Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces

Details Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces Vojta's Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces

2010-11-26

arxiv.org/abs/1011.5825v1

Mathematics

Number Theory

10 pages

Scientific paper

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces.

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