Distributions of rational points on Kummer Varieties

2012/11/19 by David Holmes, Holmes, David, René Pannekoek +1
Mathematics · #11G35 #11G50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1211.4597

openalex publication_date 2012/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Rational curves and points on K3 surfaces
• Arithmetic of rational points and zero-cycles on Kummer varieties
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• On the distribution of rational points on ramified covers of abelian varieties
• Counting rational points on smooth quartic and quintic surfaces