On symplectic automorphisms of elliptic surfaces acting on CH0

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Aut(s)(S) of symplectic automorphisms acts trivially on the Albanese kernel CHo(S)(alb) of the 0-th Chow group CHo(S), unless possibly if the geometric genus and the irregularity satisfy p(g)(S) = q(S) is an element of {1, 2}. In the exceptional cases, the image of the homomorphism Aut(s)(S) -> Aut(CHo(S)(alb)) has the order at most 3. Our arguments actually take care of the group Aut(f)(S) of fibration-preserving automorphisms of elliptic surfaces f: S -> B. We prove that if sigma is an element of Aut(f) (S) induces the trivial action on H-i,H-0(S) for i > 0, then it induces the trivial action on CH0(S)(alb). As a by-product we obtain that if S is an elliptic K3 surface, then Aut(f)(S)(boolean AND)Aut(s) (S) acts trivially on CH0(S)(alb).