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Let E be an elliptic curve over a number field F, A the abelian surface E x E, and T_F(A) the F-rational albanese kernel of A, which is a subgroup of the degree zero part of Chow group of zero cycles on A modulo rational equivalence. The first result is that for all but a finite number of primes p where E has ordinary reduction, the image of T_F(A)/p in the Galois cohomology group H^2(F, sym^2(E[p])) is zero; here E[p] denotes as usual the Galois module of p-division points on E. The second result is that for any prime p where E has good ordinary reduction, there is a finite extension K of F, depending on p and E, such that T_K(A)/p is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory., Comment: Twelve pages; Submitted to a special issue of Indigaiones Mathematicae in honor of Murre, edited by Jan Nagel and Chris Peters