Abstract view
A Reduction of the BatyrevManin Conjecture for Kummer Surfaces


Published:20040901
Printed: Sep 2004
Abstract
Let $V$ be a $K3$ surface defined over a number field $k$. The
BatyrevManin 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 $UZ_U$ 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 BatyrevManin conjecture for $V$ can be reduced to the
BatyrevManin 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, BatyrevManin conjecture, Kummer, surface, rational curve, abelian surface, height
rational points, BatyrevManin conjecture, Kummer, surface, rational curve, abelian surface, height
