Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O-K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface E x E. The results for the odd-order torsion also apply to the Brauer group of the K3 surface Kum(E x E). We describe explicitly the elliptic curves E/Q with complex multiplication by O-K such that the Brauer group of E x E contains a transcendental element of odd order. We show that such an element gives rise to a Brauer Manin obstruction to weak approximation on Kum(E x E), while there is no obstruction coming from the algebraic part of the Brauer group.