# Second descent and rational points on Kummer varieties

## TL;DR

This paper extends a descent method to analyze rational points on Kummer varieties, demonstrating that the Brauer-Manin obstruction is the only barrier to the Hasse principle under certain conditions.

## Contribution

It introduces a second descent step into the existing method for studying rational points on Kummer varieties, under the assumption of finite Tate-Shafarevich groups.

## Key findings

- Brauer-Manin obstruction is the only obstruction to the Hasse principle on certain Kummer varieties.
- Extension of descent method to include second descent.
- Conditional results assuming finiteness of Tate-Shafarevich groups.

## 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.