Automorphisms of even lattices arising in connection with automorphisms of algebraic $K3$ -surfaces

Let $H(X)$ denote the group of all automorphisms of an algebraic $K3$-surface $X$ acting trivially on algebraic cycles. This group is cyclic and we denote the order of $H(X)$ by $m(X)$. We derive some results concerning possible values of $m(X)$. These results follow from some more general theorems on even lattices and their automorphisms.