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 \(X=E_1\times E_2\) 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.