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

Abstract

In the 1980's Colliot-Thélène, 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 × E 2 of two elliptic curves. In the special case when X = E ×E is the self-product of an elliptic curve E over 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.