Weak Approximation for $0$-cycles on a product of elliptic curves

In the 1980's Colliot-Th\'{e}l\`{e}ne, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for $0$-cycles on arbitrary smooth projective varieties over a number field. We give some evidence for these conjectures for a product $X=E_1\times E_2$ of two elliptic curves. In the special case when $X=E\times E$ is the self-product of an elliptic curve $E$ over $\mathbb{Q}$ with potential complex multiplication, we show that the places of good ordinary reduction are often involved in a Brauer-Manin obstruction for $0$-cycles over a finite base change. We give many examples when these $0$-cycles can be lifted to global ones.