## Algebra & Number Theory

### Hasse principle for Kummer varieties

#### Abstract

The existence of rational points on the Kummer variety associated to a $2$-covering of an abelian variety $A$ over a number field can sometimes be established through the variation of the $2$-Selmer group of quadratic twists of $A$. In the case when the Galois action on the $2$-torsion of $A$ has a large image, we prove, under mild additional hypotheses and assuming the finiteness of relevant Shafarevich–Tate groups, that the Hasse principle holds for the associated Kummer varieties. This provides further evidence for the conjecture that the Brauer–Manin obstruction controls rational points on K3 surfaces.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 4 (2016), 813-841.

Dates
Revised: 8 February 2016
Accepted: 12 March 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842516

Digital Object Identifier
doi:10.2140/ant.2016.10.813

Mathematical Reviews number (MathSciNet)
MR3519097

Zentralblatt MATH identifier
1369.14032

Subjects
Primary: 14G05: Rational points
Secondary: 11J95: Results involving abelian varieties

Keywords
Kummer varieties Hasse principle

#### Citation

Harpaz, Yonatan; Skorobogatov, Alexei. Hasse principle for Kummer varieties. Algebra Number Theory 10 (2016), no. 4, 813--841. doi:10.2140/ant.2016.10.813. https://projecteuclid.org/euclid.ant/1510842516

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