location:  Publications → journals → CMB
Abstract view

# A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

Published:2004-09-01
Printed: Sep 2004
• David McKinnon
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating rational curves such that the density of rational points on $U-Z_U$ is strictly less than the density of rational points on $Z_U$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets $Z_U$ for successively smaller sets $U$. In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of
 Keywords: rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height
 MSC Classifications: 11G35 - Varieties over global fields [See also 14G25] 14G05 - Rational points