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

TL;DR

This paper investigates local-to-global principles for 0-cycles on products of elliptic curves, providing evidence for conjectures and exploring Brauer-Manin obstructions, especially in cases involving potential complex multiplication.

Contribution

It offers new evidence supporting conjectures on 0-cycles and analyzes Brauer-Manin obstructions for products of elliptic curves, including special cases with complex multiplication.

Findings

Places of good ordinary reduction often involved in Brauer-Manin obstructions

Many 0-cycles can be lifted to global 0-cycles

Evidence supports conjectures for products of elliptic curves

Abstract

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.