K3 surfaces with interesting groups of automorphisms

By the fundamental result of I.I. Piatetsky-Shapiro and I.R. Shafarevich (1971), the automorphism group Aut(X) of a K3 surface X over C and its action on the Picard lattice S_X are prescribed by the Picard lattice S_X. We use this result and our method (1980) to show finiteness of the set of Picard lattices S_X of rank $\ge 3$ such that the automorphism group Aut(X) of the K3 surface X has a non-trivial invariant sublattice S_0 in S_X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic lattices S_0 it has been proved by the author before (1980, 1995). Thus we extend this results to negative sublattices S_0. We give several examples of Picard lattices S_X with parabolic and negative S_0. We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. These results are important for the theory of Lorentzian Kac--Moody algebras and Mirror Symmetry.