_2\) be two binary quartic forms. We consider the diophantine equation \(q_1 (x, y)= q_2 (z, w)\) from the geometric view point. Under a mild condition we prove that the \(K3\) surface defined by the above equation admits an elliptic fibration whose Mordell-Weil group over \(\mathbb{C} (t)\) has rank at least 12. Next, we choose suitable \(q_1\) and \(q_2\) such that the Mordell-Weil group contains a subgroup of rank 12 defined over \(\mathbb{Q} (i)\) and a subgroup of rank 8 defined over \(\mathbb{Q}\).