Distributions of rational points on Kummer Varieties

TL;DR

This paper investigates the distribution of rational points on Kummer varieties, providing new results that depend on the Parity Conjecture and demonstrating the density of rational points in specific cases.

Contribution

It offers new unconditional and conditional results on rational points on Kummer varieties of any dimension, including examples of dense rational points on certain K3 surfaces.

Findings

Conditional density of rational points on odd-dimensional Kummer varieties over Q

Construction of an infinite family of K3 surfaces with dense rational points

Some results are unconditional, others depend on the Parity Conjecture

Abstract

We prove several results on the number of rational points on open subsets of Kummer varieties of arbitrary dimension. Some of our results are unconditional, and others depend on the Parity Conjecture (a corollary of the Conjecture of Birch and Swinnerton-Dyer). As examples, we show that (conditional on the Parity Conjecture) all odd-dimension Kummer varieties over Q which are quotients of absolutely simple abelian varieties have dense rational points, and we construct an infinite family of K3 surfaces with dense rational points.