Weak approximation versus the Hasse principle for subvarieties of abelian varieties

TL;DR

This paper investigates the relationship between weak approximation and the Hasse principle for subvarieties of abelian varieties, showing that finite descent and Mordell-Weil sieve obstructions explain failures of rational points density.

Contribution

It demonstrates that the failure of weak approximation can be fully explained by finite descent or Mordell-Weil sieve obstructions under certain assumptions.

Findings

Finite descent obstruction accounts for failures of weak approximation.

Mordell-Weil sieve obstruction similarly explains rational point failures.

Results depend on the assumption that these obstructions are the only ones.

Abstract

For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational points should be dense in the set of (modified) adelic points surviving finite descent. We show that this follows from the a priori weaker assumption that finite descent is the only obstruction to the existence of rational points. We also prove a similar statement for the obstruction coming from the Mordell-Weil sieve.