Automorphisms of algebraic K3 surfaces which act trivially on Picard groups (Q1187568)

Let \(X\) be an algebraic \(K3\)-surface defined over \(\mathbb{C}\). The Picard group \(S_ X\) is a sublattice of \(H^ 2(S;\mathbb{Z})\), and let \(T_ X\) be the orthogonal completion of \(S_ X\), with respect to the cup product. Let \(H_ X=\text{Ker(Aut}(X)\to\text{Aut}(S_ X))\). Nikulin proved that \(H_ X\) is a finite cyclic group and \(\varphi(m_ X)\) divides \(\text{rank}(T_ X)\), where \(m_ X\) is the order of \(H_ X\) and \(\varphi\) is the Euler function. The author studies here the case where \(T_ X\) is a unimodular lattice. He shows first that \(m_ X\) divides 66, 44, 42, 36, 28 or 12. Then he shows that \(m_ X\) must be one of the above six values if \(\varphi(m_ X)=\text{rank}(T_ X)\). Moreover, for each of these values, there is a unique \(K3\)-surface with the given \(m_ X\) such that \(\varphi(m_ X)=\text{rank}(T_ X)\). The proof is based on the observation that \(X\) admits an elliptic fibration with a section in such classes.