A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

McKinnon, David

Canadian mathematical bulletin, Tome 47 (2004) no. 3, pp. 398-406

Résumé

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.

Détail

DOI : 10.4153/CMB-2004-039-6

Mots-clés : 11G35, 14G05, rational points, Batyrev-Manin conjecture, Kummer surface, rational curve, abelian surface, height