Bloch's conjecture for (anti-)symplectic automorphisms of K3 surfaces

We prove Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex K3 surfaces of Picard number two. Together with the recent result for abelian surfaces and the known K3 cases, this settles the remaining case of Bloch's conjecture for (anti-)symplectic automorphisms of smooth projective complex surfaces with trivial canonical bundle. More generally, we prove Bloch's conjecture for (anti-)symplectic derived autoequivalences of projective K3 surfaces whose Picard number is at least three, or is two with anisotropic Néron--Severi lattice. The key new result is the congruence property of the subgroup generated by reflective involutions in $O(U\oplus N)$ for anisotropic even lattices $N$ of signature $(1,1)$. Combined with the local generation theorem and Chow-theoretic criterion of Li--Yu--Zhang, this gives the required reflective decompositions and their consequences for zero-cycles.