A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

• Autores: David McKinnon
• Localización: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 47, Nº 3, 2004, págs. 398-406
• Idioma: inglés
• DOI: 10.4153/cmb-2004-039-6
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• Resumen

  • 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 ZU of accumulating rational curves such that the density of rational points on U-ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU 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.