  According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface
X that acts as the identity on the transcendental lattice should act trivially
on CH^2(X). We discuss this conjecture for symplectic involutions and prove it
in one third of all cases. The main point is to use special elliptic K3
surfaces and stable maps to produce covering families of elliptic curves on the
generic K3 surface that are invariant under the involution.
