Second descent and rational points on Kummer varieties

2017/03/15 by Yonatan Harpaz
Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Descent (aeronautics) #Fibration #Genus #Geometry and complex manifolds #Hasse principle #Rational point #math.AG #msc:11G10 #msc:14G05

paper · pdf · doi:10.1112/plms.12191

arxiv created 2017/03/15 · openalex created_date 2017/04/07 · openalex publication_date 2018/09/19 · arxiv updated 2018/09/26 · openalex updated_date 2026/08/05

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 the study of 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 relatively mild additional hypotheses.

Citations

• Ranks of twists of elliptic curves and Hilbert’s tenth problem
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• Failures of weak approximation in families
• Néron Models