Title: Second descent and rational points on Kummer varieties,
Abstract: A powerful method pioneered by Swinnerton-Dyer allows one to study rational
points on pencils of curves of genus 1 by combining the fibration method with a
sophisticated form of descent. A variant of this method, first used by
Skorobogatov and Swinnerton-Dyer in 2005, can be applied to study rational
points on Kummer varieties. In this paper we extend the method to include an
additional step of second descent. Assuming finiteness of the relevant
Tate-Shafarevich groups, we use the extended method to show that the
Brauer-Manin obstruction is the only obstruction to the Hasse principle on
Kummer varieties associated to abelian varieties with all rational 2-torsion,
under mild additional hypotheses.