Rational points on elliptic fibrations (after Swinnerton-Dyer)

Yonatan Harpaz

In this series of four lectures we will describe an approach, originally due to Swinnerton-Dyer and further developed by several authors, to the study of rational points on surfaces fibred into curves of genus 1, and more generally, on pencils of torsors under abelian varieties. Under suitable hypotheses (and possibly assuming powerful number theoretical conjectures), one can use this method to show that if the Brauer-Manin obstruction controls the Hasse principle for most of the fibers in the fibration, then the same holds for the total space. Results obtained in this manner are currently some of the only evidence for whether the Brauer-Manin obstruction controls the Hasse principle for K3 surfaces, for example.