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A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces
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- Journal:
- Canadian Mathematical Bulletin / Volume 47 / Issue 3 / 01 September 2004
- Published online by Cambridge University Press:
- 20 November 2018, pp. 398-406
- Print publication:
- 01 September 2004
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- Article
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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 Batyrev-Manin type.
