  This paper concerns complex algebraic K3 surfaces with an automorphism which
acts trivially on the Neron-Severi group. Complementing a result by Vorontsov
and Kondo, we determine those K3 surfaces where the order of the automorphism
is a 2-power and equals the rank of the transcendental lattice. We also study
the arithmetic of these K3 surfaces and comment on mirror symmetry
