Principe local-global pour les z\'ero-cycles sur certaines fibrations au-dessus d'une courbe : I

TL;DR

This paper proves that, under certain conditions, the Brauer-Manin obstruction from the base curve is the only obstacle to the Hasse principle and weak approximation for zero-cycles of degree 1 on certain fibered varieties over a number field.

Contribution

It establishes a local-global principle for zero-cycles on fibered varieties over a curve, assuming finiteness of the Tate-Shafarevich group and local conditions on fibers.

Findings

The Brauer-Manin obstruction is the only obstruction under the given assumptions.

The result applies to varieties fibered over a curve with geometrically integral fibers.

The proof relies on the finiteness of sh(Jac(C)).

Abstract

Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of $\sha(Jac(C))$, if the fibers over a generalised Hilbertian subset satisfy the Hasse principle (resp. weak approximation), then the Brauer-Manin obstruction coming from the base curve is the only obstruction to the Hasse principle (resp. to weak approximation) for zero-cycles of degree 1 on $X$.