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A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

  Published:2004-09-01
 Printed: Sep 2004
  • David McKinnon
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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 rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height
MSC Classifications: 11G35, 14G05 show english descriptions Varieties over global fields [See also 14G25]
Rational points
11G35 - Varieties over global fields [See also 14G25]
14G05 - Rational points
 

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