On the Brauer-Manin obstruction for zero-cycles on curves

TL;DR

This paper provides an elementary proof that the Brauer-Manin obstruction is the only obstruction for zero-cycles of degree 1 on curves, assuming the finiteness of the Tate-Shafarevich group, and discusses implications for higher dimensions.

Contribution

It offers a simplified proof of Saito's result and shows that only a conjecturally finite part of the Brauer group is needed for the obstruction to be complete.

Findings

Brauer-Manin obstruction is the only obstruction for zero-cycles on curves under certain assumptions

Elementary proof reduces complexity of previous arguments

Discussion on higher-dimensional cases and conjectural aspects

Abstract

We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in higher dimensions